UDC 621.833.1 : 001.4 : 003.62 ian acute aie nv Deutsches Intute Normung 8. Begritfe und Bestimmungsgrdssen fUr Stimrader (Zylinderréder) und Stimrado verzahnung 4s It is current practice in standards published by the International Organization for Standardization (180), the comma hhas been used throughout es a decimal marker. Compared with the October 1976 edition the numbering system in this standard has been changed. The new equation numbers are numbered continuously within the main clauses and the first digit represents the number of the respective main clause in which they appear. Contents age Pave © Other relevant standards 3 25 Concepts and parameters associated with gear i teoth arising rom the postion ofthe standard basic 1 Symbols, terms, units a rack tooth profile relative to the reference cylinder 10 4.4 Symbols and terms 3 ogy ‘ead angle. y,hellx angle 10 12 Additional subscripts 5 252 Flank rection; right-handed, letthanded | 11 13 Units 5 253 Sign of the helix angle n 2 Concepts and parameters associated with 2.54 Addendum modification, addendum modtication cylindrical gear teeth 6 Cootticientxand sign of addendum modification 11 2.1 Number of teeth z and sign of number of teeth. 6 255 V-gear, zero gear tt aot pees lenragaen ana aoa ne 258 Diameters of gear teoih, angles at V-cylinder | 11 2.2.1. Standard basic rack tooth profile, basic rack. - 6 257 Gear toath heights fa 22.2 Datum line, dimensions on the standard basic 258 Tooth thicknesses, spacewidins B rack tooth prota 6 28 Geometrical limits for involute gear teeth... 13, 22.3. Sections through cylindrical gear teeth 7 261 Undercuton cylindrical gears with extornal teeth. 13 224 Gear tooth prof lank profile + 262 Pointing limit and minimum tooth thickness et 2.25 Tooth aces 7 the tip circle ofan external gear 14 226 Modulos ig ity: thy 7 263 Pointing limit and minimum spacewidth at the 227 Reference cylinder, reference circle; reterence root circle of an internal gear 14 diameter d 7 284 Range of feasible invotute gear teeth 4 228 Base cylnder, base circle; base diameter dy. 2.7. Test dimensions for tooth thickness. 16 2.3. Involute surface (involute helicold) 7 27-1 Normal chordal tooth thicknesses He 28:1 "Generator of an involute surface 7 212 Base tangent length Wi 7 282 Base helix ange fy, base lead angle § 273. Radial test dimensions for tooth thickness... 19 bers peer § 2:74. Diametral test dimensions for tooth thickness _ 23, 25a Traneverso pressure angle ata point @y trans: 27-5 Centredistence for double-lankengegementa* verse pressure angle a 8 with master gear. 24 285 Normal pressure angle at 6 point &, normal” 27.8 Tip diameter day on overcut cylindrical gears . 25 pressure angle a, 8 3 Concepts and parameters associated with 29.6 Rolling angle {ofthe Involute 8 acylindrical gear pair 26 28:7 Radius of curvature 0 of the involute, working 3.1. Cylindrleal gear pal, definitions 26 lonath Ly © a4 External gear pair 26 2.98 Involute function Inv a @ 812 internal gear pair... 26 239 Lead p, 8 313. V-geer peir : 26 24 Angular pitch and pitches 8 344 V-0 gear pair : 26 24.1 Angular piteh 9 318 Zero gear pair : 26 242 Pitches on the reference cylinder 2 3.2. Mating quantities 26 24.3 Pitches on the Y-cylinder 2 3.2.1 Gearratio uw. 26 24.4 Pitches on the V-cylinder 2 3.22 Transmission ratio i . 2 24 Pitches on the base cyinder 10 323 Line of centres, cenire distance a 27 248 Normalbese pitches py 10 324 Pitch cylinders, piten circles; pitch diameter dy 27 247 Axle pitch py 40 325 Working transverse pressure angle ayy oF a” 27 DEUTSCHE NORM July 1980 Concepts and parameters associated with cylindrical gears and cylindrical gear pairs with involute teeth re (Zylinderradpaere) mit Evolventen- Continued on pages 2 to 48 Explanations on page 48 ‘Seat hs fGeman Sanaa IN wh ea VateG OMAN eNO DIN 3960. Engl. Price group 18Page 2 DIN 3960 3.26 Working depth hy 327 Bottom clearance c, bottom clearance factor c* 3.3 Calculation quantities and factors for mating gears or 8.3.1 Reference centre distance ay a7 8.82 Centre distance modification y my centre dis- tance modification coefficient y ar 383 Centre distance modification coefficient y and ‘sum of addendum modification coefficients Bx 28 834 Calculation of the sum ofaddendum modification Coefficients Bx 28 395 Calculation of centre distance a 28 8.9.8 Addendum alteration fe my, addendum alter- ation factor ke 28 3.4 Tooth engagement 28 3.4.1 Point af contact 28 842 Plane of action, line of action 28 3.43 Zone of action, length of path of contact... | 29 8.44 Usable diameter dy; active and usable ranges of the tooth flanks 29 2.45. Transverse angle of transmission transverse contact ratio t¢ at 3.46 Overlap angle gp, overlap ratio ey at 347 Overlap lenath g5 32 3.48 Total angle of transmission g, total contact ratio 2y 32 3.8. Sliding conditions at the tooth flanks 32 3.6.1 Sliding speed u, 32 3.52 Sliding factor Ky 32 383. Specific siding # 32 4 Deviations and tolerances for cylindrical gear teeth 33 4.1 Deviations A of tooth thickness and its test dimensions. 33 4.41 Deviations of tooth thickness 4, 2.33. 4.1.2 Deviations Az of the normal chordal tooth thick nesses 34 4.1.8 Base tangent iength deviations Ay 34 4.14 Deviations Ayia of dimension over balis or pins. 34 4.41.8 Deviations Api; of radial single-ball or single-pin dimension 34 4.41.6 Deviations A, of centre distance for double- flank engagement a" with master gear 34 4.1.7 Tip diameter deviation Ags with overcut cylindr- cal gears 34 42 Tolerances T 34 421 Tooth thickness tolerance 7, 35 422 Tolerance 7; on normal chordal tooth thickness 35 4.23 Base tangent lenath tolerance Tw 35 4.24 Tolerance Tja on dimension over balis or pins. 35 425 Tolerance Ty1,0n radial single-ball or single-pin dimension 35 4.26 Tolerance Ty on centre distance for double- flank engagement with master gear 36 42.7 Tip diameter tolerance Tug with overcut cylindri cal gears 35 5 Change factors A* 35 5.1 Change factor Az of normal chordal tooth thick= ness 35 5.2 Change factor Aly of base tangent length... 35 5.3 Change factor Aji of dimension over balls orpins 35 5.4 Change factor Aj, of radial single-ball or singl pin dimension Change factor A: of centre distance for double- flank engagement a’ with master gear Change factor Aj, for tip diameter of overcut cylindrical gears 6 Deviations of individual parameters of cylin- drical gear teeth . 6.1. Circular pitch deviations 1 Adjacentpitch errors fy. 2 Cumulative circular pitch errors Fx 3 Cumulative ceular itch emer Fp 4 Cumulative pitch error Fy 5 16 1 1 1 1 815 Range of pitch errors R, Difference between adjacent pitches f, 2 Norma base pitch errors foe 6.3 Flank deviations 63.1 Deviations of transverse profile 632 Deviations of tooth traces 63.3 Deviations of the generator 6.4 Radial run-out 8.4.1 Radial run-out F of a tooth system 64.2 Radial run-out of tip circle fe 6.5. Position deviation of gearcutting axis 65.1 Eccentricity fy 652 Wobble Fo 6.8 Range of errors R 66.1 Range of tooth thickness errors Ry, range of normal chordal tooth thickness errors Re 662 Range of base tangent length errors Rw 663 Range of errors Ry, for radial single-ball or single-pin dimension 6.6.4 Range of errors Ru, for dimension aver balls or pins. 665 Range of errors for the centre distance for double-flank engagement 6.7 Contact pattern 7 Composite and cumulative errors. 7.1. Single-flank working test 74.4 Tangential composite error F 71.2 Cumulative working piteh error F, 7.1.3 Individual working error /, 7.1.4 Tangential tooth-to-tooth composite error 7.18 Transmission deviation of a multi-stage gear mating g 7.2. Double-flank working test 721 Radial composite error F 722 Working radial run-out Fj 723 Radial tooth-to-tooth composite error ft 8 Deviations of the axial positions of a cylindri- cal gear pair 8.1. Deviations from parallellam 8.1.1. Inclination error of axes fy 8.1.2. Deviation error of axes fay 8.2. Deviations and tolerance for position of gear axes 82.1 Centre distance deviations Ay 822 Centre distance tolerances T, 823 Tolerance space for position of gear axes 9 Backlash j 94 Ciroumferenti 92 I backlash j, Normal backlash ji, 9.3 Radial backlash j, 9.4 Range of errors for backlash 2 10 Alphabetical index 44 44 44 44 45 45 45 45 45 45 45 450 Other relevant standards DIN. 867 Standard basic rack tooth profile of cylindrical gears with Involute teeth for ‘general and heavy engineering DIN 868 General concepts and parameters for gears, gear pairs and gear trains DIN 1901 Part t Units; names, symbols DIN 1318 Angles; concepts, units DIN 9961 Tolerances for cylindrical gear teeth bases DIN. 3967 ‘System of gear its; backlash tooth thick ness deviations, tooth thickness toler: ‘ances; bases DIN. 3972 Standard basic rack tooth profiles of gear-cutting tools for involute tooth systems according to DIN 867 DIN 3992 ‘Addendum modification of externa cylindrical gears DIN 3993. Geometrical design of cylindrica inter: nal involute gear pairs DIN. 9999 ‘Symbols for gear teeth DIN 7182 Part 1 Tolerances and fite; fundamental con: cepts DIN 58.400 Standard basic rack tooth profile for cylindrical gears with Involute teeth for fine mechanics DIN 68412 Standard basic rack tooth protile for {gear tools for fine mechanics; involute gears according to DIN 58400 and DIN 867 1 Symbols, terms, units 1.4 Symbols and terms This standard uses the following symbols and terms: 4 centre distance ay reference centre distance a" centre distance for double-flank engagement b facewiath by, contact line overlap or width of gauge head for measuring base tangent length © bottom clearence c bottom clearance coatticlent @ ‘reference diameter 4, tip diameter day measured value of tip dlameter of overcut eylindri= cal gears dbase diameter 4; root diameter dy virtual reference diameter 4, Vecircle diameter dy pitch diameter 4 Y-oircle diameter dk diameter of circle through centre of ball 4x, dlameter of a measuring circle (at point of contact, with measuring instrument) ay Usable diameter ing usable tip diameter dys usable root diameter xz usable root diameter at generated wheel (usable flank) © —_spacewidth on the reference cylinder —spacewidth on the tip eylinder @ base spacewidth (on the base oylinder) & —_spacewidth on the rect cylinder DIN 3960 Page: spacewidth on the V-cylindar spacewidth on the Y-cylinder individual error base circle error eccentricity profile form error profile waviness tangential tooth-to-toath composite error radial tooth-to-tooth composite error adjacent pitch error normal pitch error axial piten error lea error Individual working error tip circle radial run-out difference between adjacent pitches generant form error generant angle error profile angle error tooth alignment error pressure angle error helix angle error longitudinal form error longitudinal form waviness deviation error of axes Inclination error of axes path of contact length of recess path Iength of approach path length of path of contact (total) length of addendum path of contact, length of dedendum path of contact, distance of a paint ¥ from pitch point C overlap length tooth depth (between tip line and root line) addendum dedendum working depth tooth depth of standard baslo rack tooth profile chordal height 5, constant chord height 5; transmission ratio (to be read as: involute) involute function backlash normal backlash radial backlash circumferential backlash tip diameter modification coefficient number of teeth spanned when measuring base tangent length ‘umber of teeth or pitches In a span modula basic module number of revolutions (rotational frequency) umber of revolutions (rotational frequency) of driving gear number of revolutions (rotational frequency) of driven gear pitch on the reference cylinder pitch on the base cylinder normal base pitch pitch span pitch on the V-oylinderPage4 DIN 9960 x Py Ps axial pitch pitch on the Y-cylinder lead reference circle radius tip radius base radius root radius radius of the V-circle radius of the working piteh circle radius of the Y-circle tooth thickness on the reference cylinder tooth thickness on the tip cylinder bbase thickness (on the base cylinder) tooth thicknass on the V-cylindar tooth thickness on the pitch cylinder tooth thicknass on the Y-cylindar normal chordal tooth thickness constant chord gear ratio linear speed sliding speed sliding speed at the addendum sliding speed at the dedendum circumferential speed, tangential speed ‘addendum modification coefficient ‘addendum modification coefficient at undercut limit ‘generating addendum modification coefficient ‘mean ganarating addendum modification coefficient ‘addendum modification coetticient for double-flank engagement centre distance modification coefficient umber of teeth number of teeth of driving gear number of teeth of driven gear virtual number of teeth for addendum modification calculations Virtual number of teeth for ball or roller dimensions virtual number of teeth for base tangent length cat culations deviation deviation of centre distance deviation of centre distance for double-flank en: agement tip circle diameter deviation for overcut cylindrical gears tooth thickness deviation (on the reference cylinder) tooth thickness deviation on Y-oylinder deviation of normal chordal tooth thickness deviation of dimension over balls or pins deviation of radial single-ballarsingle-pin dimension base tangant lengtn deviation starting point of meshing starting point of single-tooth contact on driving gear pitch point bail or pin diameter fend point of single-tooth contact on driving gear end point of meshing composite and cumulative errors total profile error ‘tangential composite error radial composite error cumulative pitch error cumulative circular pitch error over k pitches Ta ‘cumulative circular pitch error over 2 sector of 2/8 pitches cumulative working pitch error radial run-out of @ toath system radial run-out at rear of gear radial run-out at front of gear working radial run-out total generant error total alignment error wobble sliding factor sliding factor at tooth tip sliding factor at tooth root distance between points of measurement working length from involute origin to tooth tip working length from involute origin to tooth root working length to point ¥ generator test range bearing centre distance on a gear axis profile test range tooth trace test range dimension over balls, dimension over pins radial single-ball dimension radia! single-pin dimension contact point (e.9. between test balland tooth flank) range of errors range of errors for backlash range of pitch errors range of tooth thickness errors range of normal chordal tooth thickness errors. tange of errors for dimension aver balls or pins range of errors for radial single-ball or single-pin dimension range of base tangent length errors tolerance centre distance tolerance tolerance on centre distance for double-flank en- gagement tip diameter tolerance for overcut cylindrical gears tooth thickness tolerance tolorance on normal chordal tooth thickness tolerance on dimension over balls or pins tolerance on radial single-ballorsingle-pin dimension base tangent length tolerance base tangent length over k measured teeth or measured spacewidths any point on a tooth flank pressure angle at the reference cylinder reseure angle at the tip cylinder pressure angle at the V-cylinder pressure angle at the pitch cylinder working transverse pressure angle pressure angle at the Y-cylinder pressure angle at circle through centre of ball pressure angle at measuring circle pressure angle for double-flank engagement with master gear helix angle at reference cylinder base helix angle helix angle at V-oylinder helix angle at pitch cylinder helix angle at Y-cylinder€ fu y % e ¢ « a ™ 1 ne Nw ny é & & @ Guo o ° % a % % ’ We Yo tn te ty ty a aw ap ax ze heilx angle at measuring circle lead angle at reference cylinder base lead angle contact ratio transverse contact ratio overlap ratio. total contact ratio specific sliding specific sliding at end point of path of contact spacewidth half angle at reference circle base spacewidth half angle spacewidth helf angle at root circle spacewidth helf angle at V-circla spacewidth half angle at pitch circle spacewidth half angle at Y-circie rolling angle of the involute rolling angle of the involute at tooth tip end rolling angle of the involute at tooth root end rolling angle of the involute at point Y radius of curvature, rounding radius tooth crest rounding radius in normal section of cylindrical gear tooth crest rounding radius on tool radius of curvature of involute at point Y angular pitch angle at the centre, contact angle angle at the centre betwen maximum values for radial run-out Fry and Fray transverse angle of transmission overlap angle total angle of transmission tooth thickness half angle at reference circle tooth thickness half angle et tip circle base thickness half angle viral tooth thickness half angle tooth thickness half angle at V-circle tooth thickness halt angle at pitch circle tooth thickness halt angle at Y-circle angular velocity angular velocity of driving gear angular velocity of driven gear difference length difference in base tangent length meesure. ment angular difference sum of addendum modification coetficients sum of numbers of teeth For further symbols for gear teeth see DIN 3999 In adaition, tho following standards also deal with symbols: DIN DIN DIN 1902 Mathematical signs and symbols, 1904 General symbols for use in formulae, 1913. Physical quantities and equations; concepts, methods of writing, DIN 66030 (at present at draft stage) Representations for names of units to be used in systems with limited graphic character sets, 1.2. Additional subscripts The symbols indicated in aubolause 1.1 may be supple- mented by the following additional subscripts: b for the driving gear for the driven gear for the upper limit of a quantity DIN 3860 Page 5 i for the lower limit of a quantity | for “left-hand” of “in the direction of a left-hand helix” m fore mean value for quantities in a normal section for right-hand’ or “in the direction of a right-hand helix {for quantities ine transverse section or in the tan- gential direction Xx for quantitios in an axial section ar in the direction of the gear axis, L for designating let flanks or a master goar M for designating a measured value P —forquantities of he standard basicrack tooth profile R for designating right flanks max fora maximum value min fora minimum value zul_— for designating a permissibie limiting value 0 for quantities associated with the generating tool 1 for quantities associated with the smaller gear of & eer pairing 2 forquantities associated with tho larger gear of agear palcing ‘or designating a factor with which a quantity Is ex- pressed infractions or multiplas ofthe normal module (eg.c=c*- mor tay = hip m,) oF of the number of teeth (0.9 2 OF zie OF 2h), OF for designating a change factor (see clause 5) 1.3 Units The following standards or guidelines must be observed DIN 1901 Part 1 Units; unit names, unit symbols DIN 1315 Angles; concepts, units VDE/VD! 2605 Circular pitches and plane engles; basic concepts for angular dimensions, angle measurements, angie standards and their errors. Accordingly it Is convenient to state the quantities dealt within this standard in the following units: ‘modules and all lengths in millimetres (mm), deviations in and tolerances on length in micrometres (ym), angles whereby size is denoted in degraes (°) (eg. pressure angle, helix angle, rolling angle); for calculations with computers it is advantageous to se decimal subdivision of the degrea; ‘angles used in equations in radians (rad) (eg. angular pitch =, tooth thickness half angle 4, spacewidth halt angle 7, overiap angle 9); ‘angular daviations in millradians (mrad) or in microradians (ured) (see subclauses 6.1.4, 63.1.6, 63.24,63.25); angular velocities in radians per second (rads); ‘number of revolutions (frequency of rotation) In one per second (1/5) (instead of the unit 1/min hitherto mainly used, the ew systom of units adopts the unit 1/s, see DWN 1909, For angular units the folowing relationships apply 2nrad = 360° mm tod 2100 mm _ um 1 eed BEPage 6 DIN 3960 1 rad 1 mrad = 3,437 7'~ 206,26" 200" * = rad = 0.017453 rad 180 m G " 0,291 