UDC 621.833.1 :621.753.1 DEUTSCHE NORMEN System of Gear Fits Back1 ash Tooth Thickness Allowances Tooth Thickness Tolerances Princbles DIN August 1978 3967 Getriebe-Passystem; Flankenspiel, Zahndickenabmasse, Zahndickentoleranzen, Grundlagen To facilitate use of this Standard, the calculation of tooth thickness allowances has been included as Appendix A. Information on converting the allowances for the various measuring methods is add& in Appendix B. The DIN backlash system of fits for gear pairs allows the limiting allowances of tooth thickness to be defined with attention given to all effects occurring in the operation of a gear transmission, and to all deviations throughout the gearing. The system of fits therefore consists on the one hand of the allowances and tolerances of the gear teeth, referred to their prevailing mounting arrangements, and on the other hand of the allowances and tolerances of all the other com ponents of the gear transmission in so far as they determine the position of the teeth relative to one another. These values defined for a reference temperature vary in operation through temperature changes in the upward or downward direction, through elastic deformation under load and possibly through swelling or contraction. The system of fits is defined a a tooth thickness system of fits in the normal section on the reference cylinder, i. e. all s allowances, tolerances and operationally induced alterations in the gear transmission are treated as tooth thickness alterationsand require to be converted to the nonnal section. The normal section was chosen because the production effort, i. e. the necessary tooth thickness tolerance in the normal section, is indëpendent of the helix angle. The normal section was also chosen for metrological reasons, since the normal chordal tooth thickness and the base tangent length are measured in the normal section. The calculation of the allowances however is made over the transversesection, since on the finished gear transmission the backlash is measured as circumferential backlash (see Appendix A). The system of fits provides for safeguarding the minimum backlash and limiting the maximum backlash. The reference basis of the system of fits is the zero-play condition at the nominal centre distance, with nominal addendum modification and with error-free components. The necessary negative allowances of tooth thickness can be produced by an additional addendum modification in the negative direction Ax. This however is not taken into account in the nominal addendum modification. Whether the weakening of the tooth thickness needs to be taken into Consideration in calculationsof IoadCarryng capacity is something which has to be decided for the case concerned. In any event this should be done whenever *sni > 0.005. tooth qualities demand given tooth thickness allowances in order to ensure the requisite or permissible backlash. The minimum backlash i s determined by the upper allowances. However it does not correspond.to the sum of the upper allowances because a whole series of factors alters the backlash (see Appendix A). The maximum backlash is determined by the lower tooth thickness allowances which result from the upper allowances and the tooth thickness tolerances. This also does not correspond to the sum of the allowances because here again a series of factors alters the backlash. 1 Other relevant Standards DIN 3960 Definitionsand parameters for cylindrical DIN 3961 DIN 3962 gears and cylindrical gear pairs with involute teeth Tolerances for cylindrical gear teeth; principles Part 1 Tolerances for cylindrical gear teeth; tolerances for deviations of individual parameters Centre distance allowances and shaft position tolerances of housings for cylindrical gear transmíssions Symbols for gear teeth DIN 3964 DIN 3999 2 Backlash The backlash value says nothing about the quality of the gear teeth although, on the other hand, the different gear 2 Theoretical backlash 1 The theoretical backlashit results from the tooth thickness allowances converted to the transverse section and from the converted allowances of the centre distance. Continued on pages 2 to 23 Explanations on page 24 !lesale rights of German Standards(DIN-Nonnen) are with Beuth Verlag GrnbH. Berlin 30 11.91 DIN3967 engL Preikgr. 7; Verir.-Nr. O 1 12 www.bzxzw.com Page 2 DIN 3967 2.2 Acceptance backlash The acceptance backlash is the backlash obtained with the unloaded gear transmission a t reference temperature when one of the gears is rotated against the other. It is uwally smaller than the theoretical backlash, since the backlash-reducingfactors generally outweigh the factors tending to increase the backlash. Backlash-reducing factors are, for example, deviations in the gear teeth and also form and position deviations, see Appendix A. 2.3 Working backlash The working backlash is the backlash resulting when the gear transmission i s operating. It i s not constant. During the starting up of the gear transmikion in particular it is possible for the more rapid temperature rise of the gears compared with the housing to bring about larger changes in the working backlash. It is generally larger than acceptance backlashwhen the linear coefficient of expansion of the housing is greater than that of the gears. Shaft deflection and displacement also affect it. 3 Tooth thickness allowances and tooth thickness tolerances Normally the tooth thickness allowances and tooth thickness tolerances can be found directly from Tables 1 and 2 on the basis of existing experience, such that, as a rule, the