UDC 621.8334 in be pe cece pies ternen rege Sica ata 1001.4 : 003.62 DEUTSCHE NORM July 1980 Concepts and parameters associated with cylindrical gears and cylindrical gear pairs with involute teeth Sogritte und Bestimmungegrbssen fOr Stinider (2ylindarrader) und Stirnracpaare (Zylinderrednaara) mit Evolventen- verzahnung [As It Js cuvrent practice In standards published by the International Organizetion for Stendazaization (ISO), the comme as been used throughout as a dacimel marker. ‘Compared with the October 1978 edition the numbering system in this standard hes Been changes. The new equation numbers are rumaered continuously within the rain clauses and the fist dialt repracents tha number of the respective ‘main clause in wl for ore, en Contents 0 Other relevant standards 1 Symbols, terms, units 4.1. Symbols and torms 1.2. Additional subscripts 1.8 Units... # 2 Concepts and parameters associated with cylindrical gear teeth 24 Number of taath2 and sign of number of teeth 2.2 Reterance surfaces and datum lines 224 222 223 224 225 228 227 228 23 234 232 233 234 @ 23s 236 237 eprdcton 238 238 Standard basic rack tocth profile, basio rack Datum line, dimansions on the stendare besic rack tooth profilo. « Sections through éylindtical gear oath Gear tooth profi lan profile Tooth traces 5 Modules my, ry Hx my - Reterence cylinder, reference circle; reference diameter d Base cylinder, base circle; base diamotor dy, Inyolute surface (involute helicoid) . ‘Generator ofan involute surtace aco nolx angle fy, dase lead angle yt Contact line Transverse pressure encle at a point ay, trans vyorse pressure angi.@, Normal prascura angio at & point ayy, pressure angle @ Rolling angle ¢of the involute Radius of curvature @ of tho involute, working length Ly Inyolute function Inv a Lead p, 2 Angular pitch and pitches 244 242 243 24a 246 246 247 eng te Fe ®e ‘Angular pite! Fitohes on the Feterence cylinder Pitches on the Y-cylinder Fitches on the Veoytinder Pitches on the base cylinder . Normal base pitenes 7, Axial piten javghie ol Garman Scare (BI Narman) are win BouTnVatag Gmbh, BOHngO ese 2.5 Concepts end parameters associated with gear teeth arising from the position of te standerd besio rack tooth profilo relative to the reference cylinder Lead angle, nlx angle Flank alraction, right-handed, left-handed Sign of the hel anc ‘Addondurm modieatio ‘coeifialent.xand sign of addendum macification V-gear, zero gear Diameters of gear teeth, angles at V-cylinder. 257 Gear tocth heights 258 Tooth inioknesses, spacewidths 2.6 Geometrical limits for Involute gear teeth 26.1 Underout on eylindrical gears with external teeth 262 Pointing limit and minimum tooth thickness et the tp clrole of an external gear... .« Pointing limit ang minimum epacewrdth at the root cirele of an internal gear 264. Range of feasibie involute gear teeth 2. Test dimensions for tooth thickness, 27.1 Normal chordal tooth thicknesses 272 Base tangent lengih Wi... - 273 Radial test dimensions for toath thickness 27.4 Diametcal test aimensions for tooth thickness 278 Centredistance fordouble-flankencagemente" with mastergear.. 278 Tip dlameter day on overcu’ cylindrical gears 3 Concepts and parameters associated with a cylindrical gear palr . 3.1. Gylindrical gear pal, definitions 8.1.1 External gear pair 3.12 Internal gear pair 313 V-geer pair Sid V-0 gear pair 81.5 Zero near pair 3.2. Mating quantities 321 Goarratiow 322 Transmission fatio’! 823. Line of centres, centre distance a... - 324 Pitcn cylinders, pitch circles; pitch’ diamstar dy 828 Working transverse pressure angle cyt oF @” 251 252 253 254 255 258 263 ‘Continued on pages 2 to 48 Explanations on page 48 DIN960 Engl Price group 18Page 2 DIN 3360 326 Working depth ftw a 827 Bottom claarance bottom clearance fectore*. 27 2.3 Galeulation quantities and fectors for mating, oars ar 981 Reference centre disiance ay a7 332 Centre distance modification ym, centre dis- tanoe moditication coetticiont » 27 3.33. Cenie distance modification coetticiont j and sum of addendum modification coefficients 2x 9.94 Calculation of the sum of addendum modfication Cosfficients 2x 28 385 Calculation ot centre aistance a... 228 336 Addendum alteration f+ my, addendum alter ation factor k 28 9.4 Tooth engagement 28 84.1 Point of contact 228 84.2 Pleno of action, ino of action 28 343 Zone of action, langtn of path of conteet |. 29 344 Usabie diameter dy ace and usable ranges of the tooth flanks 29 948 Transverse angle of transmission Ye, trensverse contact ratio &. .. sige BE 348 Overlap angle ps, overlap ratio ep 31 847 Overlap lenath zy 82 848 Total angle of transmission yy, total contact fatle oy Sve ose 32 3.8 Sliding conditions at the tooth flanks. ca 35.1 Sliding speed 9. 2 352 Stiding factor K, a2 353 Specitieslidinal. .. , 2 4 Deviations and tolerances tor eyindtical gear tooth Ey 441 Deviations 4 of tooth thickness and Its test dimensions... . 83 4.11 Deviations of 120th thioxneas A, as 4.12. Deviations 4z ot the normal eherdal tooth thieke noses 24 4.18 Base tangent length dovlations Aw 4 4.14 Devistions ayy f dlmorsion over bali orping 34 4.15 Deviations Ay, of eda! single-bal or single-pin dimension 36 446 Deviations Ay» of centre distance for double: flank engagement <' with master goar..... . 34 44.7 Tipdiamoter deviation Ay, with overcul evindr- cal goers 36 4.2 Tolerances T 4 4.21 Tooth thickness tolerance 7, 35 4.22 ‘Tolorance Ts on normal chordal tooth thiaknoss 35 4.23. Base tangent length tolaranco Ty 36 424 Tolerance Tyys on dimension over bali or ping. 36 425 Tolecance Tj, 0° radial single-bell or single-nin dimension 6 4.26 Tolerance 74» 8 carte distanoe for double: flank engagement with master gear 35 427 Tip diameter tolerance Ty, with overcut cylindi= cal goore sasctae 35 8 Change factors 4” 35 5.1 Change factor A¥of normal chordal tooth thick- ness 85 5.2 Change factor Aty of base tangent length. 35 5.8 Change factor Aijq of dimension over ballgor pine 35 84 ‘all or singlo- 35 Change factor Aj-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- dricalgearteeth. . . : Circular pitch deviations 1 ‘Adjacent pitch errors. 2. Cumulative circular pitch errors Fax 3 Cumulative circular pitch error Fase Cumulative pitch error B, Rango of pitch errors Rp... ference between acjacant ritches f, | 8.2 Normal base pitch errors fas Daviations of tranavoree profile as 632 Deviations of tooth traces: 885. Deviations of the generator 64. Radial run-out 64,1 Radial run-out Fyof @ taoth system 642_Radial run-out of tip circle fy 8.6 Position deviation of gearcutting axle 65.1 Eccentricity sisters 652 Wobble F, ee : 68 Range oforors Ro... 1. 66.1 Range of tooth thickness errors K., range of normal chordal tooth thickness errors Re 66.2 Range of base tangent length errors Rw 66.3 Range ot errors Ry, for radial aingle-bail or 6.0.4 Range of errors Rygq for dimension over hells oF pins. 66.5 Range of ovrors tor the contre distance for double-fank engagement 1 1&7 Contact pattern i 7 Composite and cumulative errors. 7A Single-tlank working test 74.1 Tangantial composite error F 7.4.2 Cumulative working piteh error Fy... sss 7.4.3 Individual working error fy... ++ 744 Tangential tooth-to-tooth composite arrcr f;. 74.5. Transmission deviation of e multrstage gear mating ‘ 7.2 Double-flonk working test : 721 Radial composite error... ss 722 Working radial run-out Fl... : 723 Radial lath-to-tooth composite error [7 8 Deviations ofthe axial positions ofa eying calgearpair.. . 8.1 Deviations trom paralioliam 21.1 Inclination error of axes fre 81.2 Deviation error of exes faa 8.2. Deviations and tolerance for position of gear axes 82.1 Centre distance deviations Ay 822 Gontro distance tolerances 7. 823 Tolerance space for position of goer axes 8 Backlash j 94 Circumferential backlash j. 9.2 Normal backiash j, 9.3 Radial backlash j, 94 Range of errors for backlash i 10. Alphabetical index 41 4 a 4a 42 2 43 43. 43 43, 43 43 43 44 44 4a “a 44 45 45 45 45 45 45 48Page 2 DIN 3360 326 Working depth ftw a 827 Bottom claarance bottom clearance fectore*. 27 2.3 Galeulation quantities and fectors for mating, oars ar 981 Reference centre disiance ay a7 332 Centre distance modification ym, centre dis- tanoe moditication coetticiont » 27 3.33. Cenie distance modification coetticiont j and sum of addendum modification coefficients 2x 9.94 Calculation of the sum of addendum modfication Cosfficients 2x 28 385 Calculation ot centre aistance a... 228 336 Addendum alteration f+ my, addendum alter ation factor k 28 9.4 Tooth engagement 28 84.1 Point of contact 228 84.2 Pleno of action, ino of action 28 343 Zone of action, langtn of path of conteet |. 29 344 Usabie diameter dy ace and usable ranges of the tooth flanks 29 948 Transverse angle of transmission Ye, trensverse contact ratio &. .. sige BE 348 Overlap angle ps, overlap ratio ep 31 847 Overlap lenath zy 82 848 Total angle of transmission yy, total contact fatle oy Sve ose 32 3.8 Sliding conditions at the tooth flanks. ca 35.1 Sliding speed 9. 2 352 Stiding factor K, a2 353 Specitieslidinal. .. , 2 4 Deviations and tolerances tor eyindtical gear tooth Ey 441 Deviations 4 of tooth thickness and Its test dimensions... . 83 4.11 Deviations of 120th thioxneas A, as 4.12. Deviations 4z ot the normal eherdal tooth thieke noses 24 4.18 Base tangent length dovlations Aw 4 4.14 Devistions ayy f dlmorsion over bali orping 34 4.15 Deviations Ay, of eda! single-bal or single-pin dimension 36 446 Deviations Ay» of centre distance for double: flank engagement <' with master goar..... . 34 44.7 Tipdiamoter deviation Ay, with overcul evindr- cal goers 36 4.2 Tolerances T 4 4.21 Tooth thickness tolerance 7, 35 4.22 ‘Tolorance Ts on normal chordal tooth thiaknoss 35 4.23. Base tangent length tolaranco Ty 36 424 Tolerance Tyys on dimension over bali or ping. 36 425 Tolecance Tj, 0° radial single-bell or single-nin dimension 6 4.26 Tolerance 74» 8 carte distanoe for double: flank engagement with master gear 35 427 Tip diameter tolerance Ty, with overcut cylindi= cal goore sasctae 35 8 Change factors 4” 35 5.1 Change factor A¥of normal chordal tooth thick- ness 85 5.2 Change factor Aty of base tangent length. 35 5.8 Change factor Aijq of dimension over ballgor pine 35 84 ‘all or singlo- 35 Change factor Aj-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- dricalgearteeth. . . : Circular pitch deviations 1 ‘Adjacent pitch errors. 2. Cumulative circular pitch errors Fax 3 Cumulative circular pitch error Fase Cumulative pitch error B, Rango of pitch errors Rp... ference between acjacant ritches f, | 8.2 Normal base pitch errors fas Daviations of tranavoree profile as 632 Deviations of tooth traces: 885. Deviations of the generator 64. Radial run-out 64,1 Radial run-out Fyof @ taoth system 642_Radial run-out of tip circle fy 8.6 Position deviation of gearcutting axle 65.1 Eccentricity sisters 652 Wobble F, ee : 68 Range oforors Ro... 1. 66.1 Range of tooth thickness errors K., range of normal chordal tooth thickness errors Re 66.2 Range of base tangent length errors Rw 66.3 Range ot errors Ry, for radial aingle-bail or 6.0.4 Range of errors Rygq for dimension over hells oF pins. 66.5 Range of ovrors tor the contre distance for double-fank engagement 1 1&7 Contact pattern i 7 Composite and cumulative errors. 7A Single-tlank working test 74.1 Tangantial composite error F 7.4.2 Cumulative working piteh error Fy... sss 7.4.3 Individual working error fy... ++ 744 Tangential tooth-to-tooth composite arrcr f;. 74.5. Transmission deviation of e multrstage gear mating ‘ 7.2 Double-flonk working test : 721 Radial composite error... ss 722 Working radial run-out Fl... : 723 Radial lath-to-tooth composite error [7 8 Deviations ofthe axial positions ofa eying calgearpair.. . 8.1 Deviations trom paralioliam 21.1 Inclination error of axes fre 81.2 Deviation error of exes faa 8.2. Deviations and tolerance for position of gear axes 82.1 Centre distance deviations Ay 822 Gontro distance tolerances 7. 823 Tolerance space for position of goer axes 8 Backlash j 94 Circumferential backlash j. 9.2 Normal backiash j, 9.3 Radial backlash j, 94 Range of errors for backlash i 10. Alphabetical index 41 4 a 4a 42 2 43 43. 