mrad~0;3 mrad 1" = 4.85 prad~5 wrad 0.001? = 36" = 17,45 urad For conversions the following applies: 180. ain degrees = —- (ain radians) 87,295 780 -(« in radians) it may happen that In systems with a limited character set, (eleprinter, data handling), the unit symbols for degree, minute end second In the case of angle data cannot be represented by the superscripts °,', In these cases the {following letter symbols representating the units according to DIN 68030 shail be applied: deg or DEG for degree (engle), mnt or MNT for minute (angle), ec or SEC for second (") It is recommended to use d, m and s as further abbrevi- ations of the unit names, e.g. 17d 27m 27, 2 Concepts and parameters associated with cylindrical gear teeth All definitions in this clause relate to gears free from deviations and tolerances. Thus, the eque- tions apply to the nominal dimensions of the gear teeth; for exceptions see subclauses 2.7.5 and 2.7.6. ‘The nominal dimensions of involute cylincrical gear teeth fare determined by the following parameters which are Independent of one another: number of teeth z standard basic rack tooth profile normal module in helix angle f addendum modification coefficient x facewidtn b 2.1, Number of teeth z and sign of number of teeth ‘The number of teeth z of an external gear must be in- serted in the equations below as a positive quantity, whilst the number of teeth z of an internal gear must be inserted as a negative quantity. This satisfies the concept that during the transition from an external to an internal gear the gear diameter increases until in the first instance at d= +e a rack having z = [s reached. As the transition proceeds further, the gear diameter swings over to —= and thereafter assumes a finite negative size. By establish- ing this and the definitions contained in subclauses 2.53 ‘and 2.54 concerning the sign of the helix angle and of the ‘addendum modification it is possible to use the same equations unchanged for external gears as well as for internal gears. Hence, for an internal gear, negative values are obtained In the ¢aloulations for all the quantities depending on the ‘number of teeth — these are the diameter and radius, the ‘angular pitch, the tooth thickness angle and the space- Width angle, the radius of curvature of the tooth flanks, ‘and of the tast dimensions for toath thickness: the base tangent length, also the radial and ciametral single-ball and single-pin dimension or dimension over balls or pins. In addition, for an internal gear pair the tooth ratio and the contre distance are negative. Inproduction specifications (drawing data) however all test dimensions and the numbers of teeth, diameters etc. are always quoted as positive quantities, and this also applies to internal gears; the exception is the addendum moditi- cation, which has to be inserted with ils appropriate sign (600 subclause 2.5.4), 2.2 Reference surfaces and datum lines 2.2.41 Standard basic rack tooth profile, basic rack The standard basic rack tooth profile of cylindrical gear teath is the normal section through the teeth of the basic rack which is produced from an external gear teoth system by increasing the number of teath until 2 = = and hence the diameter until d= +=, The flanks of the standard basic rack tooth profile of involute teeth ara straight lines. In transverse sections (and only in transverse sections), the flank profiles (left flank and right flank profiles) of 2 cylindrical gear with involute teeth are portions of invo lutes to a circle (in brief: involutes). The flanks are involute helicoids in the general case, and involute surfaces in the case of straight teeth 2.2.2 Datum tine, dimensions on the standard basic rack tooth profile ‘The dimensions on the standard basle rack profile arise ut of the module m and the datum line: on the (straight) datum line, the pitch p, the tooth thickness sp and the spacewidth ep of the standard basic rack tooth profile are slated as multiples of the module m; similariy referred to the datum line are the addendum hp and the dedendum Jip, see figure 1, Addendum and dedendum together yield the profile height lip. The teeth end at the tip line, With the normals to the datum line, the flanks of the stan- dard basic rack tooth profile enclose the pressure angle ‘pi they merge via the root rounding into the reot lina. Tho pressure angle ap of the gear standard basic rack tooth profile may differ from the pressure angle apo of the cutter, standard basic rack tooth profile. ‘The standard basic rack tooth profile according to DIN 867 ‘or DIN 58400 dictates mirror-imaged identical involute ‘tooth flanks for both sides of each tooth. a Tp line hehe] heh names ner Flank angle 2 ap p=m-m pitch ep spacewidth on the datum line Sp tooth thickness on the datum tine fp profile height Fgp addendum hip dedendum ap pressure angle or root rounding radius Figure 1. Standard basic rack tooth profile of an involute tooth system2.2.8 Seetions through cylindrical gear teeth 223.4 Transverse section The sectioning of cylindrical gear teeth on a plane perpen- dicular to the gear axis yields a transverse section. A trans verse section of a rack is its intersection with a plane perpendicular to the axie of the cylindrical gear mated with the rack. Quantities in the transverse section are denoted by the subscript 22.3.2 Normal section The sectioning of involute helical gear teeth by a surface disposed perpendicular to the tooth traces of the involute helicoids yields anormal section. The surface of the normal section is curved three-dimensionally. In the case of a spur ‘gear the normal section and transverse section coincide. ‘Anormal section of a rack is its intersection with a plane perpendicular to the tooth traces of the teeth Quantities in the normal section are denoted by the sub- seript n ‘The subscriptsn and tare not applicable to spur gear teeth, 2.2.3.3 Axial section The sectioning of ylindrical gear teeth by @ plene contain- ing the gear axis yields an axial section. An axial section fof @ rack is its intersection with a plane perpendicular to the rack datum plane (see subclause 2.2.7) which contains the axis of the cylindrical gear mated with the rack. Quantities in the axial section are denoted by the sub- script x ‘Axial sections and the stating of gear tooth quantities in such sections are not mesningful where spur gear teeth are concarned. 2.24 Gear tooth profile, flank profile ‘A gear tooth profile results as the line of Intersection of the gear teeth with a plane, A flank profile is the line of intersection of tooth flank with a plane. Right flank profiles and left flank profiles are ta be distinguished where necessary. 2244 Transverse profile The transverse profile is the gear tooth protlle lying in a transverse section The transverse profile of spur gear teeth and the transverse Protile of the gear mating with such teeth lle in the same: plane. 2.24.2 Normal profile, virtual spur gear teeth Because of the curved normal section it is only possible in the case of a spur gear wilh involute helical teeth to indicate an approximate normal profile lying in a tangential plane or in an osculating plane to the normal section. Geometrical studies are therefore often based on virtual spur gear teeth lying in the tangential or osculating plane, their datum line being a circle of curvature on the normal section of the reference surface of the helical teeth (seo subclauses 27.1.1 and 2.7.2) 2.24.8 Axial profile The axial profile is the profile of the gear teeth lying in an axial section, 228 Tooth traces The tooth traces are lines of intersection of the right and foft flanks with a cylinder the axis of which coincides with the gear axis. Hence, right tooth traces and left tooth traces are to be distinguished, ‘The reference tooth trace (reference cylinder tooth trace) is the line of intersection of the flank with the reference cylinder (see subclause 2.27). The base tooth trace is the DIN 3960 Page 7 line of intersection of the~possby imegined as extended — involute flank with the base cylinder (see subclause 2.2.8) the base tooth traces are the haices of the involute heli- code (see subclause 231). The tip footn trace (root tooth {race is the line of intersection ofthe ~ possibly imagined a8 extended ~ involute flank with the tip evlinder (root eyiinder The tooth traces are helices in the case of helical gear teeth and straight lines in the case of spur gear teeth 2.26 Modules a My ‘The modula m ofthe standerd basic rack tooth profi is the normal module {module in the normal section), of the gear teeth Inatransverse section the transverse module mis found as oe en For a helical gear the axial module my in an axial section is found as Smt PI wsy “Tanta cee Fora spur gear f ‘The basic module ms i rmiy= myl [tan ay + 608" 2s) ‘and the module Is mt (neq = 22.7 Reference cylinder, reference circle; reference diameter d The reference cylinder isthe reference surface for the gear teeth. its axis coincides with the locating axis of the gear (gear axis). Hence in the case of arack the reterence plane is the rack datum plane. Quantities on thereference cylinder are stated without eubscript The reference circle is the intersection of the reference cylinder with a plane of transverse section. The reference dlameter dis determined by zon dezem= Ete (24) cos B Note: In the case of an internal gear the reference dlam- eter is a negative value, see subclause 2 1. 228 Base cylinder, base citcle; base diameter di, The base cylinderis that cylinder coaxial with the reference cylinder that is determinative for the generation of the involute surfaces (involute helicoids). Quantities associated with the base cylinder ara denoted by the subscript b. The base circle isthe intersection of the base cylinder with 2 plane of transverse section; the involutes of the base circle form the usable parts of the tooth profiles. The base diameter dy is given by 2 my cos a cos B = 2: mpl \Tan®ante0e B= z>my (25) dy=d-c08 a= 2 +1 C05 a= Note: In the case of an internal gear the base diameter Js a negative value, see subclause 2.1. 2.3 Involute surface (involute helicoid) 23.1 Generator of an involute surface In developing the envelope of the base cylinder, an enve~ lope line of the base cylinder describes an Involute surface of @ spur gear; itis the generator of the Involute surface. straight line inclined to the envelopeline in the developed envelope surface is the generator of an involute surface (involute helicoid) of a helical gear having its origin on the base cylinder in the helix, see figure 2.Page @ DIN 3960 Involute of base cylinder / Hole Involute helicold jase cylinder envelope line Developed Base cylinder inyolute line dy base diameter base helix angle Yo base lead angle TS Developed bese eyinder envelope | (base cylinder tangential plane) Inyolute of base cylinder Figure 2. Base cylinder with involute helicoid and generator 2.8.2 Base hellx angle fy, base lead angle yy, The acute angle in the angled base cylinder envelope be- tween the generator and an envelope line is the base helix angle 6, see figure 2 and equations (2.90) to (2.32). The complement angle of the base helix engle fy is the base lead angle yy I ypt = 90° ~1 fh 26 Both angles have the same sign. With the development of the base cylinder envelope first in the clockwise direction, involute surfaces curved in ‘opposite directions result and from these the left and right, flanks of the gear teeth are derived. 2.3.8 Contact line The two base cylinders of a gear pair have two common tangential planes which intersect in the instantaneous axis. These are the planes of action of the gear teeth. Each plane of action cuts the associated tooth flanks (lank and mating flank) in the contact line corresponding to the working position. Each contact line is the common gene- rator of flank and mating flank; its prolongations are tan: gents to the base cylinder. Onrotation of the base cylinders about their axes and simultaneous winding up and un- winding of the tangential planes the contact lines travel through their zone of action, see subclause 2.4, 2.8.4 Transverse pressure angle at a point ay, transverse pressure angle «;, The involute (always lying in @ transverse section) is in clined at the arbitrary point Y by the transverse pressure angle ata point ay relative to the radius passing through Y, see figure 8, The angle of Inclination at the point of intersection of the Involute with the reference circle is the transversepressure angle a, cos esas” an (2.8) See also equations (2) and (2.10) T point of contact of tangent with base circle U_ origin of involute Y arbitrary point on involute Fr refarence radius ry base circle radius ry Y-oirele radius 4, transverse pressure angle 3, transverse pressure angle at point Y Figure 2. Parameters relating to an involute 23.5 Normal pressure angle at a point i, normal pressure angle c;, In the normal section through an involute helicoid the tan gent to this surface at an arbitrary point Y is inclined to the radius through Y by the normal pressure angle at a point aq. The corresponding angle of inclination at the reference cylinder is the normal pressure angle dy this is ‘equal to the prassure angle ap of the standard basic rack tooth profile.tan ay = tan ay cos tan gyq © £80 ays * COS By For a spur gear B= 0 and ay = 2.3.6 Rolling angle £ of the involute ‘The angle of the centre defined by the origin U ofthe invo: lute and the contact point T of the tangent from point Y_ to the base circle is the rolling angle ¢, of the involute, see figure 3, The base circle arc UT is equal to the tangent portion YT, hence & + tan ay, ay) 2.8.7 Radius of curvature 0 of the involute, working length Ly ‘The tangent portion YT Is the radius of curvature g, of the involute at point ¥ and at the same time the working length LL, belonging to point, Lo. the developed base circle arc from the involute origin U. In the triangle OTY itis the side opposite the transverse pressure angle ay, at the centre of the circle O, see figure 3. yt ro Gyr tam ay, Note: In the case of internal gear the radius of curvature is @ negative value, see subclause 2.1, 2.3.8 Involute fun ‘The angular difference ¢—qistermed the involute function of angle a and is denoted by inv a (to be read as: invo: lute a), see figure 3, inv ay (2.13) Note: Ithas previously been customary to use the symbol ev @ for the Involute function, In conformity with {SO 701~ 1978 however itis recommended that in future the symbol inv @ should be used. 239 Lead p, The lead p, (of an involute helicoid, of a tooth flank) is the portion of an envelope line of a oylinder concentric with the gear axis botwaon two successive turns of an involute helicoid (of a tooth flank), se figure 4. The lead is inde- pendent of the diameter of the oylinder. Zlemgim Lzlemcn sinlpl tani fl iy~ ay taM ey = ay pm zhpe (14) Plane of standard basic rack tooth profile, datum line \ Reference cylinder envelope line, gear axis L —p, Figure 4. Helical gear: lead p,, helix angle f, lead angle y 2.4 Angular pitch and pitches 2.4.1 Angular pitch + The angular pitch + is that angle lying in a transverse section which results from the dividing of the complete Periphery of a circie into z equal parts, DIN S960 Page 9 In radians (2.18) r= 52 inegrees 2.16 Note: In the case of an Internal gear the angular pitch is @ negative value, see subclause 2.1 Figure 6. Helical gear: diameter, angular pitch, pitches 24.2 Pitches on the reference cylinder 2424 Transverse pitch (pitch) The transverse pitch (pitch for short) p, (p in the case of ‘a spur gear) is the length of the reference circle arc be: twaen two successive right or left flanks, see figure 5. dex ete pi 2.17 242.2 Normal pitch py ‘The normal pitch pss the longth of the helix arc between two successive right or left flanks on the reference cylinder inthe normal section of the gear teeth, Pn ™ Pr cos pb 242.3 Pitch span pe The pitch span ps the reference circle arc between two right or let flanks separated from eachatherby k reference circle pitches such that 1< k <2, see figure 6.It isthe sum of reference circle pitches. Penk p no (2.18) 2.19) 2.4.3 Pitches on the Y-oylinder On @ cylinder with the arbitrary diameter dy (Y-cyiinder) the following pitches are present: 243.1 Y-cirole pitoh py. ‘The Y-cirola pitch py. (py in the case of a spur gear) is the lenath of the arc with diameter d, between two successive fight or left flanks, see figure 5. (2.20) Panty 243.2 Y-cylinder normal pitch Psp, The Y-cylinder normal pitch pyais the length of the helix arc fon the Y-cylinder between two successive right or left flanks in the normal section of the gear teeth, Pin = Py C08 By (221 244 Pitches on the V-oylind ‘On the cylinder with diamster dy (see subclause 2.6.63) the following pitches are present:Page 10 DIN 3960 24.41 V-circle pitch py The V-circle pitch py, (py in the case of e spur gear) Is the length of the arc with diameter d, between two succes: ive right or left flanks, ay Panty - pM (2.22) 24.42 V-cylinder normal pitch py The V-cylinder pyq is the length of the helix arc on the V-oylinder between two successive right or left flanks in the normal section of the gear teeth. Pon ™ Pus 608 Be (2.23) 2.4.5 Pitches on the base cylinder 2.4.54 Transverse base pitch py. The transverse base pitch Pye (Ds in the case of a spur gear) Is the length of the base circle aro between the points of origin of two successive right or left flanks, see figure 5. yon dy i. a Pty 0s ay (2:28) 24,52 Base cylinder normal pitch pps The base cylinder normal pitch pp is the length of the helix arc on the base cylinder between the helices of two succes- sive right or left flanks in the normal section of the gear teeth, Pow ~ Bex * €08 ly = Pry“ C08 ay (2.25) 2.4.8 Normal base pitches p. ‘The distances between two successive right or left flanks In the plane of action are the normal base pitches ps. 2.4.81 Transverse pitch on path of contact pu The transverse pitch on path of contact po (normal base pitch pe in the case of a spur gear) Is the distance between ‘two parallel tangents ina plane of transverse section which contact two successive equal-handed flanks. In the transverse section of the gear teeth the transverse pitch on path of contact is the length of the transverse line of action between two successive equal-handed flanks, 00 figura 5: ‘The transverse pitch on path of contact is equal to the transverse base pitch, Per Pt C08 a, = Par (2.28) 24.6.2 Normal base pitch Pon ‘The normal base pitch Penis the distance between two parallel tangential planes which contact two successive ‘equal-handed flanks in the active area of the flanks. tis equal to the base cylinder normal pitch, Pon = Px * €08 tn = Pin (227) 24.7 Axial pitch py The axial pitch p, is the portion of an envelope line of a cylinder concentric with the gear axis between two suc- coessive right or left flanks, see figure 6. The axial pitch Is independent of the diameter of the cylinder and Is +r times the lead, i: rere! Pooms°® = Snrar~ tanTat~ 121 278) Figure 6. Axial pitch py of a helical gear 2.5 Concepts and parameters associated with gear teeth arising from the position of the standard basic rack tooth profile relative to the reference cylinder The datum line can 1) cross the gear axis with the lead angle y b) have a distance of x m, trom the reference cylinder. The following parameters for gear teeth thus arise. 2.5.1 Lead angle y, helix angle ‘The lead angie y is the acute angle at which the datum fine ‘crosses the gear axis, see figure 4. Itis also the acute angle between a tangent to a reference helix and the plane perpendicular to the gear axis through the tangent contact, point Tho helix angle f is the acute angle between a tangent to a reference helix and the reference cylinder envelope line through the tangent contact point. 