upper allowances for each gear should be at least as large (numerical value) as the lower allowance of the housing centre distance (without converting). If no empirical values are available for backlash and tooth thickness allowances, these must be calculated. A guide for this purpose will be found in Appendix A. The calculated values are usually rounded and then likewise taken from Tables 1 and 2. I f exceptionally small amounts of backlash are necessary for functional reasons, calculation i s indispensable. Table 1. Upper tooth thickness allowances A , t in pn I I I Referencediameter imm) Allowance series 10 10 - 100 - 85 150 200 - 70 125 170 230 310 - 58 - 48 - 40 70 - 33 - 22 - 10 - 5 7 9 12 17 22 30 41 O O O 50 125 -135-110- 95- 7 5 - 6 5 -54 -44 -30 - 40 -14 50 125 280 560 1000 1600 280 560 1000 - 180 250 450 820 - - 105 85 140 -115 -155 -210 260 - - 60 95 - 80 -110 -145 - 19 26 35 48 - 56 75 - 330 - 280 - 190 -130 - 370 500 680 920 1600 2500 4000 - 600 - 420 - -175 -240 -320 -430 -100 -135 -180 -250 -330 - - O O O O 2500 -1100 -1500 - - 560 - 760 -1020 - 340 -290 460 -390 620 -520 840 -700 -200 -270 -360 -480 - 64 - 85 -115 -155 O O O 1 I 4000 6300 I 6300 -1250 - 56 75 -5m 10000 ~ - 2 0 0 0 ~ - 1 6 5 0 ~ - 1 3 5 0 ~ - 1 1 5 0 ~ 1 - 7 8 0 1-640 1-450 1-210 1 - 1 0 0 -940 1 O 1 i Table 2. Tooth thickness tolerances TSnin pm I I I Reference diameter (mm) over up to 10 50 21 22 23 8 24 12 20 Tolerance series 25 26 30 27 50 80 28 29 130 200 10 50 280 560 1000 I I 30 200 300 400 600 800 3 5 5 8 2 0 30 80 330 1 12 50 1 125 I 6 10 12 I 10 16 20 25 1 16 I 25 40 50 I 40 I 60 100 130 160 I 100 160 200 250 I 160 I 250 400 I 560 1 000 25 30 40 50 60 60 80 100 130 160 250 300 400 500 600 800 500 600 800 1600 2 500 4 000 16 20 25 30 60 80 100 1000 1300 1600 2000 1600 2500 4000 30 40 50 200 250 300 300 400 500 1000 1300 6 300 80 130 200 I I loooo I 40 I 60 I I 160 I 250 I 400 I 600 I lm I 1600 r2400 I www.bzxzw.com the tooth thickness tolerance series and the letter symbol of the series for the upper tooth thickness allowance.. although it should be borne in mind that the tooth thickness tolerance must be at least twice as large as the permissible tooth thickness fluctuation R. for example.``. With the closer tolerance zones it is therefore recommendable when calculating different test dimensions and their allowances t o apply appropriate corrections which take account of the influence of the individual deviations on these test dimensions empirically (statistically). Their choice is largely independent of the gear tooth quality.. 3.`. has to be watched for functional reasons. pin dimension measurement)will show that the tolerance is not fully complied with. This i s the tooth thickness in the normal section which.. since the individual measured quantities are affected differently by the individual deviations of the gear teeth a purely mathematical conversion of the tooth thickness allowances does not necessarily guarantee the required backlash. that acceptance testing by a different measuring method (e..DIN 3967 Page 3 3.. g. For an error-free gear there are mathematical relationships connecting the different measured quantities.. is not directly measurable. However. the tolerance series have been given the numbers 21 to 30. Quite generally it should be noted that small tooth thickness tolerances unfavourably affect the maintaining of gear tooth quality. As a rule for transmissions of the same kind it is possible t o choose the upper allowance for pinion and gear in all cases from a single allowance series.`. DIN 3966 Part 1. however. For calculating the allowance factors according to DIN 3960. g. calculation according to Appendix A will be necessary. Example: 27cd. Therefore indirect measurements are made by various methods. g. = . VDINDE 2608). see DIN 3960. The preferred series are 24 to 27. The tooth thickness tolerances are to be found from Table 2.com . this designation yields.`. 33 Tooth thickness tolerances .70 pm and A. see --``.`-`-`.`. since they unnecessarily limit the correction possibilities during manufacture (see. Sections 4..1 Upper allowances The upper allowances are to be taken from Table 1 independently of the reference diameter and the allowance series. in the case of tooth thickness tolerance zone 26e or coarser) the tooth thickness allowances can be converted directly into given test dimension allowances (e. it is also permissible however to select values from different allowance series. October 1976 edition.4 Information in drawings The limiting allowances can be indicated in the drawing either directly or by means of an code designation. the mean generating addendum modification coefficient xh.`. according to DIN 3962 Patt 1 I f a maximum backlash . - 4 Converting the tooth thickness allowances for the different test methods The system of f i t s is referred to a theoretical value. = 170 pm. It may then happen.bzxzw. The lower allowances are obtained by combining the upper allowances with the tooth thickness tolerances. In order to distinguish them clearly from the gear tooth qualities.`--- www.`. corresponding to the mean allowance should always be used. Where adequate experience i s available (e. the limiting allowances A . Guidance on determining correction values is given in Appendix B. however. 32 Lower allowances .. The symbol consists of the number of ..1. f o r d = 100 rnm for example.`.```. Since the upper and lower allowances are always negative the amount of the tolerance has to be deducted from the upper allowance. Their choice is largely independent of the gear tooth quality and should be governed by the manufacturing facilities.3 and 5. base tangent length allowances) and these used for acceptance testing the gear.