43 43, 43 43 43 44 44 4a “a 44 45 45 45 45 45 45 480 Other relevant standards DIN. 867 Standard basic rack tacth profile of cylindrical gears with Involuts teeth for ‘goneral and heavy engineering DIN. 868 General concepts and parameters tor ‘Gears, gear pairs and gear train DIN. 1301 Part! Units; names, symbole DIN 1315 Angles: concepts, units DIN. 3861 Tolerances for cylindrical yaar ‘eoth; bases DIN. 3867 ‘System of gear fits beckdash, tooth thick: ress deviations, tooth thickness toler- DIN 9972 ‘Slanciard basic rack toath ear-cutting tools for involute tooth systeme according to DIN 667 DIN 9002 Addendum modification of axtemna oylindrical gears DIN. 9988 Geometrical design of cylindrica! inter nel Involuts sear pairs DIN. 9990 Symbols for gear toeth DIN 7182 Part1 Tolerances end fits; fundemental con: opts DIN 58400, ‘Standara besic rack tooth profile for cylindrical goars with invelute teeth for sine mechanics DIN 58412 Standard basic rack tooth profile for ‘92er tcols for fine mecharieg; involuta fgeers eccording to DIN 58400 and DIN 867 1_ Symbols, terms, units 14 Symbols and terms ‘This standard uses the following symbols and terms @ centre distance 41 reference centre distanoo 2” centre distanoo for double-flank engagement > tacowietr bye contact tine overlag or width of gauge head for measuring base tangent length © bottom clearance bottom clearance coefficient 4 feference diameter 4 tip diameter ent mossurac value of to alameter of evercut olin cal gears 4, bese diameter de root dlameter i, vietual rafarence diameter Voir diameter 4, pitch diamater 4, Y-cirale diamstor dx dlameter of circte through canta of bail yy diameter of a measuring circle (at point ot contset ‘with measuring instrument) dy Usebie diamatar ‘dye usable tip diamater yy usable oat diemotar ‘nie usable root ciamoter at generated wheel (usable flank) @ —_Spacamlsth on the roferance cylinder % — spacewicth on the tip oylindar #4 babe epacewicth (on the base cylinder) % space width on the roct cylinder DIN 3960 Page: spacewidth on the V-ovlinder specewidth on the Y-eylinder individual error bese circle errar eccentricity profile form error profile waviness tangential tooth-to-tooth comaosite error radial tooth-to-tooth composite error adjacent pitch error ‘normal pitch error fxal pitch error lead error Individual working error tip olrole radial run-out - (ain radiane) = 57,205 780: (@ in radians) | may happen that in systems with a limlted character set eleprinter, data handling). tho unit symbols for cegree, minute and aecand in the cass of angle data cannot be represented by the suparsoripis ,',”.In these cases the following letter symbols representating the wnitsaccording to DIN 88030 shall be applied: deg or DEG for degrae (angle), mnt of MNT for minute (angie), ‘sea or SEC for second (9, tig recommended to use 6, m and s as further abbrevi- ations of the unit names, €9. 17d. 27m 27s. 2 Concepts and parameters associated with cylindrical gear teeth Alidetinitlons in this clause relate to gears free from deviations and tolerances. Thus, the equa: tions apply to the nominal dimensions of tho gear tooth; for excaptions eco cubclauces 2.7.8 and 2.7.6 The nominal dimensions of invalute eylindrical gear teeth are determined by the following parameters which ara Independent of one anather: number of teath z stancard besic rack tooth prota normal macula i, hal angle addendum moditicati facewidth & 24 Number of teeth z and sign of number of teeth ‘The number of teeth z of an external gear must be in: sorted in ha equations below as a positive quantity, waist the number of testh 2 of an internal gear must be inserted 5 a negative quantity. This satisfies the concept that during the traneition from an external to an intornal gear the gear Glameter increases until in the first instance at 2a rack having 2 == Is reached, As the transition proceeds further, the geer diameter swings over to ~~ tnd thereafter assumes a finite negative size. By establish: Ing this ana tho defintions contained in subclauses 253 and 25.4 concerning the sign of the hallx angle and of the addendum modification It 's 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 calculations tor all the quantities depending on the ‘numbar of teath — these are the diameter and radius, the angular pitch, the tooth thickness angle and the spac Width angle, the radius of curvature of the tooth flanks, ang of the test dimensions for toot” thickness: the bac tangent longth, also the radial and diametral singla-ball and single-pin dimension or dimansion aver balls ar pins. In addition, for en internal gear pair the tooth ratlo and the contre distance are negative. Inpraduction specifications (drawing deta) however sl test dimensions and the numbers of teetn, diameters etc. are always quoted as positive quantities, and this also applies to internal gears: tho exception is the addendum modif- cation. which has to bo inserted with ke appropriate sig” (eo subclause 2.54), 2.2 Reference surfaces and datum lines 2.2.1 Standard basic rack tooth profile, basic rack ‘The stendard basic reck tooth profile of oylindrical gesr teeth is the normal section through the teath of the basic reckwhien ls produced from an external gear toath system by increasing the numbor of tecth until z = ~ end hence the diameter until d= +22. Tho flanks of the standard basic ‘eck tooth profile of involute teeth are straight lings. {In transverse sectons (and only in tranavorse eoctions), the Hlenk profiles (let fank and right flank profiles) of cylindrical gear with involute teeth are portions of invo- {utes to 2 cirae (in brief: volutes). The flanks ere involute helicoids in the general case, ana involuve surfaces in the ease of straight teeth 222. Datum tine, clmensions on the ‘standard basic rack tooth profile ‘The dimensions on the standard basic rack profile arise ‘out of the module m and the datum line. on tha (straight) datum line, the pitch p, the tooth thickness sp and the spacowidth op of the standara basie rack tooth protila are slated as multiples of the module ym; similarly raferred to the detum line are the addendum /igp and the dedendum is, 360 figure 1. AdGendum and dedendum togethor yiels the profile haight ip. The toath end st the tip line. With the normals to the datum line, the flanks of the atan- darci basic reck tooth profile encioee the pressure angle ‘x; ay Merge via the Foot rounding into the root lina. The pressure angle ap of the gear standard basic rack tooth profile may differ from the pressure angle apy ofthe outter, slangard basic rack tooth profi. “The standard basic rack tooth profile according to DIN 887 097 DIN 68400 dictates mirror-imaged Identical involute tooth flanks for both sides of each tooth. —pem | Ley 2 SAE oon tensa freee) | Lor atin Flank angle 2 ap pem-m pitch & ‘spacewidth on the detum tine sp tooth thioknese on the datum tine fe crofile height Typ addendum Ap dedendum ae ‘proggure angle Gm root rounding radius Figure 1, Standard basle rack tooth profile of an involute tooth system22.8. Sections through cylindrical geer teeth 22.31 Transverse section ‘Tho sectioning of cylindrical gear teeth on « plane perpan- dlouiarto the gear axis yields 2 transverse saction A trans Verse section of a rack is its intersection with a plane perpendicular to the axis of the cylindrical gaer mates with the rack. ‘Quantities in the transverse section are denoted by the subscript t 2.232 Normal section ‘Tha sectioning of Involute helicat gear teath by @ surface ‘slsposed perpendicular 10 the tooth traces of the involute helicoids yields anormal section. Tha surface atthe normal ‘section is curved thrce-dimensionaly.Intha case ot a spur {gear the normal section and transverse saction coincica Arnormal section of a rack Is ils intersection with @ plane perpendicular to the tooth traces of the teeth {Quantities in the normal section are clencted by the sub= serip' ‘Tha subseripts n and {are not applicable to spur gear teeth 2288 Axial section ‘The sectioning of cylindrica! gear teeth by 2 plane contain- Ing the gear axis yiolds sn axial section. An axial section of @ rack is I's intersection with a plane perpendicular to the reck datum plane (see subclause 227) which contains the axis of the cylindrical gaer meted with the rack Quantities in the axial section are denoted by the sub- seript x. ‘Axial sections and the stating of geer tooth quantities In such sections are not meaningful whare spur gear teath ‘re concerned. 224 Gear tooth profile, flank profile ‘A gear tooth profile results as the line of intersection of the gear teoth with a plane. A flank profiles the line of intersection ef a tooth flank with a plane, Right flank proftes and left lank profiles are to be distinquisned where necessary, 224.1 Transveree profile The transverse profile is the gear tooth profile lying In = tranaverse section, The transverse profile of spur gear testh and the transverse profile of the gear mating with such teeth lie in the same plane. 2242 Normal profile, virtual spur gear teeth Because of tho curved normal section it is only possible In the case of @ spur gear with involute helical teeta to indicate an approximate normal profile lying in a tangentia! plano or in an osculating plane to the normal section, Geometrical stusles are theralore oftan based on virtual spur gear teeth iying in the tangential or osculating plane, thorr datum line boing e circle of curvature en the normal section of the reterance surface of the hellcal teeth (see subclauses 27.4.1 and 272) 224.3 Axial profile “The axle! protlie Is the profile cf the gear teeth iying In an axial section, 225 Tooth traces ‘The tooth traces are lines of intersaction of the right and left flanks with a cylincer tho axis of whieh colneldas with the gear axis. Hence, right tooth traces and left toath traces are to be distinguished. ‘The reference tooth irace (raterence cylinder tooth trace) i the line of intersection of the flank with the reference cylinder (see subclause 2.2.7), The dase tooth trace isthe DIN9960 Page 7 line of interssction of he— possibly imagined as extended — Involute flanks with the base oylindsr (see subclause 2.2.8) the base tooth fracas are the helices of the invalute nell- colds (see cubolcusa 2.33), The tip tooth trace (root tooth trace) isthe ine of intersection of the — possibly imagined fs extended ~ involule flank with the tip cylinder (root cylinder) The tooth traces are helices in the case of helical gear teeth and straight lines in the case of spur cear teeth, 2.2.8 Modules iis, ts; My The module m of the standard besle rack tooth protie is the normal module (module in the normal section) ry of the gear teeth, Ina transverse section the transverse module's foune 2s mm en cose For e helical gear the axial module min an axial section le tound ee mo” TTR” ase ~ Tan A se. Fora spur gear = 0 and the module ism (ray = m= The basic madule my ins ~ mal [ana Fe0s" A (22 22.7 Reference cylinder, reference circle; reference diameter ¢ ‘The reference cviinde Is the reference surface for the gear tacth. ite exis coincides with tho locating ax'e of the goar (gear axis), Honea In the case ot rack tne reference plane isthe rack datum plane. Quentities on thereference cylinder are stated without subscriot. The reference circle Is the Intersection of the reference oylinder with a plans of transverse section. The reference clameter d is determined by eum mx emy= Eat 2.4) TT] we Nove: in the case of an lncernal gear the reference afam- eter is @ negative value, see subcieuse 2.1, 2.2.8 Base cylinder, base circle; base diameter dy Thetase cylinder is that eytinder cosxial with therererence oylinder that Is determinative for the generation of the: Involute surfaces involute helicolds). Quantities associated with the base eyinder are denoted by the eubecript b. The base circles the intersection of the base cylinder with 2 plane of transverse section; the involutes of the base. cirele form the usable parts of the tooth profiles. Tho base. =2- mpl [tan aq t cos f= z- my en Note: in the ease of an internal gear the base diameter (sa negatve value, see subcleuse 2.1, 2.4 Involute surface (Involute helicold) 23.1 Generator of an Involute surface In doveloping the onvolope of tho baco oylindor, an enve lope ine of the base cylinder describes an invalute surtaco fof a spur gear It fs the generator of the involute surface, Aatraightline inclinedto the envelope linen the developed envelope surface is the ganeiator of an Involute surtace: {involute helicolc) ofa helical gear having its ergin on the 'bage cylinder in tho hel, 800 figure 2Pages DIN 3360 Involute of base evlinder Involute hetieakcs Develoned Jnvolute fine Matix o _-Base cylinder envelope line dy base diameter fh, base hellx angle vs base lead angle Developed bese cylinder envelope (base cylinder tangential plane) \ Invoite ol base cylinder Figure 2, Basa cylinger with involute helicold and generator 2.3.2 Base hellx angle fh, base lead angle 7 The acute angle in the angled base evlinder anvolope be- wean the gonorator and an envaione line la the base nolix angle ,, soe figure 2 and equations (2.30) to (22). The ‘complement angle of the base helix angle fis the base lead encle ys Ly! = 90°14 26) Both angles have the same sign. With the dovelopment of the bees oylinder envelope first In the clockwise direction, invelute surtacas curved in ‘apposite directions result and from these the let and right flanks of the gear teeth ara derived, 2.3.3 Contact line The two base cylinders of a gear peir have two common tangortial pianos which interaact in the inetantanaous axis, These ara the planas of action of the gesr tacth. Each plane of action cuts the associated toth fanks (lank