1Bl= 90° =I y1 (2.28) The lead angle and helix angle have the same sign. For spur, 0° and f= 0°, The helix angle fy of right flanks ‘may differ from the helix angle fj, of left flanks. Tho relationship between @ and the base helix angle fy Is found from tan fy = tan f+ C08 a, (2.30) sin Py = sin B + C08 ey 231) <8 OB cosa sina siney 208 ay * tan? an + cos™ (232) On @ cylinder with diameter dy the helix angle By is found fram cos « ay _ tan aga a sing, = sing 20822 = SiN By By = SIN * Sam Gos an (234) tan ayn | _€08 ay - 008 cosy = MA, £08 Ap E05-AY 9 ag) tan “y, 205 ayn2.5.2 Flank direction; right-handed, left-handed in the case of external and internal goar teeth the flank direction is right-handed (left-handed) ifthe helical tooth trace corresponds to a right-hand hells (left-hand helix) 2.5.3 Sign of the hellx angle If signs are needed for the helix angle (eg. for evaluating the nominal dimensions of a cylindrical gear pair) the fol lowing applies: For an external gear a helix angle belonging to a right- handed flank direction counts as positive, whilst a helix ‘angle belonging to a left-handed flank direction counts as egative. For an internat gear the signs are reversed. The Sum of the helix angles of a pair of cylindrical gears is, therefore zero. Note: In production documents only the absolute values of the helix angles are used with additional indication of the hand (t for right-handed, | for left-handed). 28.6 Addendum modification; addendum modification coefficient x and sign of addendum modification The addendum modification of involute gear teeth is the displacement of the datum line from the reference cylin: der. The magnitude of the addendum modification is ex pressed by the addendum modification coefficient x in parts of the normal module: addendum modification = + ig for spur gears: addendum modification = x: ‘An addendum modification is positive ifthe datum line Is displaced from the reference Clrcle towards the tip clrcle; as a result, the tooth thlekness in the reference circie is greater than for zero addendum modification, see figure 7 negative if the datum line is displaced from the reference: circle towards the root circle; asa result the tooth thickness in the reference circle is smaller than with zero addendum modification, see figure 8. Note: Where internal gears are concerned, this standard ‘reverses the definition of the sign of the addendum mod- fication compared with DIN 3860, August 1960 edition, I. for internal gears xm, according to this standard equals =: m,, DIN 3960 (August 1960 edition), see subclause 2.1 2.8.5 V-gear, zero gear A cylindrical gear with addendum modification is termed 22 V-geer (Veins Gear if addendum modification is positive, Vovinus dear i addendum modification is negative). cylin: drical gear with zero addendum modification is termed zero gear, 2.8.6 Diameters of gear teeth, angles at V-cylinder The positian of the standard basic rack tooth profile relative to the reference cylinder givas rise to the following cylin- drleal surfaces and diameters In respect of the gear teeth. Note: In the case of an Internal gear the diameters and radi are negative values, see subclause 2.1, 2.5.6.1 Tip cylinder, tip circle; tip diameter dy The tip cylinder is the eylindrical envelope surface at the tips of a gear tooth system; a transverse section yields the tip circle. Quantities related to the tip cylinder are denoted by the subscript a. DIN 3960 Page 11 Datum line Figure 7. Positive addendum modification on an external goar Datum tine Figure 8. Negative addendum modification onan internal gear ‘The tip diamet figure 7) to da dy amounts in the case of gear 1 (see a-dg~2-¢ Ady +2-ay mg +2 hyp t2-Re my (2.38) ‘and in the case of gear 2 (see figure 7 and figure 8) dg" 2-a-dy-2-¢ Hdyt2-xg¢mgt2-hypt2 he my (237) Note: For definitions of the quantities a, c and k see sub Clauses 9.2.3, 227 and 3.3.6 as woll as equation (2.47). if the tip dlametars are dimensioned according to these equations, the bottom clearance will bec» 0,167 + miy oF 0,25 my, depending on the standard basic rack tooth profile of the cutting tool, For detalls see DIN 3972 or DIN 58.412, For an external gear the relationship should be d,2dy 42> tity For external gears this sats a lower limit for the addendum modification coetficiont x, see subciause 2.6.4. For an internal gear the relationship must always be || > |p|. For internal gears this sets an upper limit for the addendum modification coefficient, see sub- clause 2.64. 256.2 Root cylinder, root circle; root diameter dy ‘The root eylinder is the cylindrical envelope surface at the bottom of the tooth spaces of a gear tooth system; a trans- verse section yields the root circle Quantities related to the root cylinder are denoted by the subscript fPage 12 DIN 3960 The root diameter d; amounts in the case of gear 1 (see figure 7) to dy dy~2-hewdy+ 2x, my Zhan (238) and in case of gear 2 (see figure 7 end tigure 8) to dig" dz~2+ hy dg 2°x2°Miq~2-hyy (2.39) Note: If the tooth thickness deviations Ay. are produced by deeper infeed of the gear cutting tool, then the following |g found dy=d~2-hy+ Ag cot a (2.40) In the case of ground and shaved cylindrical gears the too! addendum hyy sto be substituted for hin equations (2.38) and (2.98) and the addendum modification coefficient of the pre-cut teeth for x 2.5.63 V-circle diameter dy, V-cylinder The V-cylinder contacts the datum ling, see figure 7 and figure 8, Its diameter (V-circle diameter) is a a+ 2ex-my, sa(ie2 ‘Quantitios related to the V-cylinder are denoted by the subscript v. cos) (2a 2.5.8.4 Pressure angle ay, helix angle fy ‘The following angles are present where the V-cylinder intersects an invalute tooth flank hate angle fy z42-x-cons tan fy = 7 PEF tan g (2.42) transverse pressure angle a a point ax cota = cos s 24s) normal pressure angle ata polnt en tan Bx 0s fy eas £05 ap * COs B £08 en cos) (2.45) 2.8.7 Gear teeth heights Note: The working depth resulting trom meshing with @ mating gears defined in subclause 3.2.6. 2.8.74 Tooth depth h ‘The tooth depth i of cylindrical gear teeth results from the. ‘tooth depth hy of the standard basic rack tooth profile and the addendum alteration it~ nty, see subclause 3.3.6. He hot Remy (2.48) 2.6.7.2 Addendum hy, dedendum hy The addendum fy and the dedendum of a cylindrical gear are stated on the basis of the reference circle: Mig? hap #2 thy + eo ty, (ean ig hyp mq (2.48) 25.8 Tooth thicknesses, spacewidths Figure 9, Helical gear: tooth thicknesses, spacewidths ‘and their half angles 25.8.1 Transverse tooth thicknesses 5, 5y.5aNd Sy. Tho transverse tooth thickness s, (tooth thickness sin the caso ofa spur goar) is the length of the reference circle are between the two flanks of a tooth, Feta xe my taney x Sn me (Gaze tanen)= See (249) The transverse tooth thickness 5, (tooth thickness syin the case of a spur gear) is the length of the circular arc of diameter dy between the two flanks of a tooth, sua dy: (J+ inves ~ inves) The transverse tooth thickness sy, (tooth thickness sy in the case of a spur gear) is the length of the V-circle arc between the two flanks of a tooth, a= 8 ($+ inv ~ ivan) The base tooth thickness sy (sy in the case of a spur gear) Is the base circle arc between the points of origin of the Involutes of a tooth £4 tt ore) six = ay ( ag tinves) (282) 28.8.2 Tooth thickness half angles Gy ty and dy Angles at the centre in a transverse section which are enclosed by radii bounding the tooth thicknesses s,. 5, Sut OF sq are tooth thickness angles. The corresponding tooth thickness half angles are: aeéea-ton -7- (2.53) to Boos inve tray sn ten Fee wt inva - inven (285)The base tooth thickness half angle yy is Soy Wor gy 79 + inva: (2.58) Note: In the case of an internal gear the tooth thickness half angles are negative values, see subciause 2.1. 28.8.3 Spacewidths ee 61 and ey The spacewidth , (ein the case of a spur gear is the length of the raterence circle arc between the tooth fianks en- closing a tooth space. The tooth thickness s, and space- width 2, together yield the transverse pitch step (287) Pi em B-Boy tana, wmo(J- 2:4: una) The spacewidth ey (ey in the case of a spur gear) is the length of a circular aré of diameter d, between the tooth flanks enclosing a tooth space, The tooth thickness sy, and spacewidth ey together yield the Y-circle pitch py (259) (2.58) tang: The spacewidth e,: (¢, in the case of a spur gear) is the length of the V-oircle arc between the two flanks enclosing a tooth space. The tooth thickness s., and spacewidth oy, together yield the V-circle pitch py. Pu (281) x tan ay ‘The base spacewidth ey (ey in the case of a spur gear) isthe base circle are between the points of origin of the two involutes enclosing a tooth apace. The base tooth thick: ness sy, and base spacewidth ey, together yield the base circle pitch py. Su # 01 ™ Pou (2.69) lanes 28.84 Spacawidth half angles 7, ny My and mp Angles at the centre in a transverse section which are enciosed by the radii bounding the spacewidths &, ey 4 0F ey ara spacawidth angles. The corresponding space: width haif angles are: a 4ex-tan ta Cc) ny = Be =~ inv a, + In ay (2.66) peer eeet asn DIN 3960 Page 13 The base spacewidth half angle ny Is mo Goan inva, (268) Note: In the ease of an internal gear the spacewidth hat angles are negative values, see subclause 2.1 Reference: For spacewicth helf angles at deviation of tooth thickness A, see equation (4,9) in subclause 4.1.4 25.8.5 Normal tooth thicknesses Sq, Sy, 5yn aNd Son ‘The normal tooth thicknesses are the tooth thicknesses in {normal section of the gear teeth; they are the lengths of the helical arcs on the respective cylinders between the tooth flanks of a tooth. They result from the standardbasic. rack tooth profile and the addendum modification: =m ( +22 tance) (269) syn 608 Dy 270) faa = 08 em Sha = Sox COS fy (2.72) 25.8.6 Normal spacewidths ey, eq, €yy ANd ey The normal spacewidths are the spacewidihs in a normal section of the gear teeth; they are the lengths of the helical arcs on the respective cylinders between the two tooth flanks enclosing a tooth space. Tey result from the stand- ard basic rack tooth profile and the addendum modifi- cation: og e:-c0np = BL 2-e-my ten ~m (§-2-* tne) ers een PO 2.6 Geometrical limits for involute gear teeth 2.6.1. Undercut on eylindrleal gears with external teeth When the teath of an external gaar are being produced undercutting of the tooth flanks occurs if the path of the corner of the cutter tip cuts into the involute portion of the tooth root flank of the workpiece during the rolling action. By suitable choice of gearcutting data, particularly as regards addendum modification, helixangle, addends and, pressure angle, underout can be avoided, For an external gear with standard basic rack profile according to DIN 867 produced by @ rack cutting tool the following relationship exists between the cutting data si ay | hyo ~ gap (1~ sin ay) core Tit 7 ‘here hp is the addendum and gp is the tip corner round- Ing radius of the tool. In the case of an external gear with standard basic rack tooth protie according to DIN 58 400, 0 (L~ sin a) In equation (277) must be replaced by the bottom clearance Sain 7mPage 14 DIN.3960 The value spin Calculated in this way can usually be further reduced by a small amount (not exceeding 0,17) it a small degree of undercut, which generally does not have any harmful effects, is acceptable. 2.6.2 Pointing limit and minimum tooth thickness at the tip circle of an external gear Tho teoth of an external gear become pointed at the tip, circle if =p + iv ay inv ayy = 0 (2.78) The pointing timit resulting in this way is considered in subclause 2.64 At the tip cylinder the tooth thickness sy, should not have a value less than 0,2 my. This provides an upper limit for the addendum modification of an external gear which is suitable for practical application, see subclause 2.6.4 The undercut and pointing limit set limits in the downward direction to the number of teeth which itis practicable to cout in an external gear. For spur gears and the standard basic cack tooth profile according to DIN 867 the following apply 2min taking x= +0,57 as the pointing limit, 02 2nin = 9, taking = + 0,45 for san see figure 10, if @ small amount of undercut is accepted, the following is. still feasible Znin= 7, taking x= + 0,41 2.6.3 Pointing mit and minimum spacewidth at the root ‘cirole of an internal goar The tooth space of an interne! gear becomes pointed at the root circle if n= inva + inv ag = 0 (2.78) The resulting pointing limits considered in subclause 2.64. At the root circle of the internal gear the spacewidth e%, should have @ value not less than 0,2 - m. This provides ‘an upper limit for the addandum modification of an internal {gear which is suitable for practical application, see sub- clause 2.6.4. The tip cigcle dlamater limit and the pointing limit for the tooth spaces jimit the feasible number of teeth for an Internal gear in the downward direction. For spur gearing and the standard basic rack tooth profile according to DIN 867 the following applies [elon = 16 with 2 = Izlaie™= 21 with = ~0.58 fr etn 0,52 for pointing limit 02 see figure 10. Other limits arising from mating reasons are dealt with in DIN 3983, 2.6.4 Range of feasible involute gear teeth The feasibility range for involute goar teeth which is im- posed by the limits according to subclausas 2.6.1 to 263 Is indicated in figure 10 for the standard basic rack tooth profile according to DIN 867 and in figure 11 for the stand: ard basic rack tooth profile according to DIN 58.400. The rolationships for helical gear teath are shown here for virtual spur gear teeth which, for a standard basic rack tooth profile according to DIN 867 or DIN 58400, are determined by the addendum modification coefficient x of the helical goar teeth and by the virtual tooth number z cost Ay, Zax eh (280) cos A For the virtual tooth number factor zi see table 1. This relationship between the numbers of teeth is so chosen that fora helical gear tooth system and the corresponding virtual spur gear tooth system the same value xnin 1S yielded by equation (2.77). in fiqure 10 and figure 11 the addendum modifications x which are feasible each time are plotted against the virtual tooth numbers Zax Table 1. Virtual tooth number factor zis = = 35 = a8 a fetion of hall ange for cy = 20° p as a th B | Phe 8 ahs 8 ti degrees degrees degrees | egrees degrees 1 | 10008 | 11 | 1526 | 21 | 12002 | sr | 15235 | a1 | 2373 2 1oor7 | 12 | 10622 | 22 | 12911 | a2 | 15680 | 42 | 22255 3 0038 | 13 | tors | 23 | 12556 | a3 | i615 | 43 | 29200 4 | 10088 | 14 | 10868 | 26 | ree | 3a | 16663 | 44 | 24223 s | roos | 15 | 11006 | 25 | 13100 | 356 | 17206 | 4s | 25002 6 1.0153 16 11181 26 1,3400 36 ar7e7 7 | to2e | 17 | stat | 27 | var20 | 37 | 1.8400 8 | ioe | ta | i143 | 28 | i4oes | a8 | 19078 2 | tose | to | 11668 | 29 | 14428 | 39 | 19788 wo | tows2 | 20 | 1158 | so | svete | 40 | 20863DIN 3960 Page 16 » svopidos tr ssou sou yjoo} ut suoneMop Way S1E86 (e9LpUIAD 40 ese9 Out UL 68 NIG 09 = "2 01 dn ssoquind yjoot 40} lnpusppe pue “2 uysar joseauns (001 243 Suzepisuoo ino\au) OO” eS NIC 4 Bulpi0s oes o18eq puepLEIs uum yjse) 1898 etn “Ly eunbiy agsoang eesoung youn “we 72 + 9p ="p 108 -wopp ojosjo dn wnuju ox enp sies6 yeuse}xe 10} senjen Suny — ¢ Our (queued 01 fq papua}xe 100} uo UY eAn9e) SOPES NIC 01 BU -ps0908 , 4109} Jo /oquunu nuUUIL pEDUBI9}0, 40} S/EAB azeun ‘syu9H 1001 12 yno.epun 0} snp s1eeB jeuse}x9 10} Sanjen Bi ez oun, Su (0 1s yapimjse10 wines 10.9 = "5 yu) Bunulod 0} enp s1B9 jeuse}xo 40) sonyeA BuNIWT 4 Sonn x sesedou x seou yo 00} UI SUOREIAEP WIM S180B JeoUPUIAD Jo ase aM UI 8666 Nid Oste 298 S198 jeu/9qU) 10, 09: = *!2 0} dn siaquinu yoo} 40} x yua!ay!900 uoREOW/poU ‘unpusppe pue™2 yje9} jo saquinu 298 129 prepuers Un HOE ¥0 ut «(¢9**- 9) = Ys qypynegeds 1oos wnusiutw 10 9 = “Ho yu Sunuled o} np iu - 100 the upper right-hand part of the figure i applicable, being designed for virtual tooth-nummber groups 100... 108, 109... 117 Bc. rising by 9 each timo. From the corresponding table @ tabuiated value Is found fer the number of teeth spanned which, depending on the addendum modification coetticient, sther yields the value of direct. or which must bo elther decreased or increased by 1,2, 307 4 measured teoth corresponding to the ranges In the upper part of the figure In ordior to yield the number of teeth spanned kn this part Of the figure also the ranges overiee, Example: gqw = 142 (tabulated valus: 16) and x~ 0,9 ylelds the possible values k= 16 + 1= 17 or k= 16+ 2 18 or k= 16 + 5 = 18, Figure 17, _Nomogram for determining the number of teeth spanned for external gears with ay = 20° 3 €DIN 960 Page 25 6 con |! ~ AY 4 é umber St measured spaces 4 g a aemeet a aer 3 2 — eo 8 E & 3 z a ls a { : 2 Lewd = 0 “0-20-30 40-50-60 -80 0 ? —— Number of teeth 2 0 The left-hand part of the figure contains the relationship between the number of z teeth 2, the addendum modification coefficient xand the number of measured spaces ft for [2| 100. The solic-iine curves correspond to the condition that the contact points of the measuring elements are 0,5 - 1 outside tho V-circle, whilst the broken-line curves correspond to the condition that the contact points of the measuring elements are 0,5 mt inside the V-circle. ‘The limits of the tooth systems feasible according to subclause 2.6 are not indicated here, see figures 10 and 1. ‘The areas covered by @ givan measured number of spaces overlap in large parts of the diagram, so that hore seversl measured numbors of spaces are feasible for a single umber of teeth and a single addendum mouification coeficient. Example: 2 = ~48 and x= +0,9 lie in the range of the measured numbers of spaces 2, 3, and 5. Only in the shaded portions is meesurement with only one measured number of spaces possible, For example: 2 = —27 and x= —0,$ ie in the range k= 4. For |z]> 100 the right-hand part of the figure is applicable. For explanation see ro 17 Number of measured spaces 244 to -252 Bi ete. Figure 18. Nomogram for determining the measured numbsr of spaces le fr internal spur gears with «= 20° y-~ Measuring anvil pr Vhead 7 3 coarvan s base cylinder Plane of action — Figure 19. Contact line overlap by, In base tangent length measurementPage 22 DIN 3960 TS a0 Contact tine overlap by —e- rr a 6 a mn 15D Bave tangent length Wy—— Example: For Wi = 69,548 mm and by = 3,7 mm the diag- onal is longer than Wz by about 0,10 mm. the overlap by is reduced, for example by edge chamfering of the flanks and facewidth deviation by 0,84 mm, then the diagonal is. stil longer than W, by 0,06 mm. In this case the measure ‘ment can still be regarded as adequetely reliable Figure 20, Relationship between base tangent length Wy. contactiine overlap byend length difference AW dy base cylinder diameter dx, diameter of circle through centre of ball dy, diamet circle diamater) Dy. measuring ball diameter PL contact point of measuring ball with left flank Pr Contaet point of measuring ball with right flank Of cylinder on which the contact points P lie (measuring For meaning of symbols see figure 21 Figure 22. Radial single-ball dimer spur gear jon My on an internal 7 Base cylinder tangential plane for right flank Base cylinder tangential plane <“ for left flank Centre of tooth space in Section A~A. |_—Centre of tooth space in Section B—B Generator of 1 pressure angle in transverse section at circle through centre of bell 3. pressure angle in transverse section at contact points P Figure 21. Radial single-ball dimension M; in a holical external gear right flank Section C-Ctor Dy External geertocth Number of teeth 2. — Figure 23. Nomogram for determining the measuring ball diameter Dy for radial single-ball or diametral two-ball measurements for ¢, = 20° and In the case of cylindrical gears with tooth thick ness deviations xe replaces x DIN 3960 Page 23, 2.7.4 Diametral test dimensions for tooth thickness 2.74.4 Dimension over balls Max Inthe caseofan external gear the dimension overballs Myx. is the largest external dimension over two balls whilstin the case of an internal gear'tis the smallest internal dimension between two balls having diameter Dy, and in contact with the flanks in two tooth spaces at the maximum possible separation from each other on the gear, see figures 24 to 26. The two balls must be in the samo plane perpen- dicular to the gear axis; the measuring element faces (making contact externally or internally) must be held parallel to the gear teeth axis. Note: The dimension over balls Is not referred to the gear axis end hence is independent of any eccentricity of the gear teeth, For an even number of teeth, see figure 