```. ) For the particular application concerned..70 pm = . = . To cater for hardening distortion.517 1- +0. For series 27 Table 2 gives T = 1O0 pm and hence . and also to keep the grinding COR low.```.070 mm A.. (These values are algebraically smaller than the lower allowance 26 p n of the centre distance. a tolerance of 1O0 p n (Table 2 series 26) is adequate.) selected in this respect also (see Section 3 3 .com . for the gear. the tolerances are correctly . A.`.`--- www..``. --``. which meet the manufacturing requirements.. the tolerance for the pinion is made comparatively large. = 70 pm for the pinion and A.`..4000 1 I 492.`.`. Since the gear is milled.0.0.```. nesses are not unacceptably weakened - Since the tooth thickness fluctuation according to DIN 3962 Part 1 i s allowed to be 14 pm for the pinion and 25 pm for the gear. gear heat-treated and milled..`-`-`. Consequently A . the observance of a functionally imposed maximum backlash is not necessary.170 pn = .bzxzw..326 +O2389 7 I I 6 b 70 Pinion 16 MnCr 5 Gear 42 CrMo 4V ~~ Housing material Housing centre distance Housing width a GG 22 300 is 7 200 Pinion hardened and ground.Page 4 DIN 3967 5 Example Length dimensions in mrn Helical gears External Pinion I 5 Gear I I Normal module Number of teeth Standard basic rack tooth profile Helix angle Flank direction r.`.cesof tooth thickness of series cd are appropriate for this type of transmission.130 m m Apii2 =-230pm=-0.. It is assumed that it is known from experience that the upper allowan. 2 = 130 pm = .. = .`..`.230mm - - - The adoption of these tolerance zones.0. means that the tooth thick. n 2 I 1 ~ I 97 20 ~~ I ~ Gear teeth Tool DIN 867 DIN 3972 9? 53? 49? DIN 3978 Left Right ß d x I Reference diameter Addendum modification coefficient to DIN 3992 Gear tooth quality Facewidth Material 1 1 I 1 I 101.170 mm (lower allowance = upper allowance minus tolerance). According t o Table 1 the upper allowances are selected = 130 pm as A. 061 30 1.485 f 0. + 0.218 A.325 129.`--- www...bzxzw.4000 maenaum moaiTicarion Xmin I I .988 =9 i-'.3808 + 0.. 177.1935 I I I I Tooth thidcness sn mittel Sn min + 0.1399 97 8.1757 The above result in the following test dimensions with their allowances: Base taqent length Measured number of teeth Allowance factor Dimension over balls Dimension over rollers Ball and roller diameter Allowance factor Working distance with master gear Number of teeth of master gear (DIN 3970) 1 ) Allowance factor W 39..`.1 894 I +O9533 I +0..DIN3967 Page5 Number of teeth z Sn nenn 20 9. I + 0.7235 8.472 f 0.126 507.`.126 323.3670 + 0.314f0.962 f 0.1 899 9.." --``.297 1.```. .```. .2399 9.`-`-`.604 f 0.5935 8.`.047 3 0.099 K A& MdK MdR DM AMa a" ZL 9..2032 + 0.619 f 0.5435 8.`.670 f 0.`..`.2389 ~I.066 30 1...047 507.``.3099 9.`.com .940 117. .. as dictated by the slope of the tooth and the housing tolerance.`. --``..Page 6 DIN 3967 The calculated allowances of the measured values are shown in Fig. For use in . 1: these are values for ideal geometry..``..```..l and 8 3 . as dictated by the tolerance for the housing and other effects.`.) practical measurements they may need to be corrected..`..`. however. also turn out to be larger than the sum of the lower allowances.`--- Figure 1. and further effects (see Appendix A). The acceptance backlash may turn out to be smaller than the sum of t h e upper allowances. B.`.`.`-`-`. see Section 4 and Appendix B (Fig. It may. Tolerance zones of test dimensions after ideal-geometry conversion'of tooth thickness tolerance zone .```.`. 1.2 A.6 f Calculation of sum o upper allowances ZA* from minimum backlashjtmb the badclashand modifying effects Determining the upper allowances of tooth thickness in the normal section Calculation of sum of lower allowances ZA& from maximum backlashitmsx and the backlashmodifying effects Definitions Maximum backlash jt mu Calculation Determining the lower allowances of tooth thickness in the normal section Calculation of backlash from the tooth thickness allowances and the badclash-modifying effects Allowance diagram of tooth thickness in the normal section A.6. form a n d dimension deviations of components Elasticity Action of the backlash-modifying effects A.1 A.10 Acceptance backlash Example for determining the backlash to be expected A.3 Lower allowances without a specified maximum backlash A.4 A.4 A.7 Backlash modification through elasticity AiE A.1 Symbols and designations U A.6.l. Centre distance Facewidth Reference diameter Individual pitch deviation Axial skew over length LG Theoretical backlash Acceptance backlash Working backlash Maximum circumferential backlash Minimum circumferentialbacklash Module Relative water absorption (relative volume expansion) Addendum modification coefficient with mean tooth thickness allowance Centre distance allowance Upper centre distance allowance Asm Lower centre distance allowance Upper allowance of tooth thickness' in normal section Lower allowance of tooth thickness in normal section Upper allowance of tooth thickness in transverse section Lower allowance of tooth thickness in transverse section . Total profile deviation 1) Pitch span deviation over k pitches Concentricity deviation Tooth trace total deviation 1 ) Separation of bearing centres of a shaft Swelling of housing Swelling of gears 1) Measured according t o DIN 3960 in the transverse section tangential to the base cylinder.7 A.5 Backlash modification through