and ‘mating flank) in the contact line corresponding to the working position. Each contact line is tre common geno- ator of flank and mating flank; Its prolongations sre tan- {gents to the base cylinder. Cnrotation af ha base cylinders about their axes and simultaneous winding up end un- winding of the tangential planes the contact lines travel through thelt zone of action, see subciause 3.4, 2.3.4 Traneverse pressure angle at a point ap, transverse pressure angle 2, Tho involute (always lying in a transveree action) is In clined at the arbitrary point Y by the tranevorse pressure angleat point ay relative to the radius passing through Y, 506 figure 8. The angle of inctination at the point of intersection of the Invalute with the reference circiais the ransversepressure angie a en ea) See alse equations (2.9) and (2.10). T point of contact of tangent with base circle U origin of involute YY arbktary point on involute PF teferance radiun fy base circle radius fy Yeeieoie rags 4% transverse pressure angle 2% transverse pressure angle at point Y tolling angle between U and T Dy radius of curvature at point ¥ = working lenatn Ly Inv a involute tunation tor a, Inv ay, Involuta function for ay = angle at the centre be tween U ang ¥ Figure 3, Parameters relating to an involute 238 Normal prossure angle at a point ¢,, normal pressure angle c, In the normal section through an fnvolute halicold the tan= ‘gent fo this surface at an arbitrary point Y is inclined to ‘he redius through Y by the normel pressure angle at 2 Doint dq, The correspending angie of inclination at the reference cylinder Is the normal pressure angle a; this is ‘qual to the pressure angle ap of the standard basic rack tooth profil,tan gy = tam a cos # (29) tam ayn = CAM ayy“ COS My Fora spur gear B= 0.and dy = a= €, 8180 2.3.6 Rolling angle of the Involute “The angle of the contre Gefined by the origin U of the invo- Jute and the contact point T of the tangent trom point ¥ to the base circles the rolling angie of the involute, see figure 8. The base circle arc UT is equal to the tangant portion YT, hence 6 tan ay ann 2.8.7 Radius of curvature v of the involute, ‘working length Ly ‘The tangent portion YT Is the radius of curvature @y of the Involute at point ¥ and at the seme time the working length , belonging to point Le. the developed base circle are {rom the invaluta origin U Inthe triangle OTY itis the opposite the transverse pressure angle dy, at the centre of the circle 0, see figure 5 mh en tna, =: fea aencgenctiney = Vee 1D A i ca x wan snare vane AR cue saiouiee zt 2.3.8 Involute function inv & ‘The angulardlittoronce ¢—qis termed theiavolute function ft angle a and Is denoted by inv a, (to be read as: invo lute a), see figure 3. inv ay = &~ a 219) Note: Ithes previously bean customary co use the symbol fv a for the Involute functian. In conformity with ISO 701— 1976 howover ft fo rocommandad that In futuro the symbol iny « should pe used. 239 Lead p: “The lead p, (of an Involute helicold, of @ tooth flank) isthe partion of an envelope line of a cylinder concentric wth ‘the gear exie between two aucceeaive turns of an involute helicoid (of @ tooth tank), sea figure 4. The lead inde: Pendant of the diameter of the cylincer. Lezley sr Velen “sin 6)” tani fT tan aye - 6 bee slater, a4) Plane of stendard basic rack tooth profile, datum line FSy it — 1 \ Reference cylinder snvelope lino, gear axis Figure 4. Helical gear: lead p,, helit angle f lead angle 7 2.4 Angular pitch and pitches 24.1 Angular piten + ‘The angular pitch + is that angle lying in a transvorse section which results from the dividing of the complete periphery of a circle into-2 equal parts, DIN3960 Page S tea in radians 2.15) 560 125 indagiess 2.10) Note: In the cese of an internal geer the anguler pitch is nagative value, seo subclause 2.7, Figura 6. Helioal goar: diameter, angular pitch, pitches 242 Pitches on the reference cylinder 2.424 Transverse pitch (pitch! a ‘The transverse pitch (pitch for short) py (p In the case of ‘a spur gaat) is tha length of the reference circle are be tween two successive right or left flanks, see figure 6, aon ty emo 24.2.2 Normel pitch Py ‘The normél pitch py isthe length of the heb arc between two successive right or let flankson the refersnes cylinder Inthe normal section of the gear teeth. P= Pe COS f= My 242.8 Pitch span py The piten span p; is tho roference clcia are between two rignter eft lenis separated from each othe by kreference clale pitches such that 1< h- ig, For external gears this sats a lower limit for the ‘eddendum modification coetticiont , 896 eubclause 2.64 For an internal gear the relationship must always be [dal > [4s]. For interne! goers thie sats an upper timit for the addendum medification coefficient, see sub- clause 2.84. 28.8.2 Root cylinder, root circle; root dlameter dy ‘Ino root oylinde isthe eylindrical envelope surface at the bottom of the tooth spaces of gear tooth system: a trans verse section yields the root circle ‘Quantities ralata to the root cylinder are denoted by tha sudscriptPage 12 DIN 3960 The root diameter a; amounts in the cage of gear t (see figure 7) to d= dy ~ Doha dy+2exymy= Phim 58) and in case of gear 2 (see figure 7 and figure 8) to peda 2 N= byt 2X0: Mg 2g 39) Noto: ifthe tooth thiokness devietlons Ay aro produced by deeper infeed of the gear cutting took, then the following 's found din d 2 het Ag cat a (240) In tho aso of ground and shaved oysinorieal gosrs the toc! addencium hy 1 0 be substiuted for hyo in equations (2.38) and (2.38) and the addendum modification coeificient of the pre-cut eeth for x 2.5.83 V-olrclo dlamoter a, V-oylinder The V-oviinder contacts the datum line, see figure 7 and ‘igure 8. Ite clametor (V-clrcle diamotor) is ede Mm =a(1+2-E- cost) arias rected te he Vayda denotes by the Sibert 2.5.6.4 Pressure angie ay, helix angle fy, The following angles are present where the V-cylinder intorsacts an inveluto tooth flank ean) hic angle Ay +34: com ian gpm EERE ng & tang +2E-sinp= tang Sean) transverse pressure ancle ata point ay cose = perry resp <8 ; 243) 122 -cosp normal prossure angio ata point ey tam eyg = tan ey * cos fy (244) £08 ay 208 6 C084 = corp. (1=2- 8 om) @ coby cos “cose Bap. 25.7 Gear tooth holghte Note: The working dept cesulting from meshing with rating gearis defined in subcleuse 32.6. 28.74 Tooth depth ‘The tooth depth Hof eylindrical goar tooth results from the tooth depth hi, of the stendard basic rack tooth protile and the addendum elteration &- my, see sudclouse 3.36. fw hp-+ hes my 2.6.7.2 Addendum h,, dedendum hy ‘The addendum i, and the dadendum hy of cylindrical ‘gear are stated on the basis of the reference circle: be tae bX Mig Ry (246) an bys hyn emy, 28) 2.6.8 Tooth thicknesses, spacewidths Figure 9, Helical gear: tooth thicknesses, spacewidihs ‘and their half angles 2581 Trensversc tooth thicknesses si sy1.548Nd sy “The transverse tooth thickness », (tooth thickness sin the case of a spur gear) iste length of the reference circle are, between the two flanks of a tooth. cme (Joa ectna)= city ase eer ee Say (a Inve, - inven) nig: (EEE eect) y(n ec inva) wa, (Sate Wwe ~ inva) @sn ‘The base tooth thickness sy (sy in the case of a spur gear) |e the bose circle are between the paints of oriein of the Involutes of a tooth, sendy (PEE sows) 252) 2.8.2 Tooth thickness half angles FPy vy end Hy ‘Angles at the contre in a transverse section which are ‘enclosed by radii bounding the tooth thicknesses sy sy. Sq OF Sy are Tooth thickness angles. The corresponding tooth thicknass half angles are: etd: x: taney pee Ete Ene a 6 Bwemee—i0ep ass pe Bheyetnver inves es‘The base tooth thickness half angle vy is ty Bev tare 289 Note: mine case ofan hina gears oo stress Nalarls ns negtve ves soe subse 21 2588 Spacewidths eu 2) 2 80d Oy The spacewisth a (in the case of spur geet) isthe lenath of the reference ‘circle arc between the tooth flanks en Closing @ tooth apace, The tooth thickness s, anc space- \widts @ togethor yield tha transverse pitch 2, tHe P (257) pa eye BE = 2ex omy tangy «(3-2-2 tines) The specowidth cy (e) in tho cate of s spur gear) ie the Jength ot # circular arc of diameter d, batween the toath flanks enclosing a tooth space The tooth thickness sy and spacewidth ey together yield the Y-circle pitch py. (2.58) ut ey = Pa 2 ($-inva + toven) (SAM inva riven) 280 The epacewidth c, (ein the en60 ofa spur goar) isthe tongih of the V-clecia are betwoen ihe two flanks enclosing Tooth space. The tooth thickness sv and spacewidth ey together yield the V-crcle pitch py. pei 81) 4: (4 invert inva) ay (AH invert inves) ee The base spacewidth ey, (6 jn the case of « spur gear) isthe bbaee circle are betwaan tha points af origin of tha two Invalutes anclaaing a tooth space. The base taoth thick hess §,, and base spacewidth ay, together yield the base circle pitch pee sia eb = Pox (43) acca wa) aan 25.8.4 Specewidth half angles n,m m and Mm Angles at the centre in 2 transverse section which are tnclosed by the radii bounding the spacewidths ey es 2 0f éy a's spacewidth angles. The corrseponding spaca~ with half angles ere: ena dees tan ee ee) yn Sena lavas a) nyo Fem a= inva, + inv ae er) DIN 982 Page 19 Tho base specowidth half angie.n, ie = inva, (2.68) =o ae Note: in the case of an intornal gear the spacowiath halt angles are negative values, see subciause 2.1, Reference: For spacewidth half angles at deviation of tooth thickness A, see equation (4.3) in eubclause 4.1.4, 25.45 Normal tooth thicknesses Sq. Sn, Suy 84 Syn ‘The normal tooth thicknesses are the looth thicknesses in 4 normal section of the gear teeth; they ere the lengths of the helical ares on the respective cylinders between tho tooth flanks ofa tooth. They result trom the standard basic rack tooth pratlle and the addendum modification: Beenie om-(§ -2-x-tei) (2.69) Syn = Sy C08 A, (2.70) Sen ™ Sut 608 em Stn 84) 608 fy (2.72) 25.86 Normal spacawidihs oy eq, yy AND Cy ‘The normal spacewicthe ore the asacawidtns ina normal seetion af te gear teath; they are the langtns of the halical ares on the respective cylinders between the tno tooth flanks enclosing tooth space. They result from the stand- ‘ard basic rack tooth profile and the adéendum moditi- ‘oation: fo mscoe Nida opgrtanns cme (Fae) wm 2.6 Geometrical limits for involute gear teeth 261 Undereut on cylindrical geare with external teeth ‘Whan the teeth of an extemel gear aro being produced ‘underoutting of the tooth flanks occurs It the path of tha ‘corner of the cutter tip cuts into the involute portion of the tooth root flank of the workpiece during the rolling action, By suitable choioe of geerouiting data, panticulerly as regards addendum modification, helix ancle, addorda and pressure anglo, undercut can be avoided. For an cxternel ‘goer with standard baste rack profile according to DIN 867 procuced by a rack cutting too) the following ralationship exists between the cutting data: e-sinber hyo = ay (= sin en) Troe ig \Whor9 fg the addendum andl a9 18 the tip corner rounc= Ing raclus of the toot In the case of an external gear with standard basic rack tooth profie according to DIN 53.400, yy (L~ sin ay) in equation (2.77) must bo replaced by tha bottom clearance Xan 277Page 14 DIN 3960 The value min celculetedin this way can usually be further reduced by a emall amount (not exceeding 0,17) a smal degree ot undercut, whien generally cose not navo any harmful etfects, is acceptable 2.8.2 Pointing limit and minimum tooth thickness at the tip circle of an external gear The teeth of an external gear become polnted at ths tip circle if tem ° (278) op inv a) ~ inv ey) The pointing limit resulting in this way is considered in subslause 254 [Al the tip cylinder the tooth thickness Sq should not have {a value less than 0,2 imy This provides an upper fit for the addendum modification of an extornal gear which is suitable for practical application, see subclause 2.6.4. The undarout end pointing limit ss imits in the downward direction to the number of teeth which itis precticabla to cout In an axtasnal gaar. For spur gears and the standard basic rack tooth profile according to DIN 867 the following apply nin = 8, taking z= +0,57 28 the polnting timit, 2axin = 9, taking = + 045 for S49 = 0.2 figure 10. Ifa small amount of undercut is accepted, the following Is Stil feasible 2in= taking = 40,41 2.8.3. Pointing limit and minimum spacewidti at the root circle of an internal gear The tooth space of an Internal gear becomes pointed at the root circle if = Shey invay tiny ag = (278) us apen wr Co) The resulting pointing limitisconsideredin subclause 2.64. At the root circle of the internal goor the spacawidth ey should have a value not loge than 02+ m. This provides {an upper limit for tre adcondum mosification of an interns {geet which is suitable for practical application, see sub- clause 264. The tid circle dlameter mit and the pointing limit for tha tooth spaces limit 