24, the following applies Max = dx + Du. (2.108) {for an odd number of teeth, see figure 25 and figure 26, the following apalies Max = dk * 008 5 + Dw (2.409) For choice or calculation of the measuring ball diameters Dy see subclause 2.7.3.1 Note: In the case of an internal gear the value found for Max 18 negative, see subcleuse 2.1 27.4.2 Dimension over pins Mix External gears with spur or helical teeth and internal gears with spur teeth can also be measured with pins instead of with balls, Equations (2.99) to (2.104) and (2.108) to (2.108) apply equally to the dimension over pins Mar. 2.74.21 Dimension over pins on spur gear teeth For spur gear teeth equations (2.108) or (2.109) apply. For positioning the measuring pins it Ie not possible to state ‘2 common plane perpendicular to the gear teeth axis. The measuring elements need only be swivelled sideways to find the maximum value; this is the actual size. 2.74.2.2 Dimension over pins on external helical gear teeth with even number of teeth Equation (2.108) applies in this case. The measurement is. more straightforward even than or spur geer teeth because ‘the actual size Is not a maximum value; instead, as the measuring elements are swivalled in the axial plane the ‘actual size is a minimum to which the measuring pins auto matically adjust themselves in all positions owing to the parallel measuring element faces. Swivelling sideways does not alter the measured value (disregarding form deviations on the tooth flanks), see figure 27. This simply moves the measuring pins on the same helical path in the tooth spaces.Page 24 DIN 3960 2.7.4.2.3 Dimension over pins on external helical gear teeth with odd number of teeth Equation (2.108) applies to this calculation also, Because of the parallel measuring element faces the two measuring pins are forced on a helical path out of the position ‘occupied by the measuring balls in the dimension over balls (see figure 25 and figure 28) and into positions which are exactly opposite one another over the centre point of the gear tooth system, see figure 28 (this eliminates the factor cos 5*= in equation (2.109) ané equation (2.108) applies). The dimension over pins is therefore twice as large as the radial single-pin dimension. The helical motion of the pins is opposed so that the measuring points move apart axially as a function of the helix angle fy, and the angle 5". For this reason, the measuring element faces must be held parallel to the gear teeth axis, as is also Necessary with the dimension over balls (see subclause 27.4.1), The gear tooth system must have a certain minimum Width dyn, a8 must the measuring elements so that the pins do not protrude beyond the tooth ends, see figure 28. The necessary minimum facewidth can be found with figure 29. Assuming the same ball/pin diameter (Dyi ya = Dus pin) @ known dimension over balls can be converted to the dimension over pins by means of the following equation (2.110) 2.7.8 Centre distance for double-flank ‘engagement «’ with master gear The centre distance for double-flank engagement a” with master gear, see subclause 7.2, can be used as a tooth thickness check. For a gear with number of teeth z; and ‘2 master gear with number of teeth 21, and addendum modi- fication coefficent x, the value of @" can be found from (aan) According to equation (2.48) In which s, is replaced by the value (s, + A,), the addendum modification coefficient xj governing the double-tiank engagement is found from aie (2.413) I tana, Putting A, = 0, see equation (4.1), the nominal dimensions for <{, a” and a” are found: when the mean value Ayn is Used the mean values are found; see subclause 4.1.6. In this test the effects of radial run-out and tooth trace deviation (and of damage to the tooth flanks) come into effect also. The test thus detects all backlash-constricting effects deriving from imperfections in the gear. Reference should be made In this connection to VDI/VDE 2608 “Single-flank and double-flank working test on spur and helical gears with Involute profile” and also to DIN 3967. For meaning of symbols see figure 21 Figure 24, Dimension over balls My for an external spur gear with even number of teeth For meaning of symbols see figure 21 Figure 25. Dimension over bells My for an external spur ‘gear with odd number of teeth For meaning of symbols see figure 21 Figure 28. Dimension over balls Mg for an internal spur ‘gear with odd number of teeth €Figure 27. Dimension over balls My on helical external gear teeth with even number of teeth 2.7.6 Tip diameter dy, on overcut cylindrical gears ‘Spur gears of precision machanisms produced by generat ing methods with modules m, = 1mm are often overcut in order to avoid the difficulty of measuring tooth thickness when the module is small and to allow the tooth thickness to be determined instead from the measured values of the ‘ip diameter. This production method requires special gear: cutting tools 2 sin nbs ia Measuring widtn= Margin DIN 3960 Page 25 With overcutting the actual size of both the tip circle and the tooth thickness is produced by appropriate radial in- feed of the cutter. This infeed corresponds to a generating addendum madification coefficient xp which in the case ‘of negative tooth thickness deviations A, is smaller than the addendum modification x referred to the nominal di= mensions. Assuming the use of zero-deviation generating too! with rack profile (nab) the following applies areas Doin lah ey xpext ata) Inthis case the tip diametersalso are altered, The measured values dayy (actual values) of the tip diameters are thus smailer with negative tooth thickness deviations (with in. ternal gears the value of |dyyq| is then greater than the value |d,|) than the nominal dimensions according to equations (2.36) and (2.37). From day = d+ 2-1 mtg + 2+ hap (2.418) the actual deviation of the tooth thickness is found as As (dant ~ da) + tan ay 116) When @ pinion type cutter is used the dependence of the ‘generating pressure angle an the sum of the tooth numbers and the sum of the addendum modifications has to be allowed for, since these determine the centre distance of the generating gear pair and hence the generated gear tip diameter. The corresponding equations are contained in subclauses 3.2 and 3.3. They allow calculation of the work gear addendum modification deviating from the nominal value x and hence the work gear tooth thickness as a function of the tip diameter deviation. __ RE coy NN t Supporting surface WL H Diameter at contact points Figure 28, Dimension over balls Max and dimension over pins Mga on helical external gear teeth with ‘odd number of teethE Necessary minimum facewidth Number of teeth 2 Page 26 DIN 3960 0 mo 0 woe 0 10 20 Virtual tooth number Zyyyj——= 0 2 30 405060 60 10 200 200 7 TT 10! L In the lower part of the diagram the virtual tooth number 2, Is found from the number of gear teeth z and the hell angle £. Next, in the upper part of the diagram the corre sponding minimum facewidth Dain is read off as a function of # and the addendum modification coefficient x. For negative addendum modification coefficients less than 0,15 the horizontal line belonging to x = —0,15 must be used Figure 29. Nomogram for determining the minimum face Width by for the dimension over pins accord Ing to subclause 2.7.4.2 3 Concepts and parameters associated with a cylindrical gear pair All definitions in this clause reter to zero-back~ lash mating of zero-deviation cylindrical gears. The equations thus apply to the nominal dimensions of the gear pairs. 3.1 Cylindrical gear pair, dot 3.41.1. Extemal gear pair ‘The mating of two external cylindrical gears (external gears) 8 or of an external gear with a rack gives an external gear pair. | In the case of an external gear pair the subscript 11s used. ions in equations for the smaller gear (pinion) and the subseriat 2 for the larger gear (wheel). When the gears are of the same size the subscripts can be allocated as desired, Inthe case of an external gear pair with helical gear teeth cone gear has a left-handed, the othar one a right-handed flank direction, 94.2 Internal gear pair ‘The mating of an external cylindrical gear (external gear) with an internal cylindrical gear (internal gear) gives an internal gear pair. In the case of an internal gear pair the subscript 1is used In equations for the external gear and the subscript 2 for the internal gear, see also subclause 2.1 In the case of an internal gear palr with helical gear teeth both gears have the same flank direction: both are either right-handed or left-handed. 3.4.3 V-gear pair ‘A V-gear pair is the mating of two cylindrical gears such that the sum of their addendum modifications is not equal tozero, Bx= x, +a240, One of the cylindrical gears in this case can be a zero gear. The centre distance of 2 V-gear pair is not equal to the relerence centre distance. The reference circles are not simultaneously the pitch circes. 3.1.4 V-0 gear pair ‘A gear pair at referance centre distance is the mating of two V-gears such that the sum of their addendum modi- fications is equal to zero. Beem tae0; 2 aia Their centre distance is equal to the reference centre distance; their reference circles are simultaneously their pitch circles. 84.5 Zero gear p ‘A zero gear pair Is the mating of two zero gears, x0; Their centro distance Is equal to the reference centre distance; thelr reference circles are simultaneously their pitch circles. % 3.2 Mating quantities 3.2.1 Goar ratio u The gear ratio of a gear paris the ratio of the number of teeth of the whee! za to the number of teath of the pinion 2, ae iF (ay Inall cases |u| 1Note: In the case of an internal gear pair u is a negative value, see subclause 2.1, The transmission ratio / of a gear pair is the ratio of the angular speed (rotational speed) of the driving gear (sub- script a) to that of the driven gear (subscript b). (32) Inthe case of an external gear pair the two cylindrical gears rotate in opposite directions, Le. thelr angular speeds or rotational speeds have opposite signs; the transmission ratio is negative, In the case of an internal gear the two cylindrical gears have the same direction of rotation, ie. their anguler speeds or rotational speeds have the same sign and the transmission ratio is positive It it ls necessary to make a distinction, ratios such that |i] > Lare said to be “speed reducing ratios" whilst ratios such that [i] < L are sald to be ‘speed increasing ratios’ 3.23 Line of centres, contre distance a Ina transverse section the line of centres of a geer palris the straight line joining the gear centres of the cylindrical gears mated with each other. The centre distance ais the distance between the gear axes. Note: in the case of an internal gear pair the centre distance is teken as negative, see subclause 2.1, 3.2.4 Pitch cylinders, pitch circles; pitch diameter dy. In the case of a cylindrical gear pair the pitch cylinders (pitch circles) are those cylinders (circles) about the gear axis which have the same velocities. The pitch circles divide the centre distance in the ratio of the tooth numbers. The pitch clroles established during the operation of cylindrical gear pair (cylindrical gear pair ina gear unit) are termed working pitch circles. The pitch circles estab- lished by a ganerating cutter during the generating of @ tooth system are termed generating pitch circles. 1 Fs (dys + dye) = 6s en du” Ere bare ea 2-2, cose; duo 42 SOSay) * TOS Au 38) Note: Inthe case of an internal gear patr dy. isa negative value, see subclause 2.1 3.2.5 Working transverse pressure angle a, or 2” ‘The working transverse pressure angle ais that pressure angle whose vertex lies on the pitch circle, working pitch circle) itis calculated from DIN 3960 Page 27 dyn dha | (ert eal-m 08 ay = Get = SH rr) dwt diya 2a ia G ute vey sive #2 EE tena, 7 The values inv ay, and inv @ can be found from tables of the Involute function or according to equation (2.13) In the double-flank working test with a master gear the working transverse pressure angle is denoted by a”, see subolause 2.7.5 32.6 Working depth /iy The working depth /ty of @ gear pairs the distance apart of the tip circles of the two cylindrical gears on the line of Centres, see figure 20. day + daa 2a hy ee 32.7 Bottom clearance c, bottom clearance factor c* The bottom clearance ¢ is the distance by which the tip circle of @ gear Is separated from the root circle of its mating gear, see figure 30, It is equal to the difference botween the tooth depth and the working depth iy. cHhm hy my 9) Figure 30, Working depth ity and bottom clearance ¢ of a gear pair 3.3 Calculation quantities and factors for mating gears 3.3.1 Reference centre distance ay ‘The reference centre distance ais the sum of the reference circle radii of the two gears. ditds ates age SB, FL ta (21 + 23) I cosh re) Note: Inthe case of an internal gear pair ag is @ negative value, see subelause 2.1 3.3.2 Centre distance modification y- nny centre distance modification coefficient y ‘The algebraiccitference between the centredistance aand the reference centre distance ay Is the centre distance modification ym, which is expressed by the centre dis- tance modification coefficient y in fractions of the normal module. way 12)Page 28 DIN 3960 8.3.8 Centre distance modification coefficient yandsum of addendum modification coefficients Bx Between tha centre distance modification coefficient y and the sum of tha addendum modification coetficients Zs i+ 2 the following relationship exists be (2-1): amas eara 3.9.4 Calculation of the sum of the addendum modification coefficionts Ex the centre distance a, the numbers of teeth zy and 2, the normal module m,, the pressure angle a, and the helix angle fare known, then the working transverse pressure angle 2, arising in the mating gears is given by equation (6). In accordance with the previous equations of this subciause the sum of the addendum modifications is then found trom (21 + 22) (inv ey ~ inva) a. 2 tan ay a1) ‘The way in which 2x= x; +x) Is distributed between the two ‘gears is governed by the permissibio stressing of the teeth ‘or by othor specifiad dimensions of the gear toeth, 2.9. root, ameter, see DIN 3902 and DIN 3993. 3.3.5 Caloulation of the centre distance « It the normal module m,, the pressure angle ay, the helix angle f, the numbers of teeth 2; and zp and the addendum, modification coefficients x; and s of the two spur gears: are known, then the centre distance of the gear pair is, found as r (3.15) 3.8.8 Addendum alteration fin, addendum alteration factor Itis sometimes necessary for the addenda to be altered to suit the mating conditions and the specified minimum bot- tom clearance. If the bottom clearance ¢ corresponding to the standard basic rack tooth profile of the gears is to be retained thon the necessary addendum alteration f- ma, and the addendum alteration factor &, respectively, are found from Bee my a= ay ~ My Dt (3.18) kay 2x an) The addendum alterations calculated in this way are ob- tained with correct sign, that Is to say: negpative values in the case of external gear pairs, so that, ‘he tip diameters become smaller, positive values in the case of internal gear pairs, so thal the value of the tip dlameter of the internal gear becomes smaller whilst the diameter of the pinion tip circle becomes larger. The calculated values are often so small that they are cancelled out by the deeperinteed of the cutting tool which is necessary for producing the backlash and by the negative root diameter deviations additionally produced by standard pinion type cutters, so that the remaining effective bottom clearance is altorad to only a slight extent (or within per- misstble limits) In the case of internat gear pairs It Is to be noted that the ‘addendum alterations which are always positive in this case can usually not be realized because the special engage- ment and manufacturing conditions of internal gear pals limit the usable addenda of the internat gear and pinion r details see DIN 3993, 3.4 Tooth engagement For the investigation of tooth engagement the workingpitch circles and the corresponding transverse profile are the criteria 3.4.1 Point of contact Ina gear pairs tooth fank andits mating flank contact each ‘other on the contact line, see subclause 2.33. A point of contact of a fiank profile Is the point at which it makes contact with the mating flank In any particular working position, ‘At each contact point in the transverse section the normal to the point of contact (erected perpendicularly on the fiank and passing in all cases through the pitch point C) together with the tangent to the pitch circles In the pitch point makes the working transverse pressure angle dy In the case of involute gear teeth the normal to the point of contact and the path of contact coincide, so that the pres sure angle is the same for al iank points 8.4.2 Plane of action, tine of action The plane of action is the geometrical locus of all points of contact of an involute flank with the mating flank of the ‘mating gear. It contacts the base cylinder and hence Is parallel to the gear axis and is distant from it by the radius of the base cylinder. Each gear pair has two planes of action, one for the right flanks and one for the latt flanks, soe subclause 2.33. The two planes intersect in the instantaneous axis which is parallel to tho two goar axes and the angle enclased by the two planes is twice the working transverse pressure angle ‘The lines of action of an invotute gear tooth system are the intersections of the planes af contact with transverse sections. They are the geometrical locus of all possible points of contact of the flank profiles in the transverse section of a gear pair and at the same time the projection of all possible contact lines of the tooth flanks in the direc: tion of the instantaneous axis. In the transvarse section of the goar teath, a line of action inthe case ofexternalgearsisthe commoninternaltangent, and in the case of internal gears the common external tangent, to the basa circlos of the mating cylindrical geers, Tt contacts the two base circias at points 7; and Tz, see figure 31 and figure 22. In both figures pinion 1 is assumed to be tho driver, that is to say the ratio is spoed-reducing, Its necessary to distinguish between the right flank line of action and the left flank line of action, 9.4.2.1 Special points on the line of action Special points on the line of action are as follows (sae figura $1 and figura 32) A starting point of engagement. This is the point at which the line of action intersects the tip circle of the driven gear. 8 intemal individual point of contact on the driving gear, external individual point of contact on the driven gear This is the point on the line of action which Is distant by ‘one normal base pitch from point E © pitch point aD_ external individual point of contact on the driving gear, internal individual point of contact on the driven gear. This is the point on the line of action whichis distant by fone normal base pitch from point A. E finishing point of engagement This isthe point at which the ine of action intersects the tip circle of the driving gear. T; point of contact between the line of action andthe base circle of gear Tz point of contact between the line of action and the base Circle of gear 2 84.22 Curvature radii of the tooth flanks The following segments of the line of action give rise to the curvature radii of the tooth fianks which aro the cri- toria for the contact ratio and motion conditions of a gear pair (s00 figure 83 and figure 34) TE eer = pV ab = ah dy «tan ayy 3.18) 2 fda- i (3.19) (8.20) Bur = BEI ~ Pe (321) TaD = pr = Ga2~ Pec (3.22) Ger + aca = a> sin ay 7 oa + nz = Oe + O12 (3.23) Equations (3.19) to (3.22) apply to the case when gear 1 is the driving gear and gear2 the driven gear. In theopposite case A and E as well as B and D are to be interchanged in figures 31 to 84 and in equations (9.18) to (3.22) Note: The values obtained for the curvature radii of an internal gear and for the segment T;T of an internal gear pair are negative, see subclauses 2.1 and 2.5.7. 