swelling or contraction AiQ A.3 Backlash modification through non-parallelism of bore axes Aics A.2.6.2.4 Allowances under modified conditions A.9 Example for determining tooth thickness allowances A9.7 A.4.2.8 Calculationof the backlashmodifyingeffects Backlash modification through temperature rise Aje A4.2 Lower allowances A. A.102 Acceptance backlash A.9.lO.1 General A l .1 A.2 A.4.l A.5 A.4 Backlash modification through gear tooth individual deviations AiF A.4.4.1 Upper allowances A.2.1 A. A.3 A.5 A.1 General Symbols and designations Connection between backlash and allowances Backlash-modifying effects Temperature rise Centre distance tolerance of the housing Non-parallelism of bore axes in the housing Gear tooth individual deviations Swelling or contraction Position.3 A.9.4 A.6 A. .9.2.9.4..6.2.6 Backlash modification through position.2 A23 A.2 Backlash modification through centre distance tolerance Aja A.l A.4.5.l Determining the theoretical backlash A. it. b d fD fZß it it.DIN 3967 Page 7 Appendix A Calculation of tooth thickness allowances or backlash Contents A.2 A. A. form and dimension deviations of components AiB A.5 A. A 2 6 Position.```.1. Tooth thickness alterations through form alterations resulting from the shrinking-on of toothed components have to be treated separately. A. Further subscripts: 1 For quantities on the smaller gear of a gear pair 2 For quantities on the larger gear of a gear pair K When measurement with balls R When measurement with rollers A. since according t o DIN 3964 plus/minus tolerancing is used.DIN 3967 --``. but instead the backlashmodifying effects have to be taken into account in the calculation.`. particularly when the gear transmission is started up. the temperature rise of gears and housing is different.`.2 Centre distance tolerance of the housing Through this tolerance the theoretical centre distance is reduced or increased.``.`. It is often this component which is the largest of all. In this way the backlash is reduced or increased.. any subsequent alteration is generally so slight that it can be disregarded. If the material has been suitably pretreated (pre-swelled) prior to machining. if a specific minimum or maximum backlash is required.2.7 Elasticity The effect of elasticity consists mainly of a displacement in the bearings and housings and deflection of the shafts and housing under load. water..3 Non-parallelismof bore axes in the housing The non-parallelism of the bore axes in the housing may consist of axial inclination and axial skew.. Axial skew is always backlashreducing.2. and correspondingly the tooth thickness fluctuation also.`. Thus they have both a backlash-increasingand backlash-reducing effect. is referred via the reference circle to the gear axis..4 Gear tooth individualdeviations Gear tooth individual deviations may act differently at the circumference of the gear.`.. A.`. so that a summation of the maximum allowable values never occurs. A2..```. 1 Temperature rise A change of backlash through temperature rise occurs not only when gears and housing are made of materials having different linear coefficients of expansion. Conversely. hydrocarbons or other chemicals alters the backlash. The concentricity deviation need not be taken into account if the tooth thickness is toleranced.2 Connection between backlash and allowances I n contrast with cylindrical f i t s the backlash arising with gear tooth f i t s cannot be calculated directly from the allowances.2. form and dimension deviations of components Mainly involved here are concentricity deviations of bearings (internal and external diameters] and of fixed or rotating parts mounted in one another. because the tooth thickness.`. A. These deviations are in some cases inter-related.2 Backlash-modifying effects k 2 . since various backlash-modifyingfactors are effective.. this amount cannot simply be distributed over the allowances. and do so cyclically in the case of moving (rotating) parts. In each case however a backlash reduction is effective at one or more points in the case of individual tooth trace. but also due to the fact that. A temperature difference is equivalent to a change in the centre distance of the housing.5 Swelling or contraction The swelling or contraction of plastics in damp air. A.`--- Tooth thickness fluctuation Tolerance Tolerance of two-flank working distance Tolerance of normal chordal tooth thickness fluctuation Tooth thickness tolerance in the normal section Tolerance of diametral two-ball or two-roller measurement Tolerance of radial single-ball or single-roller measurement Base tangent length tolerance Pressure angle Normal pressure angle Transverse pressure angle Linear coefficient of expansion of housing Linear coefficient of expansion of gears or gear rings Helix angle Backlash modification through centre distance tolerance Backlash modification through form and dimension deviations of the components Backlash modificationthrough elasticity Backlash modificationthrough gear tooth individual deviations Backlash modificationthrough swelling or contraction Backlash modificationthrough temperature rise Backlash mcdif ication through non-parallelism of bore axes Temperature difference of housing relative to 20°C Temperature difference of gears relative t o 20 "c Sum of upper allowances of tooth thickness of gear pair in the normal section Sum of lower allowances of tooth thickness of gear pair in the normal section Sum of upper allowances of tooth thickness of gear pair in the transverse section Sum of lower allowances of tooth thickness of gear pair in the transverse section A. profile and pitch deviations and also with tooth thickness fluctuations.. It acts nearly always to increase . Axial inclination does not need to be taken into account because it is not allowed to exceed the centre distance tolerances and is thus covered by these.