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 tho following applios [elnin = 16 with x= —0,52 for pointing tit [elon = 21 with x= —0,58 for ein = 0,2 a, ‘86 figure 10. ther timits srsing from mating reasons are dealt with in DIN 3988, 26.4 Range of feasible involute gear teeth ‘The feasibility range for involute gear teeth which is im: poses by the limite according to subclausas 26.1 10 2.5.3, 's indicated In igura 10 for the standard basic rack tooth profile according to DIN 887 and in figure 1! for the stand acd basic rack tooth profile according to DIN $8400. The folationchips tor holical gar toatn are shown hore for virtual spur geer teeth which, for a standard basic rack focth profile according to DIN 867 or DIN 88400, are determined by the addendum modification coafficient x of tha helical goar tasth and by the virtual tooth number z nett (C08? fh, c08 bes 7 (280) For the virtual tooth number factor 2, see table 1. This relationship between the numbers of taeth is so chosen that for @ helice! gear tooth system end the corresponding virtual spur gear tooth system the seme value nin 18 yoolded by equation (2.77). In rigura 10 and tigure 11 tha addendum modifications x which are feasible each time are plotted ageinst the virtuel tooth numbers ye 1 Table. Viva tooth umber factor zis <5 a9 fetion of helix angle for cy = 20° cos" fy 208 a zis a a A thy 6 ah a ahs tn in in in in degrees cearees degrees | degrees degrees 1 | xoooe | 11 | 1528 | av | 2082 | a1 | te2a8 | a1 | agra 2 | t0017 | 12 | 1069 | 22 | tom | az | 100 | 42 | 22205 a | noose | 13 | tos | 23 | tas | 33 | vers | 4a | 2an0a 4 | roo | 12 | 1oces | 24 | rata | a4 | reece | ae | 2220 s | 19106 | 18 | 1.1008 jaro | 35 | 1706 | 48 | 26ae2 6 1.0153 18 1.1161 26 1,400 98 17787 + | e200 | a7 | nisi | 27 | sara | a7 | sa409 a | toa | ta | tia | 26 | s4060 | a8 | sore 9 10348 19 11668 2 14428 33 | 197688 | so | rose | 20 | nigea | a0 | sare | 40 | 20550DIN 960 Pege 18 sr eaoejdas “eH 90) UL OREHED UN 806 (RLBUI 1 “Lh eunéiy Bunwry Yu ce + 8p =p ime -weip apo dn UNWILKO} anps1E=5 eUs9}K9 Jo} sonfeA SURI) —_@ eur (quso1ed 01 £q popuorxs joc1 vo yey canze) Sores Nia 1 Bul “plod ,y}20} JO 2quUML LIN WLILY paOUEIE|OD, JO} S80}eR UNI] gz eur suo} 3001 32 yno1opun oy snp s1008 2g our our a2 +p =p 1030 warp 9)ovo dh wunujujw oponp e166 ous24%e 104 canyon SURAT, Joyx0 10} Canyon Bu" ‘01 ounB 4 sseung youn counPage 16 DIN 3960 Note: Cures 6 in figures 10 and 11 are based on the (not reelizable) assumption that the Involuie faces go right Iarough to the botiom of the tooth; furthermore, the 15°fese at the bottom of the tooth of the standard basl¢ rack protile In the case of curves 8a and 5b in figure 171 has not been taken fnto account. It is therefore ganerally advisable or necessary to carry aut am investigation according to DIN 3993 in the case of internal gears, For an ex:emal gear the limits of the range are sat by the pointing limit or the minimum crest width according te Subclause 262 (curves 1), by the undercut according to ubclause 263 (Jing 2) and by the minimum tip circle diam tor according to eubciause 2.5.6.1 (line 3) For an interna! goar the limits of tho range aro eat by the limiting value of the tip circle diameter according to sub- clause 2.5.61 (line 4) and by the pointing limit or minimum ‘spacewidth according to subclause 2.63 (curves 5). These limita are nowover not alwaye achievabiain practice dua to mating reasons, see DIN 3993. 2.7 Test dimensions for tooth thickness The tooth thickness ie a circular or helical arc ard as such le not directly measurable, For enecking tooth thickness therefore Indirect measuring mathods are used and trom the values obtained in this way the tooth thickness is fourd by calculation. rua spur yer roth story GaN” Mone of rea seston Figure 12. Normal chordal tooth thickness, and height iy, above the chord of an external helical gear derived from the virtual spur gear tooth sysiem ‘at the curvature ellipse in the normal section 2.7.1 Normal chordal tooth thicknesses 224A Normal chordal tooth thicknosses f, and Sn nsignts he and f, above the chords ‘me norma shordal toot thickness f orf 8 the shore dlatence between tne tooth traces ofs tosth on the refer= nce cylinder and he Y-eyindey respectively. isthe only ‘Both ickness quantity cectly measurecie on a tooth, For helical gaer teeth the normal chordal tooth thicknoss at ihe reference eylindar can be calculated with adequate accurecy from a virtual spur gear tooth system cefined in ‘the area of the reference circle by the virtual reteronea i See (elrcle of curvature of the el- circle dlameter de = zodag rele of curvature ofthe o lipse in the normal section) and the virtual tooth thickness half angle ¢,= y+ cos? f, see figure 12 (On the retoroneo cylinder the following apcties - sim (+ 605°) Sn dn 8in Uo A Corrosponcingly the normal chordal tooth thiexness cn the Y-eylinder is found as dy sin (Wy *€08° By) (2a) 2.80) Bn cath cr) and on the V-eylinder as (y+ C088) wos oe By series expansion of the sin value and putting y terms of 4th and higher order equal to zero the following aporox: matfon equations are obtaned from oquations (2.81) 0 (2.83) Somsg (1-5: ene) 286) Fn™tp (: ~ 5 cot a,) aes ie sin (1g sh coer) oe) For nearly all cases arising In practice, these give the numerical values with adecuate accuracy. The height f, above the chord 5, (or fi, above By OF Fy dg? cost # (28n 1 +a dy: oh costa, (2.88) Tiywhy- emt Fay Recota, — (200) Fora spur ges with f= 0,f,=Oand fi, =0 aquations (2.8%) to (288) hold good exactly and equations (2.84) to (289) with good approximation, Note: Foren nternal gearthe values ford and pare nega tive, $06 subslauses 2 1, 25.6 and 2.5.8.2. Hence according te equations (281) to (2.86) the values tor the normal ‘chordal tooth thickness are positive. In equations (2.87) 0 (2.89) the lest term is negative for an internal gear. 274.2 Constant chord & of @ spur gear At distance fe trom the orown of the tp ciroe the teeth of 4 spur gear have a rormal chordsl tooth thicknoss the length of which 1s dependent only on the modula, the pressure angle end the addendum modification, but on the thor handis independent of the numberof teeth. All gars of the same standard basic rack tooth profile and with the same addendum modification thus have the same normal chardal tooth thickness § at thesame point fg, sea gure t3 and figure 4. For thisreasonitis termed the constant chordas cote smesota-(Ze2x ne) Goo in @ an) why 5 cosa Figure 12. Consiantchords, and eight i above the con Sant chord ofan external spu geer Figure 14, Consiantchord§, andheight abovethe con- stant chord of an internal spur gear 2.7.2 Base tangent length Wi, ‘The bao tangent langth W, of an oxtornal gear is the distance, measured over f teeth, or over i tooth spaces in tha case of internal gear, between two parallel olenas contacting a right flank and a lft Nank respectively in tha Involute part of the tooth fianks, sae figure 15 and figure 16, In the case of internal gears the base tengent length is ‘only measurable on gears of the spur type. ‘The base tangent lengtn is not referred to the gear axis land is therefore Independent of any eccentricity of the ‘gear teath, ‘The number of teeth spannsd (measured number of tooth spaces) ft must be cnoson co that the magsuring plenes contact the tooth flanks at a paint close to half the tooth depth, i. in the vicinity af the V-cylinder. Far contact in the V-evlinder the following relationship applies a ee rr (292) DIN 3960 Page 17 with ay according te equation (2.48) and cos fs according 10 equation (2:32). ‘Ag an approximation Fe\e found from the virtual spur gear tooth system as ce, 2a) (299 or with Jess good approximation trom the following ex: pression an 2 so * D-Tai ee kezaw using the virtual tooth number zy according to equation (287) or according to table 2 end ay, in degrees trom ‘equations (2.44) or (2.45). For fea fractional number ls generally obtained and for the bese tangent length calculation this must be rounded up tothe next whoe number. A value /t< 15 ¢hall bo rounded 10.2, With k= 1it would be necessary to measure near the base aylinder and such measurements are unsailsfactory and can be avoided in the majority of ceses, see figure 17 and figure 18 As & rosull of rounding the number of teeth spanned the msesuring elements no longer make contact with the tooth flenks on the V-cylincer, but instead {whan symmetrically positioning the measuring elements) on or near the meas- uring circle diameter dhyg = Veab + (Wg cos BS? lel Note: in ine case of an inanal gear dy Is « negative vee. Tho diatanco betwoon tha contact point of tye measuring gloment and the tip circie is 0,5 - (¢, — dy). |f this value Snegetive he (toretic)) contac! pat les oulade the tooth system. Altferent number of teeth spanned meas tired numb ct space) mut bo er0aen In the case of an external gear with symmetrical contact of the measuring elements fie generally permissible for the contact points on he tooth flanks toe up ta 0.7" a oviside aV/- 12+ 0,018 Wi, in mm should not be under-run: if the tooth edges have been chamfered, @ value dw > 2,0 + 0,03 - We in mm is recom: mended, Fer measuring the base tangent length in this case a minmum facewidth of DE Wy sin [fi + By cos fy (298) is necessary. The relationship betwean the length differ ence AW, the contact Ine overiap or measuring element Width by, snd the base tangent length Wi, fs shown in figure 26. when V-heads are used, see figure 12, the overlap 2+ by, is necessary for a given size AW. If the overlap is smell or known it is recommended that the V-heads should be Drought Into contact at the overlap mic-point from the end of tha tooth, Measurement by using disc-shaped measuring elements base circle dlameier Veciecla diamater 3 meaguring circle diameter for base tangent length Ws 5 measuring circle clameter for base tangent length Ws ‘normal base piton Steath onan extemal spur gear Measurement by mesns of pins dy base circle diemeter 4 Vecircle diameter dys measuring circle diameter for base tangent length W's iyj5 measuring circle diameter for base tangant length 175, Pe normal base pitoh tooth spaces) onan ¢ ¢Table 2. virtual tooth number factor ziyy= inv ay/inw ay ‘as.a function of helix angle {Yor a, DIN S960 Page 19 ‘Table 3 Virtual tooth number factor 2}, = L/cos** fas a funetion of hex angle f in| 6 ete | a | can | | caw | | cha rere) fiow) [a saw] @ [enw | a | aw] | etw | 2 | chw ca oe 1 ygcot sma] st f isise| ar | 228 2 tor ‘aot | se | san 2 | zan0 8 | toma esa] 33 | uaast| 4 | zae70 3 | hoo asst 34 | isane| ax | 2st00 5 | tote vaaar| 38 | yaro] as | aenie 8 | tone saeea] sa | yam: 7 | 40206 ‘gen | 37 | 5560 8 | 107s ‘azaa | 99 | 9508 8 | tcxo atag | 33 | 20358 1a | tose ates | an [218s ay [zis [21 [esa] at | 2ssie 1 Papa | a1 2 | gam | a2 2 |v | iz | na] a | za8ae 3 fine | aa as [tae] is [run] as | ze0re 4 [igen | an a | idan | a | esos] | nasse § [ioe | is 2 | ms | as | ens] 4 | arses [ou | aa 2 | rans | a8 | 0108 | sae | rr ay |e | 3 | 200i 1 0 aa | tsi | se | 2 sas 8 m |issn| 39 | 2200 tosis | 23 @ | toms | | 2a 2.1.3 Radial test dimensions for tooth thickness 27.8.1 Radial singlo-ball dimension Mac The radial single-ball dimension Myx is the distance be tween the gear axis and In the case of an external gaar the outermost point, Inthe case of an internal gear tne innermost point ‘of a measuring ball of diameter Dy in contact with both tooth flanka in a tooth space, see fiaure 21 and figure 22. ‘The paints of contact Py anc P, between the measuring ball gnd the tooth flanks shaillie on the V-oylincer or near it ‘The corresponding measuring ball ciameters Dyy must be ‘ohosan according to DIN 3877 (at pragent at drat staga): {or m, = Imm they oan be found from figure 23. Since the ‘measuring balls only need to contect the tooth flanks in tha vicinity ofthe V-eylinder. uss cen ba mada of evaiteble balle having dlameters ditfering eomawnat fram tha nemo: gram values or calculeted values. Inthe case of contaet in the V-oylindar the meaeuring ball diameters aro Dy 2am Pig C089 (AN Aknye— LAM ayy (298) tha virtual numbar of teeth z_y for dimensions over bells or pins being Ea cos? eo ei (2.100) a For factor ziyy of the virtual number of teeth see teble 2. The angle cy must be calculated according t0 Zap" C08 Saggy = ee e104 SATE ea aon tho andie cxau eecording to sn taneyay inv ag = AOE (9402) eam The measuring balls according to equation (299) contect the tooth tanks of spur gears exactly on the V-cirola (in the case of gears with tooth thickness deviations on the Generating V-circle if itis calculated wlth 25 according to equation (2.114) Instead of x); they contac’ the helical gear ‘flanks close to the V-cyiinder. The contact points may be checked by equation (2.106). Note: In the cass of an intemal gear the tooth numbers 2 ‘and z,yy and the bracketed expression in equation (2.99) ‘are negative. For tne angles « and the meesuring ball d- ‘meter Dy, the values are always positive 1 the mazsuring ball dlametar Dyy Is known the pressure angle at @ point mc inthe transverse section on the circle through the centra cf the ball is found from Du inves * 3m, CO a+ inv a, (2.103) ‘Tha diameter dj of the elrcle on whieh the centre of the measuring ball es is found as ar: : . 