3.4.3 Zone of action, length of path of contact The zone of action (length of path of contact in the trans: verse section) is that part of the plane of action (line of action in the transverse section) which comes into use uring the operation of gear pair. in the case of speed: reducing ratios the length of the path of contact is termi nated at its starting point A by the tip circle of gear 2 and at its finishing point E by the tip circle of gear 1, see figure 31 to figure 34. In the extreme case the length of the path of contact may extend as fer as the base circles. If the tip circle of the ‘generating cutter cuts the line of ection outside the points Tyor Ta the result is undercut and the length of the path cof contact is shortened accordingly. I the tip circle of the mating gear cuts the line of action outside the segment TT inthe case of external gear teeth, or inside the segment TT, in the case of internal gear teeth, then meshing dificulties will arse, DIN3960 Page 29 The length gq af the path of contact of two mating cylindri- cai gaars is given by: enh [Vana Be Tea] (dys + dio) tana] (3.28) Note: 22,d,o and dys are negative inthe case of an internal ‘gear, see subciause 2.1 ‘The length of the path of contact when a cylindrical gear (subscript 1) is mated with @ rack (standard basic rack tooth profile according to DIN 867) Is given by L(V dy tn) Be my > (= 1) sing ‘The length of the path of contact gq is divided by the pitch point C into the length af approach path gr (portion of the length of path of contact at the root flank of the driving ‘gear between the tip circle of the driven gear and the pitch point) and the length of recess path gy (portion of the length of path of contact at the tip flank of the driving gear between the pitch point and the tip circle of the driving gear, see figure 31 and figure 32. These portions of the length of path of contact are also termed the tip length of the path of contact and root length of the path of contact, of the gear, For the case where gear 1 Is the driving gear end gear 2 the driven gear, the following applies: The length of approach path is equal to the length of dedendum path of contact of gear 1 which |s equal to the langth of addendum path of contact of gear 2 (3.25) IC = par - 0c? “LB, a Wa Tada tanen) — 29 The length of recess path is equal to the length of addendum path of contact of gear 1 which is equal to the fength of dedendum path of contact of gear 2 a= CE= 001 ~ ocr L(V oh (VEE duane) goan For the opposite case (gear 2 driving, gear { driven} g. and ‘acs well as the subscripts A and E for the curvature radii are to be interchanged in equations (8.26) and (3.27) (see subclause 3.4.2.2) Note: Inthe case of an internal gear the number of teeth, the diameters and the curvature redil are negative values, ee subclauses 2.1 and 2.67. Note: The length of the path of contact can be shortened by tooth tip chamfering or radiusing or by tip relief. In such ‘a case the usable tip diameter dai or daa obtained from the production specifications is 10 be substituted for day and dya, respectively, in equations (3.49) and (3.20) 2s weil as in (8.24) to (3.27), see subclause 3.4.4. 3.44 Usable diameter dy; active and usable ranges of the tooth flanks The usable circles enclose the active ranges of the tooth flanks which are effective when the gears are operating, see figure 31 and figure 32. The starting point A of the pathPage 30 DIN 3960 Range of at active flank es of gear | | Range of ! active flank 2 Range of active flank of gear t Range of active flank of gear 2 Figure 31. Length of path of contact and active flank Figure $2, Length of path of contact and active flank ranges of an external gear pair with speed-re- range of an internal gear pair with speed-re- ducing ratio ducing ratio Figure 33. Curvature radii g of tooth flanks and sliding ‘Figure S4, Curvature radii @ of tooth fanks and sliding speed v, at point of contact Y of an external speed 0, at point of contact Y of an internal ‘gear palt with speed-reducing ratio ‘gear pair with speed-reducing ratioof contact determines the usable root circle of gear 1 with the usable root diameter dyn, whilst the finishing point & of the path of contact determines the usable root circle of gear 2 with the usable root diameter diz. The usable tip diameters dyai and dnea are governed by the production specifications of the gear and in general are practically equal to the tip diameters day and d,a, respectively. For ‘example, with tooth tip radiusing Qyq in the normal section the following apples dyyy™ dy ~ 2 -Qan* (1 ~ sin Gan) 3.28) Hence for a gaar pair due “ve acsinay ~ 25 VG wa di) + a (328) fam 7, VA (2-0 sinew Saka ar) +e (30 Note: Foran inforel gor palrthe conte cstance a and the diameters of the internal gear are negative, see sub- clause 2.1, For special cases arising with internal gear pairs see DIN 3993. During the production of a gear tooth system by the generating method tha usable circles corresponding to the transverse pressure angle at generation and the cutter, tip diameter come into action, These usable circles enclose the usable ranges of tooth flanks. During tooth production on an external gear with a rack cutter or a hobbing cutter without protuberance (adden: ‘dum fo, tip rounding radius Qo), the usable root diameter dys an the generated cylindrical gear sas followsif under- cutting does not occur: dye = Va sina —2- Pray ~ 86° Mn ~ Qua" (1 -Sin ay) /sin +45 sp ‘The addendum modification factor x: at generation must be determined taking into account the tooth thickness devietion A, and a grinding of shaving allowance, if re= quired. In case of an ellowance of this kind dy 1s the diameter usable curing preliminary treatment, During shaping the usebie diameter dr is found accord ing to equation (8.80), the corresponding quantities of he pinion type cutter with subscript O replacing the quantities with subscript 1 Furthermore, the relationship specified in DIN 383 must be observed in the case of Intornal geers ‘The transverse angle of transmission g of a gear ie the centre angle through which it rotates with its mating profile from start to finish of engagement of a fiank profile, see figure 85, The transverse angle of transmission of pinion and gears given as follows: bs at = = 2 (382) a= Fe = Set (3.39) DIN 3860 Page 31 ‘The transverse contact ratio ¢qis the ratio of the transverse angle of transmission @y to the angular pitch r or the ratio of the length of path of contact to the transverse normal base pitch. Pat, Pa Be me st @38) Figure 35. Transverse angle of transmission #y of an @x- ternal gear pair 3.4.6 Overlap angle #, overlap ratio °) The overlap angle gp is the angle between the two axial planes enclosing the end goints of a tooth trace, see figure 96, betanig! 2-b- sini gl i 7 Tigre 7H Mp2 (895) betanigl — 2-b-sini Al op ye PMOL 2 mip $B (a26) ‘The overlap ratio ey Is the ratio of the overiap angle p to the angular piteh 7 or the ratio of the facewidth b to the axial pitch py. age 20. Hb , besini pl te ee tin (37)Page 32. DIN 3960 Figure 38 Overlap angle gp of @ cylindrical gear 34.7 Overlap length zp ‘The overlap length gy of a hlical goaris the reference circle axe belonging to the overlap angle nt ype B-tan|f| 338) 3.4.8 Total angle of transmission , total contact ratio ¢, ‘The total angle of transmission gy is the angle at the centre of a gear through which it rotates from start to finish of contact of one of its flanks with the flank mated with it, It is equal to the sum of the transverse angle of transmission and the overlap angle. O11 = Bett 961 = 2 (239) (40) 2 = Pua * mp2 ‘The total contact ratio ey Is the ratio of the total angle of transmission to the angular pitch It is equal to the sum of the transverse contact ratio and the overlap ratio. (341) 3 conditions atthe toth anks 251 Slang speed Aa point of contact of no cynical gears iongagement the tcing soot pte cfrence tthe asses ofthe two tansaae pres proveing inte decton othe commen tangent Atte pln of contact se re 29 ad ure 2, the yo tanevere profes hve thenonmaapeeday=2- oy dy Bacause of ie sary of he speed rangle wi he ir angie ortyand'VOqts he sng speeds ound non [notion wih he curva rely a yum eguaon Gravee notes (2 -on) a ‘According to figure 89 and figure 34 the distance guy be tween ¥ and Cis Ba = hence ay-tarty: (1+) a (ex ~ Qy1) = F (ex ~ 02) (aaa Note: guy always counts as positive. Since w Is positive for an external gear pair and negative for an internal gear pair it usually follows that the sliding speed 1s greater for external gear teeth than for internal gear teeth. The sliding speed is proportional to the distance guy and at the pitch point is equal to zoro. It reaches its maximum values at the point of root er tip contact, ogeter a (144) (248) ‘gr and g, being found according to equations (3.26) and S27, Raper, 8.5.2. Sliding factor Ky ‘he sliding factor K; is the ratio of the sliding speed 1, to the velocity % of the pitch circles Pe _ 2* Boy ( 1) Rea ay Mbt ean The maximum values for K, are attained at the end points Aand E of the path of contact ata ky= 2 (142) (249) we: ky 2. (142) 249) Note: In the case of an internal gear pair u is taken as negative, see subclause 3.2.1. ‘The variation of the sliding factor along the line of action Is shown in figure 97 for an external gear pair and in figure 38 for an internal gear pair, taking u = 2 as the example. For judging addendum modifications the lines Ky: and Kye are continued beyond points A and E. 3.5.8 Specific sliding ¢ The specific sliding (is the ratio of the sliding speed to the speed of a transverse profil in the direction of the tangent to the profile, see figure 83 and figure $4, Equation (8.42) yields gei- 3% (350) an- Sa esi ‘The maximum values of (are reached at the end points ‘and E of the path of contact aA Gat Gt (ss) ate: Gq = 1-288 (3.8) using the curvature radii 94 and gg according to sub- clause 3.4.22 Note: In the case of an Internal gear pair w is taken a2 negative, s00 subclause 3.2.1 In the variation of the specific siding along the path of contact is shown in figure 87 for an external gear palr and in figure 38 for an internal gear pair using the exemple of u= +2. For Judging addendum modifications the curves of specific sliding are continued beyond points A and E.DIN 9960 Pege 33 a ine of action 4 Deviations and tolerances for cylindrical gear teeth For the mating of two gears itis necessary to have devi- ations from the zeroplay condition in order to achieve the necessary backlash, The minimum doviations depend on the backlash-reducing effects of the gear teeth and of the housing and as well on the operating conditions and the influence exerted by the design, whilst the maximum de- viations depend adaitionally on the gearcutting tolerances. These influences ere mostly independent of the helix angle. Therefore the deviations and tolerances on tooth thickness, (ust) siney Figure 37. Sliding factor K, and specific siid- ing (as & function of the normal base pitch for an external gearpalr with u= +2 lgure 38. Stiding factor K, and specific slid- ing (as a function of the normal base pitch for an internal gear pair with u=—2 ‘and its test dimensions are referred to the normal section (Gee DIN 3967), likewise the change factors. All definitions and equations in this clause hold good far zero-deviation gears, 4.1 Deviations A of tooth thickness and its test dimensions 4.1 Deviations of tooth thickness A, To achieve backlash the teeth of external goars and of, internal gears must have nagative tooth thickness devi ations. The tooth thickness deviation A, is the differencePage 34 DIN 3960 between an actual size and the nominal size of the normal tooth thickness s, on the referance cylinder. The largest Permissible tooth thickness of a gear is given by the upper tooth thickness A,. and the smallest permissible tooth thicknass by the lower tooth thickness deviation A, In the case of cylindrical gears with addendum modification ‘measurements near the middle ofthe tooth depth (ie. near the V-oylinder) are to be preferred to measurements on the reference cylinder. The tooth thickness deviations on a Y-oylinder are found from tae may (142 iw)! 4) the last part ofthis equation being applicable to the tooth thickness deviations on the V-cylinder (dy = dy). For calculations in the transverso section the tooth thick- ness deviations A, have to be converted into the corre- sponding transverse saction deviations Ay: ~ wa a2 For calculating the deviations according to subolause 4.1.3 10 4.1,7 itis advisable to calculate first with the average (Ag+ Ag) and then to tooth thickness deviation Asm 2 determine the limits with the aid of the change factors according to clause 5 4.1.2 Deviations A; of the normal chordal tooth thicknesses The deviation Az is the difference between an actual size and the nominal size of the normal chordal tooth thick- fess Sy, For the normal chordal tooth thicknesses §,, at diameter d, the deviations alter accordingly, see Sub- clause 4.1.1 The deviations A; and Az are generally only insignificantly differant from the deviations 4, and Aq, respectively, of the tooth thicknesses. They oan be equated with them except for very small numbers of teeth, see subclause 6.1 4.1.8 Base tangent length deviations Ay The base tangent length deviation Ay is the differance between an actual size and the nominal size of the base tangent length Wi, In the case of external gears negative deviations make the actual size of the base tangent length smaller then the nominal size for zero-backlash engage- ‘ment, whilst in the case of internal gears negative devi- ations make the absolute value of the actual size larger than the absolute value of the nominal siz. The upper base tangent length deviation is denoted by Aye and the lower by Ay. {tis convenient first to calculate the mean base tangent length according to equation (2.96) using for x the mean generating addendum modification coefficient xm which is obtained from equatién (2.114) with the mean tooth thickness deviation ym. Then by using the change factor according to subelause 5.2 the limits are found, 4.14 Deviations Ay of dimension over balls or pins ‘The deviation Ayg of the dimension over balls or pins My of tooth thickness is the difference between an actual size and the nominal size of the dimension over balls or pins. In the case of external gears negative deviations make the ‘actual size smaller than the nominal size for zero-backlash engagement, whilst in the case of internal gears negative deviations make the absolute value of the actual size larger than the absolute value of the nominal size, The permissible maximum value is denoted by Ange and the permissible ‘minimum value by Ay. The relationship between an actual size of the dimension ‘ver balls or pins and the corresponding tooth thickness deviation Is found from equations (2.104) and (2.108) or (2.108) on substituting in equation (2.103) the value GTA | Ra4extanay AS ye find ROA e taney As a 22 eM for the spacowidth half angle 7, Itis convenient to calculate first the mean dimenston over balls or pins according to equation (2.108) or (2.108), the value of 7 for equation (2.103) being calculated according to equation (4.8) using the mean tooth thickness deviation Asuy (from equation (4.2) with yy). The limits are then found with the aid of the change factors according to subclause 53. 43) 44.8 Deviations Ay, of radial single-ball or single-pin dimension For the deviations Ayj- the definitions corresponding to subclause 4.1.4 apply. 1 cose A, “Snax: * Cos B GE) 4.4.6 Deviations A,» of centre distance for double-flank engagement a” with master gear Daviation A,- of the centre distance a” for double-flank engagement with a master gear is the difference between ‘an actual size and the nominal size of the centre distance 4a’, see subciause 2.75. If in equation (2.113) the upper tooth thickness deviation A,. is substituted first, and fol lowing this the lower tooth thickness deviation Ay), then from equations (2.111) and (2.112) the permissible limits for the double-flank engagement a" and hence the upper deviation Avs and the lower deviation Aye, can be found It is convenient first to calculate the mean centre distance for double-flank engagement according to equation (2.110, the mean value a’, being calculated according to equation (2.112) using the mean working addendum modification Coefficient xy (from equation (2.113) with Ayq). The limits are then found with the aid of the change factor according to subclause 5.5. Aur= 4.1.7 Tip diameter deviation Au, with overcut cylindrical gears ‘The tip diameter deviation Ag, with overout cylindrical gears is the difference between the actual size (measured value dy) and the nominal size of the tip diameter, see subclause 2.7.6. ‘The upper tip diameter deviation is denoted by Agus and the lower deviation by Aya). The permissible limits are found according to equations (2.114) and (2.115) by substituting the upper or lower deviations A,. or Ay respectively. it Is convenient first to calculate the mean tip diameter ‘according to equation (2.115), for which purpose the value substituted for xz is the mean generating addendum modi fication coefficient which Is found from equation (2.114) Using the moan tooth thickness deviation Asn. The limits are then found with the aid of the change factor according to subslause 56, 42 Tolerances T ‘The tolerance is the difference between the permissible maximum value (upper deviation) and the permissible mini- ‘mum value (lower deviation) of a quantity. The deviations ‘Aang A, referred to in this subclause denote the permis sible deviations, Corresponding to the quantities considered in sub- Clause 4.1, the tolerances ere distinguished according to €subclauses 4.2.1 to 4.2.7 and for this purpose the appro- priate quantities are to be inserted in the general definition ‘The range of tooth thickness errors and the deviations of, other gear quantitios (0.9. radial run-out) affect the various test dimensions for tooth thicknoss differently. Hence the conversion of a tooth thickness tolerance according to subclauses 4.1.3 to 4.1.7 into the corresponding tolerance fon a test dimension for tooth thickness does not auto- maticelly yield the correct values. Instead, additional cor- rection values arising from the characteristic features of the various measuring methods have to be considered. For details see DIN 3967, 4.2.4 Tooth thickness tolerance 7. r, As 4s) 4.2.2 Tolerance 7’; on normal chordal tooth thickness Ae 40) 4.23 Base tangent length tolerance Ty Tw = Awe ~ Aw an 4.24 Tolerance Tiyy on dimension over balls oF pins mute ~ Ani aa) Ty 4.2.8 Tolerance Ty, on radial single-ball or single-pin dimension Thr = Ants ~ Ave 69) 4.2.6 Tolerance Ty on centre distance for double-flank engagement with master gear Ty = Ave ~ Ae (4.10) 4.2.7 Tip diameter tolerance Tis with overcut cylinerical gears Tay = Ada ~ Aasi aan) 5 Change factors A* For use in gear production it has been found expedient to calculate the ratios between the deviations As, Aw, Aya Of Ay, respectively, Ay-and Ag, on the one hand, and the tooth thickness deviation A, on the other and to define them notionally as change factors, With such change factors itis easy and convenient to compare also the deviation of Sq. Wi Mg Of Mr and a with one another. The change factors are referred to the normal section because the deviations and tolerances also are defined for the normal section, see DIN 3867, ‘The deviation A mentioned in this subclause denote the permissible deviations. 8A Change factor Afof normal chordal tooth thickness The change factor A¥ for the normal chordal tooth thick- ness §, existing on the V-cylinder is given by Avy Ay y+ €05 By a cose where fis to be calculated according to equation (2.42), a, according to equation (2.43) and yy according to equation (2.55). Note: The change factor Ag, is only significant for small numbers of teeth and is normally put equal to 1 (costs —sin tam ex ¥a) DIN 3960 Page 95 5.2 Change factor Aly of base tangent tength The change factor Aiy of the base tangent tangth Wis 62) 5.3 Change factor Ai, of dimension over balls or pins The change factor Aj of the diametal dimension Ma is iven by the following for even numbers of te8th and for Sad numbers of teeth in the case of external helical gears thon measuring with pins Ana, “Fiivayy OnF Ait 183) and for odd numbers of teeth (for helical gear teeth how- ever only when measuring with balls) by Tina, cop CS Z-z 4) This approximation formula becomes more and more in- accurate as the deviation becomes larger. For small devi- ations of the gear teeth from the zero-play condition how- ‘ever it sufficiently exact. Where relativaly large deviations fre conoarnes it is appropriate to calculate the change faotor from a tooth thickness corresponding to the mean deviation of the tolerance zone. 5.4. Change factor Aj, of radial single-ball or single-pin dimension ‘The change factor Aij,f the radial single-ball or single-pin dimension Mishel the size of the change factor according to equation (5.3) for both even and odd numbers of teeth. 