`-`-`.2. The deviations may accumulate or cancel. `.`-`-`..`.`--- Elasticity Figure A.```....`.`..l.`.`.`..```.cTemperature rise Centre distance tolerance Non-para1tel ism of bores Gear tooth individual deviations SwelIing or contraction Direction when calculating the upper lower allowances ~~ DIN 3967 Page 9 Remarks e----..``.-- t 4 for plus allowances for minus allowances e r---- QG > Q R QG < Q R Form and dimension deviations of components --``. Action of the backlash-modifying effects t 4 backlash-increasing + backlash-reducing - .. 1..3 (3) For calculating the maximum backlash the condition prevailing with perfectly parallel bore axes is the criterion In this case Aizp = O.`. I n the went of calculation for maximum badclash a different condition may arise and may result in a smaller or opposite effect. This m! be necessaw both at the reference a temperature of 20 C and also at lower temperatures.l.. only the dependence on module and gear tooth quality being taken into account. It differs according to whether the calculation has been made for minimum backlash or maximum backlash.. The backlash-reducing component AiF is therefore calculated according t o the error propagation law as follows If A j s is positive. AjF = - cos a t cos at (4) Backlash modifi&on through centre distance tolerance Aja I n the calculation the least favourable allowance has to be taken as the basis each time and given the appropriate sign. When taken into account in the calculation it results in the acceptance backlash becoming smaller.``.4.2 For the maximum backlash the least favourable case would arise if no deviations were present.4.`-`-`.. for the maximum backlash in the case of external gear A.```.1. Since the allowances are always negative this is represented in the schematic by a downward-directed arrow.. tan a. for the minimum backlash and A. see also Section A.2. The dependence of the parameter AjF on the reference diameter and facewidth is negligible. For a! = 2Q0therefore the rounded values for Ais can be taken from the Table A. InthiscaseAûG isequal t o A 6 a .```.`.`.4. cosß Backlash modification through non-parallelism of bore axes AjzB The effect of axial skew is the same as that of tooth trace angle deviations. and A . With gears however this is never the case.`. for the minimum backlash and Aai for the maximum backlash in the case of internal gear pairs. and this has to be separately calculated.3 Action of the backlash-modifying effects The action of the backlash-mcdifying effects is shown schematically in Fig. Table A.. a backlash increase takes place..`.4. pairs. This means A . For the maximum backlash therefore only '12 AjF is effective. The backlash modification is calculated from A. Aja z 2 * A .4 Backlash modification through gear tooth A..`. At best the gears have a deviation which is equal to half the deviation permissible for their quality.`.`--- . 1 Rounded values of AjF in pm --``.4 Calculation of the backlash-modifying effects Since it is necessary when fixing the tooth thickness allowances to start from a stipulated backlash (working backlash) the backlash-modifyingeffects are calculated a backlash modifications s Backlash modificationthrough temperature rise Aja The following applies with adequate accuracy: A. The worst case condition is considered each time. A. In the condition at rest a reduction of backlash can then arise.1 individual deviations AiF The following are taken into consideration: a) tooth trace deviations b) profile deviations c) individual pitch deviations It is unlikely that all three deviations will be effective at the same time t o their full value. A.Page 10 DIN 3967 backlash in the operating condition. O - A. In the case of minimum backlash each backlash reduction demands an increase in the amounts of the tooth thickness allowances. ``.`. form and dimension deviations of components AiB These act like centre distance deviations and are therefore calculated according to equation (2).`.(Ais).4. This applies in particular to the coarser qualities in which the highest spots of the tooth flanks are present only at a few points and therefore wear more quickly than in the case of finer qualities. then the relative linear expansion is approximately '3 w and the following backlash modification / arises: A.6. or with empirical values.(A~E) + A& + Aj:) (6) (For positive backlash the allowances are negative. the displacementsof the gears..`.. The sum of the upper allowances is first calculated in the transverse section.6 Backlash modification through position. so long as the condition of equation (8)i s complied with.2 Maximum backlashit. 1 +Asie 2 = U* COSß (8) A.```. therefore it is often possible to determine the sum of the lower allowances without calculation. The calculation of the backlash modification follows the lines of equation (2). It shoutifnot be chosen too small. When determining this backlash it must be borne in mind that all backlash-modifying effects are dealt with by the calcurafion. The principle to be applied here is that impairment of the strength at the tooth root is to be avoided as far as possible. This i s the smallest Circumferential backlash which must be present in the completed gear transmission in the least favourable operating condition.6. The minimum backlash can thus be kept small.