104 ax = 4° osay, ~ Tosa @104) “The relat single bal dnension's Mc = Wee + nd (2108) ‘The diameter dy ofthe cyinderon which thecontactpoints P, and Pp batwoon the moacuring ball and the twa tooth fark te fs found as dyn (2.406) the prassura angie ay an the circle having diameter dy bing calculated as D tamen stingy BH cox 07 Noto: In the case of an Internal gaer 2, 7) and my €8 weil fas d. ds end cx are negailve, see subclause 21. Hence nogetive values are also obtained for Myy and dy. The ‘moasuring bali dlamoter Dyq ard the pressure angiee ax, ‘and ayy are always posits, 2.7.3:2 Radial cingle-pin dimension Mex In the case of sour gear teeth and external helical gear teoth messuringpine atcameter Dy, carielao boused instead cof measuring balls. Equations (299) t0(2.107)apply equally to the redial single-pin dimension MxPage 20. DIN 3960, 3 | 19 {+ 18] 2 Lg { 14 3 | |8 Loa s oe =e: i 2 3 a9 2 8 = os & z Ea g 3 5 a: : : Bo 3 2 = oJ a 3 Bond g g 2 -ouk 1 € -o8t\ aera s 2 -04| Ball ¢ iT 3 a a ae ee a a eae a Tees : wa} 00 - ae 17 [is f Peto tam [te E ! Lt ia to 135/18 SE 7 6B I a - P S fs 8 - Be bs [8 ¢ F ! e me mof-—_+—t- 1 a Kor zs E a Las Re \ or er 3 ; 9 jt oe aon z & I I i | RS rf £ fees Ht Z = om lie $ I | 73 § of} — i be ee 5 i | ee ta ea [| bE) | | it | wlio hb 2 M8000 Victual tooth number 24 —= {nthe lower part ofthe aura tha virtual tooth rumber ze accor io equation 2.87) or eb 2s found iam ine numberof teats zn he {ear end the helix angle f. Example:2 = 26 and f= S0°gvae 24a ~ 2519038 30,0 The upper part of the figure shows om the lel the ralaonsiip between za. the addendum mociticelisn coeficlent send the numder of teeth spanned ffor 2y/3 100 Tho cells line curves cortesond {0 the condiilon di, dy~ 2. 0,5 my and the broxandine curves to ta canation ym dy +2: 0,7 ty The boundaries of tne tooth ayetome feasible acocrding to subeluse 26 ere nol indicated hare. see figures 10 and 11, covered by 8 gvan number of teeth spanned ovorlaa Iniarge parte ote alagrary, co thet neve several numer of teeth s ‘ara teasibie fer « singla virtual number ef foeth and a tingle addendum modiiation coetfcist. Exam: sayy = 39,1 and = In tho eree ol the numbers of teeth svannec f= 2, 3,4 und 5. Ony In ihe shaded parions is measuramant witt ony one numberof teeth spanned possinie. sample: zaw~ 2land x 0,890 n the range k~ 4 For eny > 100 the upper right-hand part of the gure is epplicabe, Being designec for virtual ‘ooth-numbor groupe 100.108, 109... LI? ic, ising Dy 8 aach time. From ths corresponding table « tabulated value fs found for the numberof leeth spenned which. depencing on {he acdoncum modification eceffcian, ethe!yloias Ine value of k direct, ot wnlen must be ether dacressed or meroasaci ty 1,2, 54 Isaeuied ject eorrasponcing to the ranges inthe upper part of tae Aglre in order to Yield tre number o* teeth sosnned hele tis part Df the gure azo the rangee ovariep, Example: Zniy = 142 (tabulatos value: 16) and x — + 0)9 yilds the possble volves k= 16-+ 1~17 or k= 16 + 2= 1B or k= 16 + 519 Figure 17. Nomogram for determining the number of teeth spanned f for external goars with 2a = 20° CDDIN S080 Page 21 i xH L i Measured spaces more than in table — 1 PAZ Z tT 3 “= Addendum modification cont “5 80 = Numoor of tooth 2 The left-hand part of the figure contains the relationship between the number of = toetn 2, the addendum modification coefficient xand the number of measured spaces ft Lia to 18) for |z| 3100. To a0 The solicine curves correspond to the concition that the contact poinis of the elements are 0,5 - m outside the V-citcle, whilst the broken-line curves correspond to the condition that the contact points of the measuring elements are 05) mi inside the V-circte ‘Tha limits of the tooth systems feasible according to subclause 26 are not indicated here, see figures 10 and 11. The areas covered by @ given measured number of spaces overlep in argo perts of the. diagram, so thel here severel measured aumbers of spaces ore feasible for a single number of teeth and a single addandum mositication coeficient. Example: 2= —48 ‘and x 40,9 lie Inthe range of ine measured numbers of spaces 2, 3,4 and 5, Only in the shaded portions Is meesurement with only one measured number of spaces possisle, For example: z= 27 and x= ~0,¢ lie in the renga k= 4 For |z|> 100 the right-hand part of the figure ie applcabla. For explanation sae figure 17, i [235 to 243] 27 Figure 18. Nomogram for determining the measured number of spaces /for internal ‘spur gears with «~ 20° Number of measured spaces j— Measuring anvil aw free - we f ge ‘on developed base cylinder {ow Plane of action -—~! evroment Figure 19 Contact line overlap dre In base tangent longtPage 22 DIN 8360 Contactiine overep by —e- oS 8 mm ‘Bese tangent length Wa—> Example: For Wig= 69,548 mm and by = 5,7 mm tho diag- ‘onelie longer then V7; by about 0.10 mm. ifthe overlap Dy For meaning of symbols see figure 24 | racuced, for example by edge chartering of the flanks ‘and facewidth devation by 084mm, then the diagonal is Figure 22. Radiet single-bal dimonsion M, on en internal stil onger than ¥¥, by 0,06mm. in this case tha measuto- spur gear ‘ment can stil be regarded as adequately rellable Figure 20. Relationship between base tangant length We. contactiine overlap byandiength difference AW Centre of tooth space In Section A~A ~Cenire of tooth space in Section BB d, base eylinder diameter di, diameter of clrele through centre of ball lameter cf eylinder on which the contact points P lie (measuring circle diemeter) Dy, meaouring bell diameter PL contaet point of measuring ball with left flank Pa conteet point of measuring ball with right flank ~Tseneratorot sy prosture angio tanaverte section et svete through centre of bl erie hit Pressure angle In tanaveres seston et contac pls P ee € Figure 21, Radial single-bail dimension M; in @ helical external gearassuring bal diameter y= Dy ——— Measuring bal diet =i tT 0 oo woe ww ‘Virtua tooth number 24 nw une om > Number oftesth 2. ——> Nomogram for determining the measuring ball ‘iemetar Dy for radial singia-bal cr diamatral two-ball measuraments for eq = 20° end y= Tm Intha case ot eylindticel gears with tooth thick ness deviations xp replaces # DIN 3060 Page 23 2.74 Diametral test dimensions for tooth thickness 2.7441 Dimension over balls Mex Inthe ceseof en extarnai gear the dimension over balls Mex is tho largest anternal dimansion over two balls whilstin tha ‘ease of an internal gear tis the smallest Internal dimension between two balls having diameter Dy, end in contact with ‘the flanks in two tocth spaces at the meximum possible ‘separation fram each other on tho gear, sae figuies 24 40 26. The two balls must be in the cama plene perpen- dicular fo the gear axis; the measuring element faces (aking contact externally or internally) must be held parallel to the gear taath axis, Note: The dimension over balls is not referred to the geer ‘axis and hence is Independent of any ecoantricity of the ear tooth, For an even number of teath, see figure 24, the following applies Mex = dx + Dy, (2.108) for en odd number of teeth, see figure 25 and figure 26, the following applias Max = d+ €0s (2.109) For choie2 or calculation of the messuring ball diameters Dy see subelause 2.7.3.4 Note: Inthe case of an intesnal gear the velue found for Mgx s negative, see subelause 2.1, 27.4.2 Dimension over pins Max External gests with spur or halal teeth and internal gears, with spur teeth can also be measured with pins instead of with balls, Equations (2.99) to (2.104) and (2.106) to (2.108) ‘apaly equally to the dimancion over pins Mae. 274.24 Dimension over pins on spur gear teeth. Fr spur gear tacth equations (2.108) or (2.109) anply. For positioning the maasuring pins i Is not possible to state a common plane perpendicular to the gear teath axis. The measuring elements need only be swivalied sidemays to find the maximum value; thie is the actual sizo, 27.4.2.2 Dimension over pins on external helical gear teeth with even number of teeth Equation (2.708) applias in thie case. The measurement ts ore straightforward even thanfor spur geer teeth because the actual size Is not 2 maximum value; insteed, as the measuring elaments are ewivelied in the axial plane the ‘ectual size is a minimum to which tha measuring pins auto~ matically edjust themselves in all positions owing to the parallel maasuring element faces. Swiveling sideways does not alter the measured value (disregarding form deviations fon the tooth flanks), sea figure 27. This simply moves the meaeuting pinson the same helical path in the tacth spaces.Page 24 DIN 9980 27.423 Dimension ovar ping on external helical gear teeth with odd aumber of teeth Equation (2.198) applies to this caiculation algo, Becsuse Cf the parallel measuring element faces the twe measuring ins ate forced on a helical path out of the position Gecupied by tha measuring bails in the dimension over balls (see figure 25 anc tigure 25) and into pesitions whieh fre exactly opposite one another over the centre point of the gear tooth system, see figure 28 (this eliminates the factor cas 5" > la equation (2.108) and equation (2.108) applies). The dimension over pins ie theretore twice az large as the radial singla-pin dimension. The naiical motion cf the pins is opposed so that the measuring paints move ‘apart axiely as a furction of the helx angle hy and the 22 must be eld parailet to the Gear testh axis, a5 is also ecessery with the dimension over balls (ese subclaute 2.7.42), The gear tooth system must have a certain minimum Width byiq) AS Must the meaauring elements so that the pins do not protrude Beyond the tooth ence, see figura 28. The necessary minimum facewidin ean be found with figure 29, Assuming the same ball/pin dlamotor (Dy, tat = Dai pind @ known dimension over balls can be converted to the ‘dimension over pins by means of the following equation Max ~ Du Max ~ angle 5". For this resson, the measuring slement faces + Due ano 2.7.5 Centre distance for double-flank ‘engagement a’ with master gear The cenire distance for double-flank engagement ¢ with master gear, see subclause 7.2, can be used as @ tooth thickness check For a gear with number of teeth z) and ‘2 master gearwith numbarottactn 2 andaddendum modi fication cosfficen’ x, the value of a” can be found trom (ated em coe m a aaa tie any the working transverse pressure angle «” being foune eccording to equation (9.7) from isan ae inva" = inva, +2 tana, (2.112) According to equation (248) in which sis replaced by the value (5,1 9, tho eddondum mocitication coetficint Nj governing the double-fark engagement ie found from fats on ee tae 2.113) ), $86 equation (4.1), the neminal dimeneione for xj, «and a’ are found; when the mean value Aga Is usad the mean values are found; see subclause 416. In thie tot the offeots of radial run-out and tooth trace eviation (and of damage to the tooth flanks) come into effect also, The last thus celacts all backlash-conatricting effects deriving from imperfections in the gear. Reference should bo mage in thie connection to VOIVDE 2603 “*Single-flank and doubie-flank working test on spur end helical geers with Involute profile” and also to DIN S867, For moaning of symbols, ‘ee figure 21 Figure 24 Dimension over balls My for an external spur ‘deer with evan number of testh For meaning of symbols ‘see Tigure 21 Figure 25. Dimension over balls My for an external spur ‘gear with odd number of teeth For meaning of symbols ss0e figure 21 Figure 26. Cimengion evar balls My 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 2.7.8 Tip diemetor dan, on ovorcut cylindrical ‘Spur gears of precision mechanisms produced by generst Ing methods with modules m, = 1mm aro often overcut ‘norder to avold the difficulty of maasuring toath thickness when the module is small and to allow the tooth thickness to be determined instead from the measured valuoe of the tio diemoter, This production motnodroquires epoc'al gesr- cutting too)s. ee i & Margin | 7 Measuring width= Margin | DINs960 Page 25 With overcutting the actual eize ot both the tip circle anc the tooth thickness Is produced by appropriate radial in- {eed of tha cutter, This infeed corresponds to.a generating addendum modification coefficient = which in tha case ‘of negative tosth thicknase deviations As is smalier than the addendum modification x referred to the nominal di- mensions ‘Assuming the use of zero-deviation gonorating tool with rack protile (hob) the following apaties A Fin xgeet any 2 Inthis case the tip diameterselso areattered The measured values d,s, (actual values) of the tip diameters ere thus mailer with negative tooth thickness deviations (with in- termal gears the Value of |dyy| Is then greator than tho value [d,)) than the nominal dimensions according to equations (2.36) and (2.97), From day d+ 2 oxy mg #2 hap ang the actual ceviation of the tooth thicknoss Is found s¢ As ~ (dane ~ 4a) * tan ene ‘When a pinion type cutter is used the dependence of the generating pressure angle onthe sum of thetooth numbers fnd the sum af the addendum modifications has to bo allowed for, since these determine the centre distance of the generating gacr pair and nence the generated geer tip dismetar, The corrasponding equations ere contained in subelauces 2.2 and 23. They allow calculstion of the work {gaar addendum mositication deviating trom the nominal value x and hence the work gear tooth thicknsss as a tunction of the tip diameter deviation, eo T + Contact points Supporting surteca LLLLE Diameter at centaei poinss Figure 28. Dimension over bells Mux ancl dimension over pins Mex on holical extarnal goar tooth with ‘906: number of taothPage 26 DIN 3360 Number of tocth ¢ o oD we ww a Virwal tooth number 2——— 20% soe Mim mo 2008 T = 0 In the lowar part of the diagram the virtual toxth number qt (8 found from the number of gear teeth z and tha nel anglo f, Noxt, in tha upper part of the diagram the corre- sponding minimum tacewth dy, i road aff as a function of f and the addendum mouification coemficient 3. For negative addendum modification coefficients less than =0,15 the horizontal line bolonging to # = — 0,15 must be. used, Figura 29. Nomoaram for determining the minimums face ‘width dyin tor tho dimansion over pins accor ing to subclause 27.4.2.3 3 Concepts and parameters associated with a cylindrical gear pair All datinitiong in this clause refer to zero-back~ lash meting of zero-deviation cylinerical gears equations thus epply to the nominal the gear pairs. 