5.5 Change factor A‘: of centre distance for double-flank engagement a’ with master gear The change factor Aj of the centre distance for double- flank engagement a” with mester gear is, Ay 1 08 a A, 2° “sing” cose This approximation formula becomes more and more in accurate as the deviation becomes larger. For small devi- ations of the gear teeth from the zero-play condition how= everitissutticiently exact, Where relatively large deviations are concerned it is appropriate to calculate the change factor from a tooth thickness corresponding to the mean deviation of the tolerance zone, The angle ais the working transverse pressure angle when engaging with the master ‘gear, see subclause 3.25. Ay 65 5.6 Change factor Aj, for tip diameter of overcut cylindrical gears ‘The change factor for overout gears made by a zero-devir ation and zero addendum modification cutter (e.g.2 hob) is, Aip= Ae OD) in GE ~ cot en Ct Inthe case of cylindrical geers of the overcut type produced with a pinion type cutter itis generally not possible to convert with the constant change factor according to equation (66), With gear teeth of this kind it is necessary to take account of the total number of teeth and the sum of the addendum modification coefficients of the generating gear pair, see subclause 276.Page 36 DIN 3860 6 Deviations of individual parameters of cylindrical gear teeth The definitions of this clause are concerned with the devi- ations of individual paramsters of cylindrical gear teeth Which are referred to the zero-deviation reference cylinder, see subclause 2.27, Individual deviations fare those deviations trom their nomi- nal values which are exhibited by the various parameters of the gear teeth, such as pitch, profile shape, base diam- eter, pressure angle, tooth traces and helix angle. A total deviation F contains the joint effects of several individual deviations The permissible maximum values of the individual dev- ations, see DIN 3961 to DIN 3964,are denoted whereneces- sary by the additional subscript ,per* For general information on deviations, see DIN 7182 Part. 6.1 Circular pitch deviations Circular pitch deviations, or pitch deviations in brief, are measured on the reference circle or on a measuring circle as close as possible to the reference circle and concentric with the gear axis, The difference between the measuring circle diameter dy: and the reference diameter d affects the measured vaiues of the deviations by the factor dyld; generally speaking these effects are negligibly small ‘The measured values are also influenced by the effects of any eccentricity of the gear teeth and by profile deviations. For a graphical respresentation of circular pitch deviations see figure 38. 6.1.1 Adjacent pitch errors / ‘An agjacent pitch error fy is the difference between the actual size ofan individusl ransverse pitch and the nominsl siz0 2. On a gear having z teeth there are 2 adjacent pitch errors of the right flanks and equally as many of the left flanks. ‘The deviations fp arise as differences between the indi- vidual measured values end the mean of all z measured values 6.1.2 Cumulative circular pitch errors Fx ‘A cumulative circular pitch arror F,, is the deviation of the actual size of a pitch span over # individual pitches from Its nominal size. It's also found — provided the measuring errors aro sufficiently small ~ as the algebraic sum of the Headjacent pitch errors contained in the span Foes 2 fy (6.1) . If the deviations of all pitch spans are measured from a aiven reference tooth flank as the starting point, or are calculated from the adjacent pitch errors fy according to equation (6.1) and then plotted versus the corresponding tooth numbers, the result obtained are the cumulative circular pitch errors according to figure 39, ‘The values plotted in figure 39b represent the position deviations of the tooth flanks on the reference circle in ralation to the notional zero-deviation circular pitch with z uniformly distributed pitch points one of which coincides with the refarence tooth flank. From the series of these values all z individual pitch deviations and all cumulative circular pitch errors can be found 2s differences. 6.1.8 Cumulative circular piteh error Fy./s ‘A cumulative circular pitch error Fp zg is the cumulative Circular pitch error existing over a sector of (approximately) ot the gear periphery (it~ 2/8) 6.1.4 Cumulative pitch error Fy The maximum cumulative circular pitch error on the gear 's termed the cumulative pitch error Fp. Itis stated without sign and Is found from the cumulative pitch errors as the difference between the algebraically largest and the alge- breically smallest value, Tapapetsfettatatota fatale epepa efatatn | ) adjacent pitch errors fy shown as vertical blocks be- eet yrange of pitch evors in iference betwoon ajacent itches (inthis case Between pitches Nos 20 and Zon ark No 20) ») cumulative circular pth errors refered to enk No 2, Shown asa stoppoetine cagram F, cumulalve pth ever 6) eumultive crutr pitch errors over a sector of = 3 Indvvelptoheseoch tne, shown as vorteal Backs n mig-pan 4) cummlatie pltch-span deviations calculated from the Span cevations of igure rolerre to flank No 21 show es astopped-lnedagram with coresponana iege step wat) Figure 39. Graphical representation of circular pitch de- viations (example: z= 21) ¢6.4.5 Range of pitch errors Ry ‘The range of pitch errors Rp Is the altference between the largest and the smallest Sctual size of the transverse pitehes p. of the right or eft flanks of a gear. ‘Tho range of piten errors R, can be determined directly from the measured values ofa circular pitch moasurement without knowledge of the adjacent pitch errors fy. 6.1.6 Difference between adjacent pitches /, A difference between adjacent pitches fy Is the difference between the actual sizes of two successive transverse pitches of the right or left flanks of a gear Differences between adjacent pitches are found direct {tom circular pitch measurements as the differences of the measured values of paits of adjacent pitches. 6.2 Normal base pitch errors fy. ‘A normal base pitch error fy is the dlference between the actual size and the nominal size of a normal base pitch p. Errors of the transverse base pitches are denoted by [yt ANG OxrOrs Of the normal Base pitches by fn Tho results of the base pitch measurements are inde- pendent of any eccentricity of the gear tooth. Baso pitch Seviations may derive both from regularity o the clreular pith as wall as trom form and position deviations of tho two tooth fianks, As a result of irregular profile or tooth trace configuration they mey differ at afferent points on the same tooth flanks For connection between base pitch deviations and trans verse pressure angle deviation ar base circle daviation, 886 subclause 63.16. 6.3. Flank deviations Flank deviations are the deviations existing within the flank test area of the gear by which tho tooth flanks depart from the involute helicoids of the nominal base cylinder when taking into account the desired deviations (eg, crowning). Flank deviations on a cylindrical gear can be determined 2) ina plane of transverse section (deviations ofthe trans: verse profile, or af the profile for short) ) on the reference cylinder orsomeother coaxial cylinder (deviations of the tooth trace) ©) in a plane tangential to the base cylinder (deviations of the generator), The records of flank deviations obtained with the aid of fiank testing machines are the flank test diagrams. In the test diagram produced by the majority of flank testing machinas the nominal involute, the nominal tooth trace and the nominal generator appear as straight lines, see figure 40, From the flank test diagram (test diagram curve) umerical values of the actual deviations on the ger are derived by the use of relationships indicated below. The flank test range generally comprises the range of the usable flank (height and width of the flank). Alternatively, it may be specially agreed for acceptance test purposes on gears. 6.3.1 Deviations of transverse profile It Is only in transverse sections that involute gear teeth exhibit Involutes of thelr base circle. Deviations of the tooth profiles rom theirnominal profileare therefore meas- tured only in transverse sections. The nominal profiles are formed by involutes of the nominal base circle with allow- DIN 9960 Page 37 ance made for intended devietions trom the involute form (eg. profile modification) For this measurement it is usual to employ measuring machines which guide the stylus according to the Involute, ‘generating law during the measurement and rece the tooth flank along @ transverse profile. When the direction of stylus motion ig at right angles to the flank the measured values must be converted to deviations in the transverse 1 section by multiplying by eq Note: Inthe test diagram traced by the measuring machine the measured values are read off at right angles to the direc tion of chart feed, that is 10 say in the case of atest diagram according to figure 40 in the direction at right angles to the line AA which corresponds to the recording of enominal, inyolute, For evaluation of the test diagram the equalizing line BE Is drawn as the actual involute. This is calculated if necessary by the "method of the least sum of the squares Of the errors". An inclined position relative to the line AA indieates the presence of a deviation from the nominal base diameter or from the nominal pressure angle. ‘The profile test range must generally be taken as the length ‘of path of contact when the gear Is meshed with its mating {gear (or if the latter is unknown, with the rack). The profile {ost range Lg is conveniently defined by stating the working length L;for the root and Ly for the tip or by tha correspond Ing working angles é; and ¢, ‘The following ere distinguished: Total profile error F; Profile form error fy Profile angle error few Profile waviness fr 6.3.1 Total profile deviation F; The total profile error F of a tooth flank is the distance between the two nominal profiles which enclose and touch the tooth flanks within the profile test range. Note: ina test diagram according to figure 40 the total profile error F;s the distance measured at right angles to the direction of chart feed between the line AA and the further ling N'A’ parallel to it which are drawn through the ‘outermost points of the test pattern inthe direction of chart feed within the profile test range. The desired deviations {rom the involute form are allowed for by appropriate devi- tions of the lines AA and AA’ from straightness. 634.2 Profile form error fr The profile form error fy of @ tooth flank Is the distance between the two involutes of the base circle which ~llow- Ing for desired deviations from the involute form — enclose and touch the actual profile within the profile test range, ‘The profile form error also comprises the wave depth of 3¢ profile waviness, see subclause 63.1.5 Note: Ina test diagram according to figure 40 the profile orm error [ys the distance measured at right angles to the direction of chart feed between the parallel ines B'S! and B°B" parallel to the averaging actual involute BB, which lines contact the test diagram within the profile test range. 63.1.9 Profile angle error fire The profile angle error fxg is the distance between the two nominal profiles which cut the involutes of the actual base circle at the starting and finishing points, respectively, of the profile test range. The profile angle error yg is usuallyPage 38 DIN 3960 Figure 40. Flank deviations Tost diagram and survey of deviations =—— Direction of chart feed Test range L Profile Tooth trace Gonorator © | Total profile error Total alignment error Fy Total gonerant error Fe @ | Protite angle error fre Tooth alignment error fxg Generant angle error fe @ _| Protite torm error fr Longitudinal form error fr Generant form error fe Test range | Profile test range Le Tooth trace test range Ly Generator test range Le 85 | Averaging actual imolute | Averaging actual tooth trace Averaging actual generator AKA | Nominal profiles Nominal tooth traces Nominal generator which envelope the actual flank 86'.5°5" | Actual involutes ‘Actual helices ‘Actual generator Which envelope the actual flank Nominal protlies Nominal tooth traces Nominal generator hich cut the actual generators or tooth traces atthe starting and finishing point, respectively, of the test range stated in terms of jm as a linear dimension assigned to the profile tast range L», see DIN 9961. The profile angle error fug is deemed to be postive if the involute of the actual base circle rises to the material-frea side, compared with the nominal profile, in the direction of increasing Working length; itis deemed to be negative If the involute of the actual base circle falls towards the material side in the direction of increasing working length, A profile angle error is caused by a deviation of the pressure angle or by eccentricity of the base circie of the gear teeth (non-coincidence of gear-cutting axis and axis of rotation) which has the same effect on the individual tooth Note: In atest diagram according to figure 40 the profile angle error fg is the distance measured at right angles to the direction of chart feed between the linas C’C’ and C°C” drawn parallel to the line AA which cut the line BB at the starting and finishing points of the profile test range. 6.8.1.4 Pressure angle error fe, base circle error fy The profile angle error fa can if necessary be converted to the corresponding base circle error fy (dference be- tween the actual base diameter and the nominal base di: ameter) or to the corresponding pressure angle error fe (difference between the actual pressure angle and the nominal pressure engle), fonds: fe 62 fite feo Tana ona If fu Is Inserted in pm and Le in mm, then fy is found in lum, ja In mrad (see subciause 13). 6.3.1.5 Profile waviness fiw A profile deviation repeating cyciically with the working angle is denoted as the profile waviness fy. Its character- ized by the wave-dopth and wave-length. 68.1.6 Connection between pressure angle error fy base circle error fy and base pitch ertOF fye The actual size of the pressure angle on a tooth flank cannot be measured directly on a cylindrical gear, but in~ stead can only be calculated after determining the actual base diameter, see equations (2.7) and (6.2), In the case of spur gears the actual pressure angle a can be calculated from the measured value pera of the normal base pitch p. by means of the following expression Pew (64) Note: This equation only applies exactly Ifthe gear teeth are free from pitch and flank deviations. ifthe value inserted for pey Is the average taken from a sufficient number of ‘measurements distributed over the gear periphery then the corresponding average will be obtained for a. For an (2.9. estimated) uncertainty Ap of the pay value the uncertainty of the ais Aa EApe/(m» m +£0,01mm and m sin a). For example if, for ‘3mm the uncertainiy is In the eace of spur gears the following relationship existe between the pressure angle error fq and the normal base pitch error fye tem > aoane (68)It Jpg Is Inserted in um and the module m in mm, then fe Is found in mrad (see subclause 1.3) In the case of helical gears itis usual to measure pen: for this purpose it is necessary to replace a by ay 2 BY tin, Pew BY Pevin 84 Foe BY Fyan In the above equations. Simiary the following relationship exlstsbetweenthe base Circe error fy and the normal piton e170" [ye dy Ios 68) to ree ‘tek ‘The equations hold good exactly for helical gear teeth if the transverse section values are used, otherwise only approximately. If the calculations made with equations (62) to (6.6) are to relate not only te a single flank but instead to several flanks or all the flanks of a geer, then the corresponding average values (2.9. over 4 or all flanks of the gear) must be used. When the average is formed over the entire periphery of the gear the influence of eccentricity of the tooth system iselminated. Otnerwise the eccentricity ean be determined by the values of fy on the two (opposite) flanks having the algebraically largest or smallest value of fie & fe 6.3.2 Deviations of tooth traces A tooth trace is the intersection of the tooth flank with & cylinder, usually the reference cylinder, see subclause 22.6. Deviations of the tooth traces from their nominal form ere measured In successive transverse sections tangentially to the base cylinder. The nominal tooth traces are formed by helices (of the raferance cylinder) with the nominal lead, allowance being made for desired variations (e.9.crowning), The measurement is made by the use of machines which simulate a nominal holix (with infinitely large lead in the case of spur gear teeth) and which cause the stylus to explore the tooth flank along a toath tace, Note: In the dlagram drawn by the test machine the ‘measured values are read off at right angles to the direction of chart feed; hence in a diagram according to figure 40, at right angles to the straight line A which corresponds to, the plot of & nominal tooth trace. For evaluation of the diagram the equalizing line BB Is drawn in as an averaging actual tooth trace, if necessary by using the “method of the least sum of the squares of the errors”, An inclined position relative (0 the line AA indicates the presence of a deviation from the nominal lead p, or from the nominal helix angle f. itis usual to specify the entire facewidth as the tooth trace: test range Ly, ‘The following are distinguished: Total alignment error Ps Longitudinal form ertor fr Tooth alignment error Fup LLongitualinal form waviness fy 6.3.2.1 Total alignment error Fy ‘The total alignment error Fp of @ tooth flank Is the distance between the two nominal tooth traces which enclose and touch the fianks within the tooth trace test range. Note: inthe test diagram according to figure 40 the total ‘alignment error Fy is the distance measured at right angles to the direction of chart feed between the line AA and the DIN $960 Page 39 line AA’ parallel to it which are drawn In the direction of chart feed within the tooth tace test range through the ‘outermost points of the diagram. Desired deviations from the helical form are allowed for by appropriate deviations Of the lines AA and A's’ from straightness. 6.3.2.2 Longitudinal form error fy Thelongitudinal form error fyofatooth lankisthe distance between the two helices with the actual lead which — with allowance made for desired devietions trom the helical form ~ enelose and touch the actual tooth trace within the tooth trace test range. The longitudinal form error jy also includes the wave-depth ofthe longitudinal form waviness, see subclause 63.2.6 Note: Ina test diagram according to figure 40 the longi tudinal form erro: fy is the distance measured atrightangles to the direction of chart feed between the lines BB! and B°B' which are drawn parallel to the averaging actual tooth trace BB and which contact the test diagram within tl wave test range. 