`. 1 + A . Hence.5 Calculation of sum of upper allowances & from minimum backlashit msn i * and the backlash-modifying effects The calculation is based on the minimum backlashjtdn.. transmissions for instrumentation purposes.(AAJ + VAjE + Aj5p + Ajil . First. are calculated. A.7 Backlash modification through elasticity AjE This component depends on the service loading and has to be determined according to the design circumstances. Narrowing of the backlash should be undertaken only if the function of the gear transmission demands t h i s (actuator transmissions. If both gears are of plastic material and if w is the relative water absorption (e. A. The rules which apply are the same as those for determining A.`--- Reference values for the relative water absorption w can be found from the data published for the material concerned by the plastics-manufacturers. for one of the gears the upper allowance O can also be adopted. The tabulated values are the upper allowancesof tooth thicknesses in the normal section and apply to all modules and all qualities.3 Calculation The interaction of the effects is shown in Fig. The selection should be made in such a manner that the amount of the sum of the selected allowances is a t least as large as the amount of the sum calculated according to equations (6)to (8). U* s .`.4. When determining this backlash it should be borne in mind that all backlash-reducingeffects are dealt with in the calculation. I . These are the same as apply to the minimum backlash (see Section A..) The individual backlash modifications are to be inserted with the sign found for them. I n the calculation the signs have to be watched: swelling i s to be taken as positive and contraction as negative. In all other cases the only determining factor is a possible reduction of the root strength of the teeth through diminished tooth thickness. Some of these components however do not a c t simultaneously to the full extent.l.DIN 3967 Page 11 Backlash modification through swelling or contraction AjQ The effect is the same as that of temperature rise.(itmio .6. w = 0..1 Determiningthe upper allowances of tooth thickness in the normal section The sum of the upper allowances in the transverse section calculated with equation (6)or (7) has to be converted to the normal section --``. which affect the centre distance deviations. A. The interaction of the different effects is shown in Fig.`. A.. A. It is immaterial how the sum of the allowances is distributed between the two gears.6 Calculation of sum of lower allowances Z A from maximum backlash jt mm and the backlash-modifying effects A.`. g.`-`-`. Similar considerations apply to the swelling and contraction of other components which influence the backlash. A . A...5).1 Definition The sum of the lower allowances of tooth thidmess of the gear pair in the transverse section is calculated from the maximum backlash which has not to be exceeded in the completed gear transmission when the backlashmodifying effects are operative.02 A 2 percent by volume).4. From Table 1 a suitable value has to be chosen consistent with the calculated sum A . gear transmissionswith non-uniform drive or alternation of loading direction).```.5 For normal cases in general mechanical engineering A ~ Q AjE = Ajjg = O can be adopted and the equation = simplified accordingly as follows: (7) A.5.. This is t h e largest circumferential backlash which may be present in the completed gear transmission in the least favourable operating condition.l. These have therefore to be allowed for according to the error propagation law. `-`-`.2. Selective assembly of gear A. the minus sign should be used before the square root.... ifthe value in quation (15) between the vertical lines is negative. corresponding to their size.`. t h e case may arise in which the tooth thickness tolerance i s no longer consistent with production requirements. The backlash modification through unequal temperature rise acts to reduce the badclash at Ajo <O (lower part of the diagram). In this case therefore a tooth thickness tolerance consistent with the production requirements frequently demands a higher accuracy class for the axial position. Since both allowances are negative.Page 12 DIN 3967 the upper allowances.```. Hence the following is obtained: and mating gear also makes possible an increase in the tolerance. Since some components may act with both backlashreducing and backlash-increasingeffect.4 Attention should be paid to the remarks concerning Aja in Section A. since in this case the tolerances do not add together in full.2 shows how the backlash and backlash modifications are constituted and how the sum of the allowances and tolerances of the two gears results therefrom.`.C A 2 (111 At the same time a check must be made t o ensure that the tolerance is equal to at least twice the tolerance for the tooth thickness fluctuation -..cosa.e.`. In the calculation the individual backlash modifications have to be inserted with the correct sign.7 Calculation of backlash from the tooth thickness allowances and the backlashmodifying effects With prescribed allowances . in accordance with the system of fits of this Standard ..`.`. With equation (9) the case may arise in which the amount of the sum of the lower allowances is larger than the maximum backlashjtmax. COS& Ti + T 5 ZAme . but instead overlap partially or completely. To arrive at a larger tolerance. otherwise the plus sign. A. = AiB = O can be adopted and the equation thereby simplified as follows A. the equation changes as follows: jt --``. the plus sign in front of the square root sign should be used.