3.1 Cylindrical gear pair, definitions 3.1.1 External goar pair ‘Thomatingof troextemal cylindrical gears (external ears) for of an external gear with a rack gives an external gear pair. In the case af an external gear pair the subscript 1is used ‘in eouations for tre smator goar (pinion) and tha aubsoript 2 for the larger gear (whee) When the gears are of the same size the subscripts can be allocated as dasired {In the casa of an extornel gear pair with helical gear teoth ‘one gear has a left-handed, the other one a right-nandos. flank direction. 3.4.2 Intemal gear palr ‘The mating of an external cylindrical ger (axlernel gear) with an intornal eylincrical gear (internal gear) gives an Internal gear patr {In the ease of an internal geet pair the subscript 1's used. iin equations for the oxtarnal goar and the eubeoript 2 tor the intemal gear, see alsa subclause 2.1 Inthe case of an internal geer pair with helical pear tocth both gears have the same flank direction: both are eltner right-handed or left-handed, 31.3 V-gesr pair A V-gear sir is the mating of two cylindrical gears such, {hat the sum of thelr addendum moditications not equal to zero. Be 1 tao, (One of the cylincricel gears in this case can bea zera gear. The centre distence of a V-gear pair Is not equal to the. reference contre distance. The reference circles are not. Simultaneously tne pitch circles, 341.4 V-0 goer pair ‘A gear pair at reference centre distance Is the mating of two V-gears suen tnat the sum cf their addendum modi- fications is equal 10 zero. Bee ny +22 wn ae Their centre distance Is equal to the reference centre are seid to be “speed reducing ratios” whilst ratios ‘uch that |i| <1 are seld to be “speed increasing ratios" 3.23. Line of centres, centre distance « Ina transverse section the line of centres ot @ gear pairs Ihe straight line joining the gear centras of the cylindrical ‘gears mated with each other, The centre distance @ is the ‘istence between the gear axes. Note: In the case of an internal gear palr the centre distance is kon as negative, see subclause 2.1 2.24 Piteh oylingers, piteh elteles; pitch diameter dy Inthe case of a cylindrical geer pair the pitch ayiinders (Giteh circles) are those cylinders (circles) sbout the gear axis which nave the samo velocities Thepitchclrcles divide: the centre distance in the ratio of the teoth numbers. The pitch circles ostablishod during the oporation of eylindrieal gat pair (cylindrical ger pair in a gear unkt) are termed working pitch circles, The pitch circles estab~ lished by a generating cutter during the generating of @ tooth systom are termad generating piten circles, ig Fide dyad =a a) Qo 2a de rag 8 TET core _ dy 4h aha 7 Taka oa) de 4. Baw da" Baye TT cose | _ dks oda tory Ta (35) Note: Inthe ease ofan interaigearpalrdys'sanogaive value, see subclause 21, 3.25 Working transverse pressure angle 1 oF 2" “The working transverse pressure angle ay, is that pressure angle whose vertex llas on the pitch circle, worklag pitch circle). It's caleulated fram DIN3960 Page 27 don don _ ley + ea) ome sew dy dys cose, 3.6) inva = inva +2- 2 tan a an ‘The values inv ag, and iny a can be found from tables of the Involute function or according to equation (2413). In the double-flank working test with a masier gear the working transverse pressure ancle is denated by a”, sce subolause 27.6. 3.2.0 Working depth hy The Working Gopth ‘ny of a gear pal Is the distance apart of the tip circles of the two cylindrical gears on the line of centres, see figure 30. gy + daa aia lye (a) 327 Bottom clearance c, bottom clearance factor c* The dottom clearance c Is the distance by which the t ircle ef @ gear is separated from the root circle of its ‘mating aoar, see figure 30. It is equal to tho difference between the tooth depth li and the working depth fy. Cnt hy ct my ee Figure 20. Working depth ty and bottom clearance ¢ of a gear pic 3.9 Calculation quantities and factors for mating 33.1 Reference centre distance 2y ‘Tha reteranca centre distance a, is the sum of the reference circla rad oF the two gears. di tde ate ag = Em, rity (21 + 22) 7 cosh 19) Note: Inthe cese of en internal goer pairag is @ negative value, see sunciause 2.1. 38.2 Centrediatance modification yn, centre distance modification coetticient y ‘The algebralcdifference between the centradistance «and the reference centre diatanaa aq is the contr distanco modification y- mt which Is expressed by the centro dis- fanco mosifieation coefficient yin fractions of the nermat module. Yon =a ~ aa an ate (co Eee (He) aaFage 28 DIN 3900 3.3.3 Centre distancemoditication coefficient yandsum of addendum modification coefficients 2x Betwoan the contre sistance modification coettciant yan ae J ven iva Bx: (Eta 3.3.4 Caleulation of the sum of the addendum modification coefficients Dx ithe centre distance a, the numbers of tasth =, and 2, tho normal module imy. the pressure angle y ani the helix angle fare known, then the working transverse pressure angle ty, arising in the mating gears is given oy equation (86). In accordance with the previous equations of this Subclause tha sum af the adcendum modificatiana is then found from fei + 22) “(Inv ew ~ Inv) mee FT ty 14 ‘The way in which 2x x) + #18 destlouted botween the two (gaars Is governed ay the permissibie stressing of the teatn or by othar specified dimensions of the gear teeth, e.g. root diameter, see DIN $992 and OIN 3098, 3.8.5 Caleulation of the contre distance « If the normal moduls mz, the pressure angle ty, the helix angle f, the numbers of teeth 2; and z2 anc the addendum modification coefficients 2 and 2 of the two spur gears are known, then the centre distance of the gear pair Is foundas 9 8 CS at my tei +22) c08e ech, @15) 2 eos eS ay, 3.3.6 Addendum alteration 2, addendum alteration factor k Its somatimes necessary for the addends to be altered to Suit thee mating conditions and the specified minimum bot- tom clearance. ifthe bottom clearance c corresponding to the standard basic reck tooth profile of the gears is to be ‘elainad then the necoesary adéendum alteration fe» 1, and the addendum alteration factor #, respectively, are found from Be tig = a= aq ~ tty BE (are kayo Bx ain The addendum alterations calculated in this way are ob tained with correct sign, that isto say: nogptive values in the case of externol gear pairs, so that the tip diameters bacoms smalir, positive values in the case of internal gear palrs, so that the valve of the tip clamotor of tho intornal gear becomes smaller whilst tne dlameter ofthe pinion tip clrcle becomes larger. ‘The calculated values are often so small that thoy are cancelled cut by the deeperinfeed of thecuttingtocl which isnecessaryfor producing the backlash andby thenegative root c:ametor deviations additionally producedby standard pinion type cutters, go thet the remaining effective Sottom Clearance is altered to only a slight extent (or within per rmigsible ims), In the ease of Internal gear pairs it le to be noted that tha addendum alterations which ara alwayspositiveinthls case can usually noi be realized because the special engays- ment and manufacturing conditions af internal gear pairs limit the usable acidonda of the internal gear and pinior, For details see DIN 3093, 34 Tooth engagement For the investigation of tooth engagement the working pch circles and the corresponding transverse profile are the citer, 3.4.1 Point ofcontact ‘na gear pair tooth flank endiits mating flank contact each ‘other on the contact line, sae subclause 283. point of contact of a flank profile 's the point at which It makes contac! with the mating tian in any particular working position ‘At each contact point in the transverse section the normal to the goint of contact (erected perpendicularly on the flank and passing in al casas thraugh the pitch point C) together with the tangant to the pitch circles in the pitch point makos the working transverse pressure angie tui, In the case of involute gaar tooth the normal to the point of contact and the path of contact coincide, so that the pres sure angle isthe same for all fank points 8.4.2 Plane ot action, ine of action The plane of ection is the geometrical locus ofall points of contact of an involuto flank with the mating flank of the ‘meting gear. It contacts the base cylinder and hence Is pparalle! fo the cear axis and is distant from it oy the radius of the base cylinder. Each gear palr has two planes of action, ane for the right flanks end one for the left flarks, see subciause 283. The two planes intereaet in the instantanoous axis which is paarallat to the two gear axes and the angle enclosed by the {wo planas is twice the working transverse pressure angle ‘The lines of action of an involute gear tooth system are tne Intersections of the planes of coniact with transverse sections, They are the geometrical lacus of all possibie pointe of contset of tne tienk profiles in the tranaverse section of a gear pair and al the sama time the projection fall possible contact lines of the tooth flanks in the direc: tlon of the instantaneous exis In the transverse section of the gear tooth, a line of action inthe case ofexternaigearsisthe commoninternaliangont fand in the case of Internal gears the common external tangant, 0 tho base circles of the mating cylindrical goars, I contacts the two base circles at points Ty and Tp, see figure 31 end figure 82. In both figures pinion tis assured to bo tho drivor, that is to sey the ratio ia speed recuina, tis necessary to distinguish between the right flank line of action anc the loft flank line of action, 34.2.1 Spectel points on the line of action Special points on the line of action are as follows (see ‘igure 31 and figure 32) ‘A. starting point of engagemant. “This is the point at which the tine of action intersects tha tip eiele of the drivan gear. B intornal ind'vidual point of contact on tho driving gear, extemal incividual polnt of contact on the driven gear Thisis the point on the line of action which is cistent by ‘one normel base pitch from point E. h point Ca1D external individual point of contact on the driving gear, Intornal indiviauai point of contact on the drivan soar This is the point onthe line of action which i distant ay ‘one normal base pitch from point A E finishing point of engegement. This isthe point at which tha ine ofaetionintersects the tip circle of the driving geer. Ty pointo! contact between theline of action andthe base Circle of gear 1. Tp pointo* contact netween thaling of action andthe base Eircle of gear 2 3422 Curvature radil of the tooth flanks ‘The following segments of the line cf action give rise to the curvature radii of the tooth flanks which are the ar: teria for the contact ratio and motion conditions of a gear pair (see figure 99 and figure 84) Te seas gv a-ak yy © tan ay) 18) TH - on 19 = 1 TE enn} 20) 0 ~ 961 ~ Pes aw TD -eor= 0ma~ Po eam 1 toca = 450 Re = oa ton ~ oe On 29 Equations (2.19) to (8.22) apply fo the case when gear t isthe driving goer andgcer 2 the driven gear.Intheopposite age A and E as wall