6.3.2.9 Tooth alignment error fp ‘The tooth alignment error fypis the distance ina transverse plane between the two nominal tooth traces whieh inter sect the helix witn actual lead at the starting and finishing points, respectively, of the tooth trace test range. The tooth alignment error frig is usually stated in um as a linear dimension assigned to the tooth trace tast range 1, see DIN 3861, Forall the definitions and calculations concerning the gear pair a tooth alignment error in the case of an external gear is termed positive if its helix is right-handed relative to 8 helix with the nominal lead; itis termed negative if its helix is left-handed relative to a helix with the nominal lead. In thecase of an intarnal gear the opposite signs apply. Hence, when cylindrical gears ara mated deviations of equal ‘magnitude but opposite sign cancel out Note: In contrast to this in gear production and for meas- urement techniques tooth alignment errors which result in @ higher value (absolute value) of the actual helix angle compared with the nominal helix angle aro termed positive (Fig positive) €00th alignment errors which result in @ smaller value of the actual helix angle are termed nogative (Fri neyative) 0 the case of spur gear teeth tooth alignment errors are always termed positive, the letters "t" (1.8, "in the sense of a right-hand helit*) or *t" (ie. "in the sonso of a leftthand helix") are added indicating the direction of the deviation, Two different definitions for the signs of the tooth align- ment errors are found from the two preceding paragraphs, Therefore the active direction of the deviations must be unambiguously specified in all the production documents, test certificates etc A tooth alignment error is caused by a deviation of the helix angle or by non-parallelism of the gear-cutting axis and axis of rotation (wobble) which manifests itself on the tcoth individually in the same way as the former. Note: Ina test diagram according to figure 40 the tooth alignment error fy Is the distance measured at right angles ta the direction of chart feed between the lines C'C’ and CC" which run parallel 10 the line AA end intersect the line BB at the end of the tooth trace test range.Page 40 DIN 3960 6.3.24 Helix angle error fp ‘tooth alignment error fg can if necessary be converted into the corresponding helix angle error fs (diference be- tween the actual hell angle and the nominal helix angle). Approximately it is fy hn ie costp en It fupis inserted in umm and Zy in mm, then sis found in mrad (se0 subcieuse 13). 6.9.2.5 Axial pitch error fpy, lead error fyx ‘A nelix angle erzor /p can be convertedinto the correspond: ing exiel pitch error fx and into the corresponding lead error fox Py sinl Ble &¢ i) fox =~ fp also Py fo fe Garp eae te) If pcand p, are inserted in mm and /p in mrad, then fy, and Jou re found in um. In the case of spur gearing conversion i$ pointless since py = = and p= =. 6.3.2.6 Longitudinal form waviness fow Longitudinal form waviness Is a longitudinal form error repeating cyclically over the facewidth. Itis characterized by the wave-depth and wave-length. 6.3.8 Deviations of the generator ‘A generator Is the line of intersection of the tooth flank with a plane developed from the base cylinder, see sub- clause 2.3.1. Deviations of the generator from its nominal form are measured in successive transverse sections tangentially to the base cylinder. The nominal generators are straight lines or lines containing straight lines and Intended deviations from straightness. The measurement is made by machines which quide the stylus in accordance with the law of generation of the involute helicold in a plane tangential to the base cylinder ‘and parallel to the direction of stylus motion, and explore fa tooth flank along a line of contact. Note: In the dlagram drawn by the testing machine the measured values are read off atright angles tothe direction of chart feed; hence in a test dlagram according to figure 40, at right angles to the straight line AA which corresponds to the plot of a nominal generator. For evaluation of tha test diagram the equalizing line BB is drawn in as.an averaging actual generator, if necessary by using the “method of the least sum of the squares of the errors". An inclined position relative to the line AA indicates the presence of a deviation from the base hellx angle 6, or from the base diameter di, itis usual to specity the length ofthe longest line of contact {8 the generator tast range Lz. On a helical tooth fiank the lines of contact are of unequal length, see figure 41 6.83.1 Total generant error Fe ‘The total ganerant error Fe Is the distance between the two nominal generators which enclose and touch the tooth flanks within the generator test range. \ “Base cylinder — Contact lines on a helical tooth flank Figure 41 ‘The following are distinguished: Total generant error Fe, Generant form error fer Generant angle error firs: Note: Ina test dlagram according to figure 40 the total generant error Fi, is the distance measured at right angles to the direction of chart feed between the ling AA and the line A'R’paraliel tol, which are drawn through the outermost points of the test diagram within the generator test range In the direction of chart feed. Intended variations from the generator form are allowed for by suitable deviations the lines AA and NA’ from straightness, 6.3.3.2 Generant form error fri The generant form error fry Is the distance between the two generators having the actual lead which — taking account of intended deviations from straightness~ enclose and touch the actual generator within the generator test range, Note: Inatest diagram according to figure 40 the generant form error fer is the distance measured at right angles to the direction of chart feed between the two lines BIB’ and BB" drawn parallel (0 the averaging actual generator BB which contact the test diagram within the generator test range. 63.33 Generant angle error fur ‘The generant angle error fixe is the distance between the two nominal generators which cut the averaging actual ‘generator at the starting and finishing points, respectively, of the generator test range. It Is usually stated in um as a linear dimension allocated to the generator test range Lp. Note: Inatestdlagram according to figure 40 the generant angle error fue is the distance measured at right angles to the direction of chart foed between the lines C'C' and C'C" drawn parallel to the line AA which cut the line BB at the starting and finishing points, respectively, of the generator test range. 6.4 Radial run-out 6.4.1 Radial run-out F, of a tooth system The radial run-out F, of a tooth system is the difference In radial position of a moasuring element (ball or cylinder fr V-head) which is Inserted successively into all tooth spaces and which contacts the tooth flanks near the V- circle, the gear being rotatably mounted for this purpose in tg guidance axis. F, denotes the largost difference be- ‘wean the measured values occurring at the gear periphery, see also subclause 6.6.3.A radial run-out Fis caused mainly by eccentricity of the ‘gear teeth in the measuring plane at right angies to tha ‘gear axis and by irregularity of the spacewidth resulting {rom pitch deviations of the right and lett flanks, 6.4.2 Radial run-out of tip circle f, Inthe case of overcut cylindrical gears (see subclause 27.6) 2 radial run-out of the tooth system may be determined bby measuring the radial run-out of the tip circle f.. This is found by moving the gear as the largest difference of position of a pressure foot pin radially movable and being in contact with the tip cylinder. 6.5 Position deviation of gearcutting axis, A position deviation of the gearcutting axis exists if the axis of the reference cylinder of tne gear teeth (gearcutting axis) and the guidance axis of the gear fal to coincide when measurements are being made or in the operating con dition (e.g. a8 a result of a position deviation during the ‘clamping of the gear blank for cutting).In the general case fof auch a position deviation the two axes intersect; their position relative to one another is characterized by the distance between the two axas in the middle of the face- width (eccentricity of the gear tooth) and by the angle of Intersection (cutting wobble). With, generally speaking, adequately good approximation a position deviation of the gearcutting axis can be deter- mined by measuring the radial run-out Fy and Fin two transverse planes (near the front V and the rear A of the ‘gear with separation 1). Form the course of the radial run~ ‘ut the averaging curves and their fundamental are deter mined for each periphery of the gear. The double ampli- tudes of the fundamantal are denoted by 2 fy or 2 fue and the centre angle between the maximum values of the fundamental by gi 6.5.1 Eccentricity /y ‘The eccentricity fof the gear tosthis equate the stance between the gearcutting axis ard the guidance axis of the earin the middie ofthe facowidth It's found as i Aa . te 30 V + n+ 2 fav fan 08 9 (6.10) 8.5.2 Wobble Fy The wobble Fy of the gear teeth is equal to the angle of intersection between the gearcutting axisand the guidance axis of the gear, itis found as fey few cose (6.11) When ge = 0 and fey # fey these two axes intersect at a point which ie usualy situated along way outside the gear body, whilst with = ¥ they intersect within the gear body. With. O and fey fer the axes are parallel io oneanother. With + Oand g, x the axes are skewed relative to one another (they cross one another) 66 Range of errors R A range of errors (span) R Is the difference between the largestand smallest measured value of measured quantities of like kind within a series of measurements. On a gear tooth system ranges of errors are found as a result of deviations in quantities measured at different points on the ‘gear teeth (e.g. for pitches see subclause 6.1.5) DIN 9960 Page 41 In the different measuring methods the ranges of tooth thickness errors and the deviations in other quantities of the gear teeth (8.9, radial run-out) manifest themselves differently. The ranges of errors of different measured quantities can therefore not be readlly converted one to another, e.g. by means of change factors. For allocating ranges of tooth thickness errors to the ranges of errors of another measured quantity it Is necessary to take into account the special features of the measuring method sed, see DIN 3967. (For range of pitch errors Ry see subclause 6.1.5.) 6.6.1 Range of tooth thickness errors R., range of normal chordal tooth thickness errors Rr The range of tooth thickness errors R, is the difference between the largast and smallest tooth thickness s of a gear. RS Sous ~ Su (6.12) ‘The corresponding range of normal chordal tooth thickness, errors Rz can generally be equated with the range of tooth, thickness errors R,. 6.6.2 Range of base tangent length errors Rw ‘The range of base tangent length errors Ry isthe difference between the largest and amaliest base tangent length Wy of a gear. Ry Wa onas — Wa is (6.13) 6.6.3 Range of errors Ru; for the radial single-ball or single-pin dimension The difference between the largest and smallest measured value of the radial single-ball or single-pin dimension My of a gearis equal to the radial run-out F,, see subclause 6.4 86.4 Range of errors Ry for the dimension over balls or pins ‘ne range of errors Ry is the difference between the largest and smallest measured value ol the dimension over balls or pins fg of a gear. Rua = Manan ~ Ma win (era) 6.6.5 Range of errors for the centre distance for double-flank engagement ‘The difference belween the largest and smallest measured value of the centre distance for double-flank engagement @ on a gear is equal to the radial composite error Fi, see subclause 7.2.1 67 Contact pattern ‘As.a result of gearcutting deviations, deviations in the gear, positions and operating factors a tooth ‘lank will not be contacted by the mating flanks in all points of its active range during the meshing action. The contact pattern denotes that area of a tooth flank in which contact with the mating flanks takes place, 7 Composite and cumulative errors In the working test gear teeth are mated with counter gear teeth and the combined effects of their individuel geo- metrical deviations (Individual errors) on the working action are determined as composite and cumulative errors. These can be assigned to one of the gears (the test gear) if the gear used as the mating gear is 8 master gear with devi- ations which are negligibly small compared with the devi- ations of the test gear. It Is common to use master gears with e quality at least three grades higher than the specified quality of the gear under test.Page 42 DIN 9980 It the deviations of the mating gear are not negligibly small, (@.g. when the working test is performed with two trans mission gears) the composite and cumulative errors can only be assigned jointly to the gear pair. See VDUVDE Code of practice 2608 “Single-flank and, double-flank working test on spur and helical gears with, involute profile’ AT ASS f LASS. ig ZT L } sana co, I coo Reterence measuring system Centre distance « set to a fixed value, ‘Single-tlank engagement of the right flanks or left flanks through torque constraint. Tangential composite errors = relative deviations of angle of rotation compared with the corresponding zero-devi- ation angular settings brought about by a reference meas- uring system, Figure 42. Measuring arrangement for the single-flank working test 1 revolution of test gear saa - 7A Single-flank working test Inthe single-flank working test two gears are meshed with fone another at the prescribed centre distance so that either the right flanks or the lef flanks remain in continuous engagement (singie-fiank engagement), see figure 42. The tangential opposite error of the right flanks of a gear are ‘generally different from those of the left flanks of the seme. ‘gear. From a given starting position the deviations ocourring in the angle of rotation, that is to sey the deviations in the ‘angular positions of the gear compared with the theoretical, positions governed by the positions of the mating gear and the ratio of the numbers of teeth, are measured. This re- uires the use of a reference measuring system (reference ‘gear unit) in which the zero-deviation angular positions (theoretical positions) are realized, The deviations are generally stated as a distance along the circumference ofa test circle, e.g the reference circle or the base circle; alternatively, they may also be stated in angular measure (eg in seconds or in milliradians or microradians). TAA. Tangential composite error Fj ‘The tangential composite error Fyis the range of errors of the actual angular positions compared with the theoretical angular positions. itis found as the difference (sum of the ‘absolute amounts) of the largest leading and the largest lagging angular position deviation compared with a given starting position within one revolution of the test gear, ‘800 figure 43, Ina circular test diagram the tangential composite error Fi Is the difference between the largest and the smaliest, ‘stance of he recorded test trace from the axis of rotation of the test diagram chart. In the case of a strip type test oS8B5 8858 1 [2s 26 2 ze a 6 IT 2 = —— Direction of chart foes 10 9 8 7 6 5 4 ToothNo i]25 ; tangential composite error F,, cumulative working pitch error Figure 43. Strip type single-flank working diagrams of a gear fi. tangential tooth-to-tooth composite error fp Individual working errordiagram the tangential composite error Fs the difference between the largest and the smallest ordinate of the test trace. 7.1.2. Cumulative working pitch error F, ‘The cumulative working pitch error Fis the long-undulation component in the test diagram of the tangential composite errors. This component can be obtained trom the test diggram produced during the single-tlank working test by drawing in an “averaging line” with which the short-undu- lation components are suppressed. Tne “averaging line” has an essentially sinuscidal characteristic of one cycie per gear periphery, see figure 43, The cumulative working pitch error Fy is the differance (determined radially or in the ordinate direction) between the highest and the lowast point of tne averaging line. 7.1.8 Individual working orror /, Individual working errors are the short-undulation com ponents in the lest diagram of the tangential composite terrors; they result trom the differences between the re- Gorded test trace and the "averaging line” eccording to subclause 7.1.2. The periodicity of the short-undulation components per gear periphery agrees in meny cases with the number of teeth of the test gear. These components ‘may however also contain the effecis of undulations inthe profile or longitudinal form errors ‘The Individuat working error fy Is the difference between the highest and the lowest point in the trace of the short- undulation components, see figure 43. The cumulative working pitch error F, and the individual working error together yield approximately the tangential composite error F 7.44 Tangential tooth-to-tooth compos The tangential tooth-to-tooth composite error fis the largest difference which occurs In the angular position deviations within an angle of rotation corresponding to the duration of a tooth engagement, see figure 43. 7.1.8 Tranamission deviation of a multi-stage gear mating The transmission deviation of a multi-stage gear mating is, the angular position deviation of the shaft at one end of the {gear train from its theoretical position of the shaft at the other end of the gear train and the ratio of the numbers, Of teeth. It is referred to the shaft at one end of the gear train. it contains the tangential composite errors of the individual gear pairs, the effects of load in the different stages of the gearing and the effects of inaccuracies of mounting, and it Is composed of two components. One component derives from daviationsin the parameters from their theoretical values whilst the other results from elastic, deformations of the transmission elements and thelr bearings. 7.2 Double-flank working test In the double-flank working test two gears are meshed with each other with zero backlash and at the same timea force acting coaxially with the line of centres is applied to keep a loft flank and a right flank of the gears simultaneously in engagement (double-flank engagement), seo figure 44 The alterations in centre distance occurring under these conditions are meesured, The contre distance established in the double-flank working test is denoted by a” and the working transverse pressure angle by a’, see also subclause 2.75. DIN 3960 Page 43, Test load = ‘The centre distance aie established through the action of 2 load acting coaxially with the line of centres and varies with the rotation of gear and mating gear. Double-flank engagement, Radial composite errors = Centre distance elterations. Figure 44. Test arrangement for double-flank working test FL radial composite error FE working radial run-out {radial tooth-to-tooth composite error a) strip chart b) circular chart Figure 48, Double-flank working diagrams 7.24 Radial composite error Fi The radial composite error Fis the range of errors of the working centre distance a’, in other words tisthe difference between the largest and the smallest working centre dis- tance within a single revolution of the test gear, see figure 45. 7.2.2 Working radial run-out F ‘The working radial run-out Fis the long-undulation com- ponent in the test diagram of the radial composite errors. ‘This component is obtained from the test diagram yielded by the double-fiank working test by drawing in the “aver- aging line” whereby the short-undulation components (the Periodicity of which per gear periphery usually agrees with the number of teeth of the test gear) are suppressed, see figure 45. The working radial run-out Fris then the distance, between the highest and the lowest point of the “averaging line’. 