```. The allotment of the lower allowances should be made in such a way that. g. and if so at what cost. For normal cases in general mechanical engineering AjQ = Aj. From Table 2 tooth thickness tolerances are selected such that the sum of the tolerances for the two gears can be A6. The starting point is zero allowance. The individual backlash modifications are to be inserted with the signs determined for them.8 Allowance diagram of tooth thickness in the normal section Fig. otherwise the minus sign. (12) -..i Fß2 cosat cosat 2 cosat The lower allowances then result from the upper allowances and the tolerances A .4. If the value between the vertical strokes under the square root sign yields a negative figure.``.the working temperature can be influenced.-. the backlash-reducing components of the minimum backlash must be reduced and likewise the backlash-increasingcomponents of the maximum backlash.m&++ja+Ajzß it- - +AjB---.. a check should be made to establish whether. The upper allowance must therefore have . the amounts of backlash (addition of the allowances) are also entered as negative values. the two gears are given tolerances consistent with the production requirements. the diagram can only serve as an example and does not apply to all cases. in other words it is necessary to weigh the cost of production of the gears against that of the housing.`.`.`--- - =- EA* Fßl 332 + Aja + Ajxp i-AiB .. If it i s impossible or undesirable to increase component accuracy any further. (IO) Determiningthe bwer allowances of tooth thickness in the normal section The sum of the lower allowances in the transverse section calculated by equation (9) i s converted according to equation (8) to the normal section. If the individual influencing factors are present as actual values.the prospective acceptanceor working backlash i s calculated with the backlash-modifying effects taken into account. = A me -T (13) If the upper and lower allowances are determined on the criterion of function.Fßl T12R. ............. ...... t* s 2 2 o O LHzAsn . .`................```. .....``. ..`-`-`..`... ........g o c - ZAsni . - ...`--- ...... --``..`.....```.`... ...`..`. 1-".. ....DIN 3967 Page 13 -L ...`. L A & = .20) 10 tan 2 0 ' * 10-61 2 cos 9 8 6 ' ..061 mm==-60pm According to equation (2) the backlash modification through centre distance tolerance is AjF = 1 ..20) 11.60) . = O. 0 Ajo = 300 [(50 .75 pm (series e) A. The sum of the lower allowances may therefore amount to 310pm if a maximum backlash of 300 pm is to be guaranteed.O - 2 fi. The upper part of the diagram shows how the allowances develop when Aja and &Q are larger than O (e.(20-(. Aj. = O is adopted. After all components have been taken into account the totals of the upper and lower allowances in the transverse section result. i...`.. The minimum backlashjtmhi s taken as 20 pm. - AjD = 300 [(80. The sums of the allowances are distributed according to design considerations (tooth thickness).9.5 pm T Aja = 2 (.045 mm = .19)2+ (.l the backlash modification through gear tooth modification is: =-0.52+ 9 3 2 .`.9 Example for determining tooth thickness allowances Same gear transmission data as in main part. According t o equation (1) the backlash modification through temperature rise is c) According to equation (2) the backlash modification through centre distance tolerance is Aja = 2 26 - tan 20" cos 9 8 6 ' .`.4.. g) For t h e backlash modification due to position. similarly backlash increases are directed towards the zero line. form and dimension deviations of the components. 19=9.(300.45 pm - . i) According to equation (9) sum of the lower the allowances in the transverse section is According to equation (3) the backlash modification through non-parallelism of the bore axes is 70 Ajxp = .2 Lower allowances a) The maximum backlashjtmau taken as 300 pm.1 g2 + O2 + 9. Section 5 .45).`.9. 1 gives AiF = 19 pm Since steel and cast iron are not subject to swelling.1 Upper allowances . is the b) According to equation (1) backlash modification through temperature rise is A. corresponding to the helix angle. Ai.99 = 19 pm.40 pm (series e) AsieZ= .. g.19 pm f) Since steel and cast iron are not subject to swelling.310 pm .99 = .`. - .- .99 .5 - A. of cast iron 15 (yG = 10 Maximum backlash 300 pm.`.``.Page 14 DIN 3967 a larger absolute value than would be necessary for the minimum backlash only.20) 1 lW6] tan 20' cos 9 8 6 ' . --``.. perature difference between housing and gears 2C a t 0 ' 70 O C gear temperature.(90.```.4 and Table A. form and dimension deviations of the components. .99 k) According to Table 1 t h e upper allowances are selected such that their sum amounts to at least 115pm: &e 1= . in the case of steel gears in plastics housings).0.20 -=200 - 7 Pm Table A. h) The calculation of shaft deflection yields a backlash modification AJE = 15 Km. the following is assumed: AiB = 15 pm. It therefore has t h e effect of increasing backlash. The component Ai+ therefore has to be indicated in the direction away from the zero line.99 =. Coefficient of linear expansion of steel (yR = 1 .(70.15 The backlash modification through elasticity A ~ E is disregarded here because it is assumed that in this case it is not backlash-reducing. but also gear with gear tooth quality 6 Maximum tem.`-`-`.20) 11.1521) = ..26) - tan 20" cos 9 8 6 ' . housing 80 O .`.