as B and D aro to be Intercnanged in figures 31 10 34 and In equations (3.19) to (8.22) Note: The values obtained for the curvature redii of an Internal gear and for the segment [12 af an intemal gear air are negaive, see subclauses 2.1 and 2.57, 34.8 Zone of aetlon, longth of path of contact ‘The zone of action (loncth of path of contact in the trans- verge eoction) ia that part of the pisne ot action (line of ation in the trenaverse section) whieh comes into use dduting the operation of © gear pair. In the ease of speed feducing ratios the lenath of the path of contact is term fated at ‘ta starting point A by tho tip citcle of gear 2 anc at He tiniehing point E by the tip circle of gear 1, see figure 31 to figure 34. ln the extreme case the length of the path of contact may’ extend as far as the base circles. if the tip circle of the ‘aonerating cuttar cuts tha line of action outside the points 1, oF Ts the result is undercut and tha length of the path of cantact is shortened accordingly. Ifthe tip cele of the ‘mating geercuts the line ot action outsidethe sagment 1:72 Inthe oaae of extornel gear tooth, orincido tha segment T\T in the case of internal gear teeth, then mashing difficulties wil arse, IN 3960 Page 29 ‘The length ga of the pat of contact of two mating cylindri= cal goars is given by: 1) /goas 2 Ber [v am ahi ~ (ay + dha) una] (29 Note: 22 and dja are negative inthe case of an internal ‘gear, see subclause 2.1. The length of the path of contact when a oylindricel gear (ubseriat 1) is mated with @ rack (standard basic rack tooth profile aacarding to BIN 887) Is given by bo 5 (Vai = ahr don a) mam) tema ‘The length of the palh of contact gai divided by the piteh point C into the lonath of aparoach path gy (portion of tha Tength of path of cantact at the root flank of the driving {gear betwean the tip circle of the driven gear and the pitch point) and the length of recess path ga (portion of the Tenath of path of contact atthe tip iank of the driving gear between the pitch point and the tip circle of the driving gear, see ‘igure G1 and figure 32. Thase portions of tho lengih of path of contact are elso termed the tip Ionath of the patn of contact and root longth of the ath of contact, or ine geet. Forthe case where gear 1s the driving gearand geat 2 tha driven gear, the following applies: The length of approach path Is equal to the length of dedendum path of contact of gear 1 which Is equal to the fonath of addendum path of contact of gear 2 ai AT = gaz - Ger oh (Ep VET HE dastanee) (aaa 2 Mei The length recess geth is eaual to the length ot addendum ppath of contact of gear 1 which is equal to the length of adendum path of Contact of gear 2 = CE 21 ec 1 ; 2) (VERE ay tame) gaan For ha opposite case (gear 2 driving, gear { driven) gy and £88 wall es the subscripts A and B for the curvature radi 2F6 to be interchanged in equations (9.26) and (3.27) (eee subclause 3.4.2.2). Note: Inthe case of an internal gear the number of teeth, the diameters and the curvature red are negative values, 496 subciauses 2.1 and 2.6.7, Note: The length of the path of contact can be shortened by tooth to chamfering oF racluging or by tip rlict I such ‘2 0220 tha usebio tip dlametor dy,) OF dup obtained from the production specifleations is 10 be substituted for dy: and da, respectively, in equat/ons (9.19) and (9.20) as well as in (3.24) to (8.27), see subolause 3.4.4, 34.4 Usable dlameter dy; active and usable ranges of the tooth flanks ‘The usable circles enclose the active ranges of the tooth flanks which are effective wien the gears are oerating, see Figure $1 and figure 32, The starting point A of the pathPage 30 DIN 3960 Range of sotive flank of gear? Range of active lank of goer 2 Figure 31. Length of path of contact and active flank ranges of an external goer pair with speed-re ‘ducing ratio. Figure $3, Curvature radii @ o! tooth flanks and sliding speed v4 at point of contact ¥ of an extornel oar pai with epocd-roducing ratio Range of eile flank \ Range of live flank of gear Figue 32. Length of path of contact and active flank range of an internal gear pair with speed. ducing retio Figure S4. Curvature radii g of tooth flanks end sliding speed v4 at point of contact ¥ of an internal ear pair with speed-reducing ratio € —of contact datermines the usable root circle of gear # with {ho usable roo! diameter dyn, whlist the fioishing point € of the path of contact determines the usable ract circle of ‘gear 2 with the usable root diamater dyj;. The usable tip Hhamatere dizi and dye are governed by the produetion specifications of the gear and in general are practicely ‘qual 10 the tip diameters dys and dyo. respectively. For exemple, with tooth tip redivsing Qua i tne normal azctian the following applies yay Gy ~ 2+ Quy * (1 = $i en) (3.28) Hence fora goer pal dn Veena 2) Vda a+ a (2.20) ane” ey aT A (ae tina VRE) a 30 Note: Far an internal gear pair the centre distance a anc the diameters of the interal goa aro nogative, 88@ SUD: clause 2.1 For special cages arising with internal geer poirs see DIN 3985, During the production of a gear tooth system by the ‘ganerating method the usable circles corresponding te the trenaverse preasure angle at ganoratian ang the cutier ‘ip diameter come into action. These ussbie circles enclose the usable ranges of tooth flanks. Outing tooth produetion on an external gear with a rack ‘utter of @ hobpIng cutter without protuberance (adden ‘dum fgg, lip rounding radius gq) the usable root diameter riyieon the ganerated cy/indrical gearieas follows it under cutting doce not ocour: ae = Vf sine —2- Chan = Atty Oro" -$in ay) /Sin es]? (aan ‘The addendum modification tactor xy at generation must be cetermined taking Into account the tooth thickness Gevaation A, and a grinding or sheving eliowarce, if re quired, In case of an allowance of thie kind dure Is the diometer usable during proliminary treatment During shaping the usabie diameter dye is found accord- Ing to equation (8:30), the corresponging que tities of tre pinion type cutter with ubscript Dreplacing the quantities ‘with eudeoript 1. Furtnermore, tne relationship specified in DIN 3993 must be observed in the case of internal gears. 34,6 Tranevoree angle of transmission (x. transverse contact ratio ¢ ‘The transverse engle of transmission g, of a gaar Is the contre angla through which It rotates with its mating profile ‘rom start fo finish of angegement of a fiank profile, soo figure 95. The tranaverse angio of tranemiseion of pinion and gear is gvon ae follows: (932) (933) DIN s900 Page 31 ‘The ransverse contact ratio ¢ai¢ the ratic ofthe transverse angle af transmiasion pe to the angular pitch ror the ratio, Of the length of path of contect to the transverse normal pace piten. ai HH Bay a Pa Pal Figure 36. Tranaverse angle of transmission ig of an e ternal gear pair 346 Overlap angle g,, overlap ratio ¢3 ‘Tha overlap angle gp is the angle between the two axial planes enclosing the end coints of a tooth traco, soo figure 96, 5 1 2b sint pt on Sey Heme (698) b- tant sbesinil exp SM, SCN eng The overlap rato gy ie tho ratio of the overlap angle ps 10 the angular pitch + or tne ratio of the facewid:h b to the ‘axll pitch p, on qePage 32 DIN 9360 Figure 96. Overiap angle wp of a evindricel gear 94.7 Overlap lensth sp The overlap lanath gp of ahalicl gear isthe reference cicle tare belonging to the overlap angle yp po opm B tan iB (3.98) 34.8 Total angle of transmission 4, total contact ratio cy ‘The total angle of transmission i, is the angle at the centre of @ geer through which it rotetes from start to finish af contact of one of Ie tlanka with the flank mates with it. (CIs equal to the sum of the transverse angie of transmission and tne overiap angle eu Pat + ORLA we (3.29) on 27 2 + Onn = 2 240) ‘The total contact ratio &, Is the ratio of the total angle of transmission to the angular pitch. itis equal to the sum of the tranavorea contact ratio and the overlap ratio. on te Beste ean 25. Siding condtion tthe tooth Hanks 481. Sikg coed Aintolnot curator rocfndr gen negacermet Hees eats Sete rine ne too teanmene pcan paving the even of he Sceatan tae Athen of contcl ae fre Send Mur he a7 fonseencttenmete malts Sacause of the senility ofthe epecd tinales with the tr angles ¥O7; and ¥Os"9 tha sling speed found in ean- Junction wth the carvare ral gy, and, (Soe equation @i2yas veeen (28a) an According to figure 33 and figure 24 the distance ga, bo tween ¥ and Cis Bg = * (Oct ~ Out) =F leca ~ oy) (943) nonce arte ty (14h) ou Note: ay always counts as positive. Sinee w Is positive Tor an exterael gear pair and negative for an intemal gear pair i usually follows thet the siding speed is grester for ‘xtornal gar 19919 than for internal gear teeth, ‘The sliding speed is proportional to the distance guy and Al the pitch point is equal to zero, It reaenes its maximum values at the point of roat or tin contact, wen seeu=ae( ied) a5 eesavnr (ied) (3.48) [grand g, delng tound according 16 equations 826) and (327, respectively 3.5.2 Sliding factor Ky The sliding feetor Kgs the ratio of the sliding spood y, 10 tha velootty 2 of thé pitch clrtas: Kyo t= 2 tee (1+2) (3.47) ‘The maximura values for Ky are altained at the end paints ‘Aand E of the path of contact ata kee 4H (143) (248) ate ye 3H. (143) 49) Note: in the case of an intemal gear pair u is taken as negative, see subclause 32.1 Tho variation of tha sliding factor along tho ling of action |e shown In figure 37 for sn external gear pair and In figure $8 for an internal gear pal, taking u = +2 as the example. For judging addendum modifications the lines Ks tnd iyo ate continued beyond points A and E. 35.8 Specific siding ¢ ‘The specific slicing ¢is the ratio of the sliding speed to tha speed of a transverse profile in hedirection of the tangent to the profil, see figura 83 and figure 4, Equation (8.42) yields (50) (st) ‘The maximum values of Care reached at the end points And of the path of contact ou at A: 52) esi aE G1 53) using the curvature radli gq and gg according to sub- clause 34.22. Note: in the case of an internal goar pair u i taken as negative, ses subclause 3.2.1 In the variation of the specific sliding along the path of ventact is shown in figure 37 for an external gear pair and In figure 38 for an Internal gear pair using tne exemsle of = £2 For judging addendum moaifiestions the curves of spesitic sliding are continued beyond points A and E.= ine of action —— 4 Deviations and tolerances for cylindrical gear teeth For the mating of two goars It is necessary to have devi ations from tne zaro-pley condition In order to achieve the necessary backlash, The minimum deviations depend on the backlash-reducing effects of the gear taeth and of the rousing and as well an the operating conaitions and tne lnfluonce exerted by the design, whilst the maximum ce- vations clepend additionally on the geeroutting tolerances. ‘These influances are mostly independent of the helix angle Therefore the deviations and tolerances on tooth thickness DIN 3080 Page 82 Figure 87. Sliding factor Ky and specific sild- Ing (ae s function of the normal hase pile’ for an external gesr pair with w= +2 Figure 38. Sliding factor Ky and specific slid- ing Ces a function of the normal ‘base pitch for an internal gear pair with w= —2 and its test dimansions ara referred to the normal section {(ge2 DIN 8867), likewise the change factors. All definitions and equations in this clause hold ‘goad for zero-deviation gears 41 Deviations A of toath thickness: and its test dimensions 4.1.1 Deviations of tooth thickness A, ‘To achieve backlash the taoth of oxtornal gears and of Interna! gears must have negative tooth thickness deve ations, The tooth thickness deviation A, is the differencePage S4 DIN 3960 otween an actual size and the nominal size of the normal tocth thickness son the reference cylinder, The largest permissibie tooth thickness of a gear is given by the upper tooth thickness Ay, and the smallest permissible tooth thickness by the loner tooth thickness deviation Ay, In the easeof cylindrical gearswith addendum modification mansurements near the midsle of the tooth depth (ie. near ‘tne V-eylindan are te be preferres to moacuramante enthe ‘eference cylinder. The tooth thickness deviations on a Yeeylinder are found from ice wa: (142-4) wn the last part ofthis equetion being applicable to tha tooth thickness deviations on the V-evindet (a, = dy) For calculations in the transverse section tho tocth thick ness deviations 4, have to be converte into the corre sponding ttensvetse section deviations Ay: Au” op aa For calculating the deviations according to subclause 4.13 {0 4.1.7 it ls acvieable to caleulate first with the average 1 3 se Aa) and then to determine the limits with the aid of the enenge factors according to clause 5. 