7.2.3 Radial tooth-to-tooth composite error ff The radial tooth-to-tooth composite error ff is the largest difference in the working centre distance which occurs within an angle of rotation corresponding to the duration of one tooth engagement, see figure 45,Page 44 DIN 9960 8 Deviations of the axial po: of a cylindrical gear pair ‘The axes of @ cylindrical gear pair may exhibit variations from parallelism and a deviation from the nominal centre distance. These quantities ere stated as variations or as a deviation of the axial position of gear 2 relative to the reference gear axis 1. When the two cylindrical gears are ‘mounted in the housing different variations or a different deviation may occur compared with those obtained when mounted in a test machine ions 8.1 Deviations from parallelism The deviations of the axial positions of a cylindrical gear palr trom parallelism are stated in the following manner, see figure 46. Gear axis 1 (reference axis) and point Or, which bisects the distance L. between the bearing centres of gear axis 2 determine the plane of centres |. The perpen- dicular dropped from O, to axis 1 lies in the plane of centres | and meets axis 1 at point 0). The plane Il which also passes through point O2isat right angles to plane ang parallel to gear axis 1, Planes | and Il intersect one another in the straight line 1-1" which is parallel to gear axis 1 In the event of deviations from parallelism goar axis 2 (hich alse passes through Oz) makes an angle with the parallel line 1-1. By projecting gear axis 2 on to planes | ‘and Il this angle can be resolved into two components; the two components are generally stated as portions of lines referred to the distance between the bearing centres, 0° the permissible deviations in DIN 3964 tet gear axis 1 2-2 gear axis 2 Oz point of bisection of gear axis 2 ©; meeting point of perpenciuiar rom O wth gear 1 plane of centras of geer axes " Plane perpendicular to plane of centres and par= Biel to 1-1 Yt! line of Intersection of planes | and I 2-2' projection of 2-2 onto plane | 2!-2" projection of 2-2 onto plane Il Figure 48. Deviations of axial positions from parallelism 8.1.1 Inclination error of axes fos The inclination errar of axes is the deviation of gear axis 2 from parallelism to gear axis 1 in the plane of centres | In plane | of figure 46 fz is the vertical distance to the line of section 1-1’ of tha boaring centres of gear axis 2 pro- Jected on to the plane of centres. 81.2 Deviation error of axes fap The deviation error of axes isthe deviation of gear axis 2 {rom parallelism to gear axis in the plane Ii perpendicular to the plane of centres. In piane It in igure 48 fap the vertical distance to the line of section 1-1 of the bearing centres of gear axis 2 pro- Jected on to plane I 82 Deviations and tolerance for position of gear axes 8.24 Centre distance deviations A, The centre distance deviation A, is the difference be- tween the actual centre distance of the gear pair and the nominal size To avoid unduly large deviations of the axial positions from the nominal sizee In the caso of multi-stage gear pairings, and to enable the same production specifications to be used for both external gear pairs and internal gear pairs, the permissible limiting values of the centre distance are specified symmetrically to the nominal size, so that the permissible centre distance deviations have the sign 2. enveloping right parallelepiped double-cuneiform tolerance space determined by Ta and fog C-C-C tolerance restricted from above and below by ics Figure 47. Tolerance space for position of gear axis 2 in relation to gear axis 1 ¢ ¢8.2.2 Centre distance tolerances 7, ‘The centre distance tolerance T,{s the difference between the upper centre distance deviation Aye and the lower centre distance deviation Ay Ty = Ae ~ Ani en 8.2.3 Tolerance space for position of gear axes ‘The tolerance space for gear axis 2, which is aligned with gear axis 1 (reference axis) and is not referred to the housing, is given by the positions of gear axis 2 relative to gear axis 1 which are possible es a result varlations or deviations. The tolerance space Is enclosed by a right, parallelepiped the iongitudinal axis 1-1’ of which is paralle to gear axis 1 and is distant from it by the nominal centre distance a, see figure 47. The right parallelepiped has length Lo, height 7, in the plane of centras | (see figure 46), land width fo in plane Il (see figure 46). in figure 47 the right parallelepiped Is denoted by the points A Form the centre distance tolerance T, and the permissible deviation error of axes fap therefore a tolerance space according to figure 47 is obtained which has the shape of a double wedge (denoted by points A and 8} the knife: edges B-8 of which face one anather and are coincident with the perpendicular 0,02 (see figure 46). Through the ‘existing (as maximum: permissible) inclination error of axes fzs the tolerance space is additionally restricted, In the vertical direction wedge-shaped elements are lost at the ends of the tolerance space, thase having in each case the height fz; and an end face of the double wedge, whilst at the opposite end face of the double wedge they finish at a knife-edge. This tolerance epace thus has the {form of a bevelled double wedge the knife-edgos of which have the height T, ~ fsa and the end faces the height T, —2 - fox, This tolerance space is denoted in figure 47, by the points C. Itis the minimum tolerance space for full utlization of the permissible inclination error of axes. With smaller inclination error of axes the tolerance space is correspondingly larger; with /z,~= 0 itis equal tothe double wedge ABA. The tolerance spaces of further gear axes are to be formed similarly. Their planes of centres contain the reference axis and lie within their angular tolerance. 10 Alphabetical index DIN 3960 Page 45 9 Backlash j On zero-deviation cylindrical gears the backiesh is the dis tance, stated in a given direction, between the non-working flanks of the teeth of a gear pair when their working flanks are in contact. The size of the backlash depends on the deviations of the tooth thicknesses of the two gear tooth systems and of the centre distance, as well as on deviations of profile form, tooth trace, pitch and concentricity and on temperature differences between the gear teeth and the housing, and possibly also on effects resulting from swelling af the materials used and elastic detarmation. See DIN 3967 9.4 Circumferential backlash /, The circumferential backlash j; is the length of the pitch circle are theough which each of the two gears can be rotated, whilst the other is held stationary, from the point where the right flanks are in contaet to the point where the left flanks are in contact, Its magnitude is represented in the transverse section. 9.2 Normal backlash j, The normal backlash j, Is the shortest distance between thenon-working flanks of the teeth of agear pairwhen their working flanks are in contact. Jn fr €08 ay 08 f i c0S.a 0S hy — (0.1) 9.3 Radial backlash j, The radial backlash jeis the difference inthe centre distance betwaen the working condition and that of zera-backlash engagement. in 2 tam aus ce 9.4 Range of errors for backlash R, The range of errors for backlash Rj (circumferential range of errors for backlash Rj, normal range of errors for back= lash Rijn) Is the difference between the largest and the smallest backlash occurring in the case of a gear pair in mounted state. The figures following the terms in the alphabetical index are the clause numbers. Active range of tooth flanks 3.4.4 ‘Addendum 2.8.7.2 ‘Addendum alteration 3.36 ‘Addendum alteration factor 3.36 ‘Addendum modification 2.5.4 ‘Addendum modification coefficient 2.5.4 Adjacent pitch errors 6.1.1 Angles at V-cylinder 2.5.6, 2.5.64 Approach path 3.4.3 Axial module 2.26 Axial piten 2.4.7, Axial pitch error 63.2.5 Axial profile 2.2.4.3 Axial section 2.23.3 Backlash 9 Base circle 22.8 Base circle error 63.1.4, 63.1.6 Base cylinder 2.28 Base cylinder normal pitch 24.5.2 Base cylinder tooth trace 2.2.5 Base diameter 22.8 Base helix angle 23.2 Base load angle 2.3.2 Base spacewicth 258.3 Base spacewicth halt angle 2.584 Bese tangent length 27.2 Base tangent length deviations 4.1.3, Base tangent length tolerance 4.2.3 Base tooth thickness 25.8.1 Base tooth thickness half angle 2.58.2 Basic module 2.26 Basic rack 22.1 Bottom clearance 32.7 Bottom clearance factor 3.2.7 Contre distance 3.23, 3.35 Centre distance deviation 8.2.1 Centre distance in the double-flank working test 7.2 Centre distance for double-flank engagement 2.7.5, Centre distance modification 33.2 Centre distance modification coefficient 3.3.2, 33.3Page 46 DIN 3960 Contre distance tolerance 82.2 Change factors 5 Circular pitch deviations 6.4 Circumferential backlash 8.1 Composite and cumulative errors 7 Constant chord 2.7.1.2 Contact line 2.3.3, Contact lina overlap 2.7.2 Contact pattern 67 Cumulative circular pitch errors 6.1.2, 6.1.3, Cumulative pitch error 6.1.4 Cumulative working pitch error 7.1.2 Curvature radii of tooth flanks 3.4.2.2 Cylindrical gear pair 8, 3.4 Cylindrical gear teeth 2 Datum lines 2.2, 2.2.2 Dedendum 2.5.7.2 Deviation error of axes 8.1.2 Deviations tor positions of gear axis 82 Deviations trom parallelism 8.1 Deviations of axial postions & Deviations of centre distance for double-flank engagement with master gear 4.1.6 Deviations of cylindrical gear teeth 4 Deviations of dimension over balls 4.1.4 Deviations of dimension over pins 4.1.4 Deviations of generator 63.3, Deviations of individual parameters 6 Deviations of normal chordal tooth thickness 4.1.2 Deviations of radial single-ball dimension 4.1.5 Deviations of radial single-pin dimension 4.1.5 Deviations of tooth thicknass 4.1, 4.1.1 Deviations of tooth traces 63.2 Deviations of transverse profile 63.1 Diameter of gear teeth 25.6 Diametral test dimensions for tooth thickness 2.7.4 Difference batwoen adjacent pitches 6.1.6 Dimension over balls 2.7.4.1 Dimension over pins 2.7.4.2 Distance between bearing centres of a gear axis 8.1.1, 81.2823 Double-flank engagement 7.2 Double-flank working test 7.2 Eccentricity 65.1 End point of engagement 3.4.2.1 External gear 2.1 External gear pair 3.1.1 Facowidth 2, 2.7.2, 346,347 Flank deviations 6.3 Flank direction 2.5.2 Flank profile 2.2.1, 224 Flank test range 6.3 Goar ratio 3.2.1 Gear teeth haights 2.5.7 Gear tooth profile 2.2.4 Gonerant angle error 63.33, Gonerant form error 6.3.3.2 Generating addendum modification coofticient 2.7.6 Generating pitch ciroles 3.2.4 Generator 23.1 Generator test range 63.3, Height above chord 27.1.1 Holices 2.25 Holix angle 2.6.1, 2.563 Helix angle error 6.3.2.4 Inclination error of axes 8.1.1 Individual deviations 6 Individual point of contact 3.4.2.1 individual working error 7.1.3 Internal gear 2.1 Internal gear pair 3.1.2 Involute function 238 {nvolute helicoid 23 Involute surface 23 Lead 299 Lead angle 2.6.1 Lead error 63.2.5 Left tlanks 28.2 Left tooth trace 22.6 Length of path of contact 34.3, Lenath of recess path 3.4.3, Line of action 3.4.2 Line of centres 3.2.3 Longitudinal form error 63.2.2 Longitudinal form waviness 6.3.26 Mating quantities 32 Measured number of spaces 2.7.3.1 Measuring ball diameter 2.7.3.1 Measuring circle diameter 2.7.2, 6.1 Measuring element width 2.7.2 Measuring with master gear 2.7.5 Module 226 Nominal dimensions 2 Normal backlash 2.2 Normal base pitch errors 6.2, 63.1.8 Normal base pitches 2.4.6, 2.4.6.2 Normal chordal tooth thickness 27.1 Normal module 22.6 Normal pitch 24.2.2, 243.2, 244.2 Normal pressure angle 2.3.5 Normal pressure angle at a point 23.5, 25.63 Normal profile 22.4.2 Normal section 22.3.2 Normal spacewidth 2.5.86 Normal tooth thickness 2.5.8.5 Number of teeth 2.1 Number of teeth spanned 2.7.2 Overcut cylindrical gears 27.6, 4.1.7, 427,58 Overlap angle 3.4.5, 3.4.6 Overiap length 3.4.7, Overiap ratio 3.4.5, 3.4.6 Pitch cylinder 3.2.4 Pitch diameter 3.2.4 Pitches 2.4 Pitch point 3.4.2.1 Pitch span 24.2.3 Plane of action 233, 8.4.2 Plane of centres 81 Pointing limit 2.6.2, 263 Point of contact 8.4.1 Position devietion of gaarcutting axis 6.5 Pressure angle 23.4, 23.5, 256.4 Pressure angle at @ point on circle through centre of ball 27.3.4 Pressure angle at a point on measuring circle 2.7.3.1 Pressure angle error 63.1.4, 88.1.6 Profile angle error 6.3.1.8 Profile form error 6.3.1.2 Profile height 22.2 Profile test range 6.3.1 Profile waviness 6.3.1.5,Radial backlash 93 Radial composite error 7.2.1 Radial run-out 64 Radial single-bell dimension 27.3.1 Radial single-pin dimension 2.7.3.2 Radial test dimensions for tooth thickness 27.3 Radial tooth-to-tooth composite error 7.2.3, Radlus of curvature of involute 23.7 Range of errors for base tangent length 6.6.2 Range of errors for centre distance for double-flank en- ‘gagement 685 Range of errors for dimension over balls 6.8.4 Range of ercors for dimension over pins 6.8.4 Range of errors 88 Range of errors for backlash 9.4 Range of errors for feasible involute gear teeth 2.64 Range of narmal chordal tooth thickness errors 6.6.1 Range of pitch errors 6.1.5 Range of errors for radial single-ball dimension 863 ange of errors for radial single-pin dimension 663 Range of tooth thickness errors 6.6.1 Reference centre distance 3.3.1 Reference circle 22.7 Reference cylinder 22.7 Reference cylinder tooth trace 2.2.5 Reference diameter 2.2.7 Reference surfaces 2.2 Reference tooth trace 2.2.5 Flight flanks 23.2 Right tooth trace 22.6 Rolling angle of involute 2.36 Root circle 2.5.6.2 Root cylinder 2.5.62. Root diameter 2.5.6.2 Root length of path of contact 34.3 Root tooth trace 22.5 Sections through aylindrical gear teeth 2.23 Sign 21,253, 254 Single-flank engagement 7.1 Single-flank working test 7.1 Sliding factor 3.5.2 Sliding speed 3.5.1 Spacewidth half angle 2.6.84 Spacewidths 258,258 Span 66 Special parts on the line of action 3.4.2.1 Specific siding 85:3 Standard basic rack tooth profile 22.1, 22.2 Starting point of engagement 3.4.2.1 ‘Sum of addendum modification coefficients 333, 334 Sum of reference clrcie pitches 24.2.3 Tangential composite error 7.1.4 ‘Tangential tooth-to-tooth composite error 7.1.4 Test dimensions for tooth thickness 2.7 Tip circle 25.6.1 Tip circle radial run-out 6.4.2 Tip comer rounding radius 2.6.1, 3.4.4 Tip cylinder 2.5.8.1 Tip diameter 2.5.6.1, 27.6 Tip diameter deviation 4.1.7 Tip diameter tolerance 4.2.7 Tip length of path of contact 349 Tip tooth trace 2.2.5, DIN 3960 Page 47 Tolerance for positions of gear axis 82 Tolerance of canire distance for double-fank engagement with master gear 4.2.6 Tolerance on dimension over balls 4.24 Toleranea on dimension over pins 42.4 Tolerance an radial single-ball dimension 4.25 Tolerance on radial single-pin dimension 4.2.5 Tolerance on normal chordal tooth thickness 42.2 Tolerances for oylindrical gear teeth 4 Tolerance space for axial positions 823 Tooth alignment error 63.2.3 Tooth depth 2.57.1,3.26 Tooth engagement 3.4 Tooth thickness angle 258.2 Tooth thicknesses 2.58 Tooth thickness half angle 2582 Tooth thicknass tolerance 4.21 Tooth traces 2.2.5 Tooth trace test range 63.2 Total alignment error 63.2.1 Total angle of transmission 24.8 Total contact ratio 3.4.8 Total deviation 6 Total gonerant error 6.3.3.1 Total profilo deviation 6.3.1.1 Transmission ratio 8.2.2 Tranamission deviation 7.1.5 Transverse base pitch 24.5.1 Transverse module 2.2.6 Transverse normal base pitch 24.6.1 Transverse pitch 2.4.2.1, 2.4.3.1, 244.1 Transverse pressure angle 23.4 ‘Transverse pressure angle at a point 23.4, 256.4 Tranaverse profile 2.2.4.1 Tranavarse section 2.23.4 ‘Transverse tooth thicknesses 25.8.1 Undercut 2.6.1 Usabla diameter 3.4.4 Usable range of tooth flanks 9.4.4 Usable root diameter 34.4 Usable root diameter at generated wheel 3.44 Usable tip diameter 3.44 Vecircle diameter 25.63 V-cylinder 25.63 V-gear 255 Vegear pair 3.1.3 Virtual number of teeth 2.6.4, 27.2 Virtual referenca circle diameter 2.7.1.1 Virtual spur gear tooth system 27.1.1 Virtual tooth thickness halt angle 2.7.1.1 V-0 gear pair 3.1.4 Wobble 65.2 Working centre distance 7.2 Working depth 3.2.6 Working length 28.7, 63.1 Working pitch circles 32.4 Working radial run-out 7.2.2 Working transverse pressure angle $2.5, 34.1 Zero gear 255 Zero gear pair 3.1.5 Zone of action 34:3,Page 48. DIN 3960 Further standards and documents Supplement 1 to DIN. 3960 Concepts and parameters for cylindrical gears and cylindrical gear pai classification of the equations DIN 9962 Part 1 Tolerances for cylindrical gear teeth; tolerances for deviations of individual parameters DIN. 3962 Fart 2 Tolerances for cylindrical gear teeth; tolerances for tooth trace deviations DIN 3962 Part Tolerances for cylindrical gear teeth; tolerances for pitch-span deviations DIN. 3963 DIN. 3964 DIN. 3977 on cylindrical gears DIN. 3998 Part 1 Tolerances for cylindrical gear teeth; tolerances far working deviations Deviations of shaft centre distances and shaft position tolerances of casings for cylindrical gears (at present at draft stage) Diameter of measuring element for test dimension of the tooth thickness Designations on gears and gear palrs; general concepts DIN 3998 Part 2 Designations on gears and gear pairs; cylindrical gears and gear pairs Supplement 1 to DIN. 3998 Designations on gears and gear pairs; alphabetical index of equivalent terms DIN 8000 Parameters and errors of hobs for involute cylindrical gears; fundamental concepts DIN 58405 Part 1 Cylindrical gear drives for fine machanics; scope, classification WDE/VOI 2605 Circular pitches and plane angles: basic concepts for anguler dimensions, angle measurements, angle standards and their errors VDIVDE 2608 —_Single-flank and double-flank working test on spur and helical gears with involute profile voi 3336 Production of cylindrical gear teeth with involute profiie, cutting methods Explanations This standard contains terminological principlos for manu- facturers and users of geer transmissions and for other sstandarés on tooth eystems. Compared with DIN 960, August 1960 edition, concepts. and parameters have been fully revised and new concepts, required by the state of the art have been included. Certain obsolete terms such as “limiting number of teeth” and, “shaft angle error", have been omitted. The symbols and designations have been brought into line with the basic, standards DIN 3998 and DIN 3999 and have thus at the same time been made conform to international practice. ‘Compared with DIN 9980, October 1976 edition, some printer's errors and mistakes (e.g.in the ception of figure 17, in figure 95 and in subclause 4.1.6) have been corrected and desirable supploments added; in subclause 63.23 reference was made to the use of the symbols of tooth, alignment errors used in gear measuring technique ditfer- ing from the specifications in this standard. Subciause 25.6.3 has been divided into 2.5.68 and 2.5.6.4, subclause 9.4 has been newly added. In subclause 2.7.3.1 the virtual tooth number zn determined empirically has been intro- duced which yields better approximation values for Dya with a large f than zyy in the case of hellx gear teeth; the corresponding equations have been simplified. They hold good exactly for spur tooth systems and approximately “for helix tooth systems, Compared with the October 1976 edition the number and order of the equations have been changed because eque- tions have been either added or transposed. Toavold errors when using the previous edition and this revised edition simultaneously, the numbering system has now been ‘changed: the new equation numbers include the numbers ff the main clauses in which they occur. The grouping now adopted in DIN 9960 is as follows: Symbols, terms and units are covered in clause 1 In clauses 2 and 8 the definitions and parameters for zero-deviation and zero-deviation cylindrical gear teeth and for zero-backlash mating of zero-deviation cylindrical gears are presented. Clauses 4 and 5 desoribe the deviations and tolerances land the change factors for zara-deviation cylindrical gear teeth Tne deviations of individual parameters relating to oylin- drical gear teeth are dealt with in clause 6, the composite ‘and cumulative errors in clause 7, the deviations of axial positions of a gear pair in clause 8 and backlash in clause 9. This standard is concerned with cylindrical gears with Involute teeth. Definitions and parameters for bevel gears ‘and hypold gears as wellas for cylindrical worm gear trans- missions will be dealt with in DIN 3971 and in DIN 3975, rospactively. General definitions and parameters of overall coverage for gears, gear pairs and gear transmissions are grouped in DIN 868, Tne composition of symbols Is explained in DIN 3999, The terms collected in ISO Recommendation R 1122-1969 are listed in DIN 3098 Part 1 and Part 2 in German together with the corresponding English and French terms in so far as the nomenolature concerns this standard.