44.2 Deviations Azof the normal chordal tooth thicknesses The deviation Ayis the difference between an actual size {and the nominal size of the normal chordal tooth thick nove #5, For tho normal chordal toath thicknesses Sp, at diameter dy the daviations alter accordingly, gee SuS- clause 4.17 ‘The deviations 4zand Aig are genarally only insignificantly sifferent from the deviations 4, and A., respectively, of the tooth thicknasses. They can be equated with them ‘exoept for very small numbers of teeth, see subclause 5.1 tooth thickness deviation Aun 4.1.8. Base tangent length deviations “Ay Tha base tangent length deviation Ay is the difference: between an etual size and the nominal eize of the bas> tangent length Wi, n the ease of extarnal gears negative eviations make the actual size of the base tangent length smailor thon the nominal size for zero-backlash engage ‘ment, whist in the caso of internal goare negative devi- ations make the absolute valua of the actual size larger then the absolute value of the nominal siza, ‘The upper base tangent length doviation is denoted by ‘yrs and the lower by Ay It 1s convenient first to calculate the mean base tangent length according to equation (2.96) using ter x the moan generating addendum modification coefficient xq whieh ‘© obtained from equatién (2.114) with the mean tooth Uhiekness deviation ay,. Thon by using the change factor 40 subelause $2 the limits are found. 41.4 Deviations Ay of «dimension aver balis or pins The daviation Ayyg of tha dimension over balls or pins My of tooth thickness is the ditterence between an actual cize and the nominal size of the cimansian over balls or pins, In tha case of external gears negative deviations make the ‘actual size smeller than the norrinal size for zer0-backlash fengegerent, wnilst in the case of intornal goars nective deviations maxe the absolute valua of the actual size larcer than the absolute value of fhenominet size. The permissible maximum value is denoted by Axde and the permissible minimum value by Ayes ‘The relationship between an actual size of the dimension over balls or pins and the corresponding tooth thickness Goviation is found from oquationé (2.104) and (2.108) or (2,109) on subsituting In equation (2.103) tne value iy is the cumulative Circularpitch error existing aver asactor of fepproximately) wot the goat peripnery (R= 2/8). 6.1.6 Cumulative piteh error F, ‘The maximum cumulative etcular pitch error on the gear is tormad the cumulative pitch error Fp. Itis stated without sign and is found from the eumuiative pitch arrors ae tho difference between the algebraically largest and tne alg braically smallest velue. Pawnee] 3] 151517] Spee perape s efrf ate tape 4 + 8) adjacent pitch errors fy shown as vortical blocks bo- {ween the flank numbers range of pitch errors fy differance between adjacent pitchas (in thie ease between pliches Nos 20 and 2, Le. on flank Ne 20) b) cumulative ciroular pitch errors referred to flank No 21, shown as a stepped-line diagram F,, cumuletive pitch error ©) cumulative circular plich errors over a sector of k= 3 indivdtuel pitches each time, shown as veriical blocks in rmid-span, 4d) cumulative plich-span deviations calculated from the span daviations of figure c, referred to flank No 24, shownase stepoed-iine diagram [with correspondingly large step wiath) Figure 39, Graphical reprasentation of olroular pitch oe- ations (example: ¢= 21)6.4.5 Range of piteh errors Ry ‘The range of pitch errors isthe difference between the latgast and the smallast actuel size of the transverse pitches p, of the right of left flanks of geer. ‘The range of pitch errors R can he determined divectly tram the measured velues of circular pitch measurement without knowiedae of the adiacont pitch errors fy. 6.1.8 Ditference between adjacent pitches /, Aditferenoe between adjacent pitches j is the diference botwaen the sotual sizes of two successive traneverso pillehes af the right ar left flanks af a gear Differences between adjacent pitches are found direst from circular pitch measurements as the differences of the measured values of pairs of adjacent pitches. 6.2 Normal base pitch errors f,. A normal nase pitch error fy. Is the difference between the actual size and the nominal size ct @ normal base pitch py. Ercore of the transverse base pitchos are denoted Dy foe: Bnd orrore of tho normal baeo pitcheE DY jyen- The results of the base pitch measurements are inde- ondont of any eccontricity of tho gear teeth. Base piten deviations may derive both trom irregularity of the circular pitch as well as from form and position deviations of the {wo tooth flenks. As a resut of irregular profile or tooth trace configuration tney may citfer at different paints on the same tooth flanks For conneetion betwoen beso pitch caviations and trans- verse pressure angie deviation or base circle deviation, 80 subclause 63.16. 63 Flank deviations Flank daviations are the deviations existing within the Nank {ast area of the gear by which the tooth flanks depert from the Involute ralicoids ef the nominal bese cylincor whon taking into account the desired deviations (e.g. crowning) Flank devitlans on a cylindrical gear can be determined 4) in aplane of transverse section (deviettonsof tne trans vetse profile, or of the profile for short) bb) onthe reterence eylinder orsome other coaxial cylinder (deviations of the tooth trace) ©) ina plane tangential to the dace cylinder (deviations tf the generator). The records of flank deviations obtained with the aid of fiank testing machinoe ato tho flank tact ciagrams. in the test diagram preduead ay the majority of flank testing machines the nominal involute, the nominal tooth trace land the nominal generator appear a3 straight lines, 9e¢ figure 40. From tha flank ‘ost diagram (test diagram curve) numerical velues of the actual deviations on the gear are derived by the use of relationships indicated below. ‘Tha flank test rango gonerally comarises the range of the Usable flank (height gnc width of the fark). Alternatively, it may be specially sgreed for acceptance test purposes on gears 63.1. Deviations of tronaverse profile It ig only In transverse sections that inyolute gear Leeih exhibit involutes of thelr base circle, Deviations of the tooth oroflles from their nominal profileare therefore meas- tured only in tranaverso sections. The nominal proties are formed by Involutas of tre nominal base cle with allow~ DIN 3980 Page 37 lance made for intended deviations trom tho involute form (@.. protilo modification) For thie mecsuremant it Is usual to employ measuring machines which guide the stylus according to the involute ‘generating law curing the measurement and trace the tooth flank slong a transvoteo profile, Whan the direction of blylus motion [s at right anglas to the flank the measured values must be converted to deviations In the transverse. secon by matipyng by Note: Inthe test diagram traced by the measuring aching tha measured values are read off ai right angles to the direc: tion of chert feed, thats 10 say in ine case of atest diagram aeeorcing to figure 40 in ihe direction at right angias to theline Aa which corresponds to the recording of aniominel involute, Fer evaluation of the test diagram the equalizing line BE is drawn a8 the actual involute. This is calculated if necessary by the “method of the feast sum of ina squares of the errors”. An inclined position relative to the line AA indicates the presence of a deviation from the nomine! base diameter or fram tne nominal eressure angie ‘The profi test range must generally be taker as thelengih of path of contac! when the gear is meshed with its mating gear (or If the letter is unknown, with the rack). The profile tostrango L, is ecnveniently defined by stating the working length £;for ihe root and L.for the tiper by the correspond- ing working angles ¢ end és. The following are distinguished: Total profile error Fj Profle form error fr Profile angio orror fix Profile waviness fry 6.8.1.1 Totel profile deviation F The otal profi error Fy of a tooth flank Is the distance beiween the twa nominal profiles which enclose and touen the tooth flanks within the profile test range, Note: In a test diagram according to flgure 40 the total proiile error Fy is the distance measured at right angles to the direction of char: feed botwosn the ino AA and tno further fina ®’N’ paraliol to mt wmnioh are drawn through tne utermast points of the test pattern in the diraction of chart feed within the profile test renge. The desired deviations from the involute form are allowed for by appropriate dey'- ations of the lings AA and AN from straightness. 681.2 Profile form error fr The profile form error fof @ tooth flank is the distance between the two invoiutes of the base cirale which—allow- ing for desired deviations from the Invotute form —enciose ‘and touch the actual profile within the profile test range ‘Tha profile form error also comprises the wave depth of the profile waviness, see subslause 83.1.5. Note: Ina test alagram according to figure 40 the profile Jour sor fis the distance messured at right angles to the direction of chart feed between the paralie! nes BB end B'3" paral! lo the averaging sotval ‘avoluie 8B, which lings conte! the test diagram within the protie test rango. 63.1.3 Profile angie error ine ‘The profile angle orror fxg Is the distance between the two omninal pratilas which cut the involutes of tne actual base circle at the starting and finishing points, respectively, of the profile test range. The profile angle errer fire fs usuallyPage S8 DIN S980 ——oi ireotion of onan feed Test onge | +] igure 40. Fane deviations Test diagram enc survey of deviations Prato Tooth trace Gonorator © | Total pote enor F Total alignment ear Total ganerant error Fy @ | Profile ance ero fan Tooth alignment rrr fp Generantenale ero fixe @ | Protie foun enor f Longitudinal form eer ge Genorant fea etr fer Testranae | Profile test conge Le Tosth trace teat range lp Generator test range Le 28 | Averaging acialinvolute | avaraging actual tootn trace ‘Averaging actual generator ASA’ | Nomina proies Nomina tooth traces Nominal generator which envelope the actu lnk 88, 8B" | Actual iment | notua\ helices Actual generator hich onvolope the actual lank 60,076" | Nemina prottes Nominal oth ices Nominal generator whlen eu the actual gnerstors ortoatn tres a the starting and fishing pain, respectvoy, of the test ange stated in terms of jim as a linear dimension assigned to the profile test range La, see DIN 3961, The proiile angle arror fixe is deemed to be poeitve if the involute of the actual Base ofrolo rises to tho material-troe sido, compared ‘nith the nominal profile, in the direction of increasing ‘working length; itis Geemed to be negative if the involute of the eectual base circle falls towards the meterial side in the direction of inereasing working lonath A profile angle error is caused by adeviation ofthe pressure angle or by eccentricity of the base circle of he gear teeth Inon-colneidence of gear-culting axie ans axis of rotation) ‘hich has the same effect an the Individual tooth, Note: ina test clagram according to figure 40 the proftie angle 0r70r ja Is tho dlstance measured 2: right angiae 10 the alrection of chart feed between the lings C'C! and CC" drawn pareltel to the line AA which cut the line BB af the Starting and finishing points of the profile test range. 8.3.14 Pressure angle error fy, bese circle error fy The protle angle arto jy can If nacessery be converted {a the corresponding bate ctcle error jt (lference be~ twesn the actual base diamotor and tho nominal bese ameter) or to the corresponding proscure angio orror fy 162) fi Ty tana, If fig te Inserted in ym endl Lin mm, then fp is found in um, jg In mrad (gee subciause 1.3). (63) en 634.8 Protile waviness fig A profile deviation repesting cyclically with the working angle is danotad as the profla waviness fry Itls cheracter- ized by the weve-depth and wave-length. 63.1.6 Connection between pressure angle error fabase circle error fy and base pitch O11 0F Moe The actual size of the pressure angle on & tooth flank cannot be measured cirectly on a cylindrical gear, but in~ toad can only ba calculated after dotermining the actual ‘base diamater, see equations (2.7) and (6.2), In the case of spur gears the setual pressure angle a can bbe calculated from the measured value p,y ot the normel ‘base pitch p, by means of the follawing expression Post cosa = 4) Note: This equation only applies exactly ifthe gear teat are free from pltci and flank deviations. Ifthe value Inserted {or poy IS he average taken from a sufticient number of measuremonis distributed over ine gear priphary then the corresponding average will be abtsined for a. For an (e.g. estimated) uncertainty 4 of the pest Value the uncertainty ofthe a is S¢~ =t4p,/(m» m- sin a). For example if, for SE0(0Lmm and m= S mim the uncercainiy 1s In the caso of spur gears the following relationship exists between the pressure angle error f, 2nd the normal 6680 pitch error fye fos Tae es] id