```.117 pm j) The conversion to the normal section according to equation (8)yields Ume.115 Dm = . the following is assumed: AjB = 15 pm. e. The difference of t h e two calculated sums i s the sum of the tolerances of the two gears which is distributed with due consideration of production requirements.0-0 + v(. For the backlash modification due to position.117 * cos 9 8 6 ' = .(.`--- i) According to equation ( 6 ) the sum of the upper allowances in the transverse section i s EAe5 . transmission temperature a t full C load: gears 90 O C . like any other backlash reduction also. AjQ = 0 i s adopted. The sums of the allowances in the normal section are smaller.5 d) According to equation (3) the backlash modification through non-parallelismof the bore axes is Ajzp = O e) According to Section A.7)*+ 1g2 + 1g2 + (- 1512) = . In the example therefore 20 + 60 = 80 pm.15 = 330 pm.174 pm). A.8969" =0. this curcumstance must be taken into account.5 Acceptance backlash If checking of acceptance backlash is proposed.52 = .~E therefore the maximum acceptance backlash is 300 + 45 . A& = A 6 6 = 50 O C will apply. the condition according to equation (121 is fulfilled.. A. of must be taken into account.9. i.141 mm= 141 pm Hence. the permissible maximum backlash can be exceeded in the cold condition by the corresponding Aje although. As regards the maximum acceptance backlash.5 1O-6) 2 tan 20" cos 9. .100 = 175 pm. 3 shows the size of the backlash and backlash modifications. it must be borne in mind that when the gear transmission is cold then the minimum acceptance backlash must be larger than the minimum backlash by Aja. so that C A . with vehicle transmissions. the conditions are changed fundamentally.- - Hence the sum of the upper allowances in the transverse section is BA~=-(20+1/(-19)2+(-7)*+192+192+(-15)2) = . I n the example course. 1= > > This however would make the sum of the lower allowances (. so that backlash is still present a t this temperature. according to equation (9) the sum of the lower allowances is E A & = .1521) T 28 prn T 36 pm i 2 On the basis of production requirements tolerance series 27 is chosen. = Equation (13) gives the lower allowances as A. . 2 = 235 pm - A. 1O-6 is used. so that there i s no tolerance. 40 60 = 100 pm and Asni2 = 75 . The worst condition for minimum backlash therefore occurs a t reference temperature 20 OC.- - - A.1g2+ O 2 +9.!j2 + 9.30 O C . for example. 2 = 100 pm ' 1 T2= 160 pm From equation (13) the following are found A.310 cos 9.9.8969' = . the tolerances are selected freely according to equation (12) and Table 2.3 Lower allowances without a specified maximum backlash If no maximum backlash is specified.140 pm A. The backlash modification through temperature rise Aje acts to increase backlash. Here A j e = O. is changed to 177 pm. as may be the case. For this case therefore the specification for the maximum backlash needs to be checked and a design modification undertaken if necessary. - - - .9. - d i .DIN3967 Pagel5 j) The conversion to the normal section on the lines of equation (8) yields XAe = . I n this case therefore the minimum backlash must be made a t least 140 pm.70 11. The backlash modification through temperature rise from 30 O C to + 20 "C alone amounts to 138 Frn.5 7 pm If the gear transmission i s to be exposed to relatively low temperatures in the idle condition.124 pm . Fig.4 Allowances under modified conditions If instead of a grey iron housing a light metal housing with linear coefficient of expansion (Y = 24 . A. For calculation of the lower allowances there is a backlash modification through temperature rise Ajo = 300 (60* 24 . At a temperature of .(300-141 15 CAsiel=l-305-(-115)1=190pm From Table 2 the following are selected from tolerance series 26 for the pinion and gear 2 = 60 pm and T2= 100 pm 1 ' Ti+ T2= 160 pm According to DIN 3962 Part 1 the tooth thickness fluctuation is allowed to be R S 1= 14 pm R S 2= 18 pm The toterances are thus more than twice as large as the tooth thickness fluctuation. e.305 pm k) According to equation (11) the sum of the tolerances for both gears is 5"1+2'2=IU.124 pm) larger than the sum of the upper allowances (. 3. Backlash.Page 16 DIN 3967 Caiculation Execution 1 I Figure A. backlash modifications Allowances and tolerances for the worked example (the components under the root sign of equations (7) and (9) have been combined to a single amount) . 8969' tan 20' = 3 Therefore according to equation (15) t h e maximum ..10.(..26 pm and A .3 Ajzß = O 2.2.`.(+ 26) cos 9.(70) (.10.203) + k 19) = 184 pm . = = .9.1 5 p m 2. According to Section A.cos 9. According to equation (2) the backlash modification due to the centre distance allowances A ..``.712 + (19)2 + (19)* + (+ O = 166 pm U& = - A.1 the following apply Ajzp = .`. According to equation (14) therefore the minimum backlash is found as U = .4.```.406 pm cos 9.406) =-19pm =+19pm - --``.`-`-`.t .8969" 400 CA. Conversion to the transverse section according to equation (8) yields -*Oo =-203pm cos 9. According to Section A.1 the theoretical backlash is jtmm = . = + 26 pm is calculated as A j = 2 (..(- 203) 15)2 -fi. = (.10. Backlash modificationdue to non-parallelism of the bore axes according to Section A.`.. Ai.406) + (19) =425 pn +0=426pm The theoretical backlash of it = 1 8 4 pm to 425 pm is modified by the backlash-reducingeffects to become it= 166 pm to 426 pm....`. + Ud = .2 the following apply AjF1 = AjFZ = 19 pm AjB = 1 5 p m 3.200 pm .2 Acceptance backlash Aja.1912 + (.DIN 3967 Page 17 A. = . According to Section 2.400 pm 2. AjE are left out of consideration when determining the acceptance backlash.10 Example for determining the backlash to be expected System of fits DIN 3967: pinion 27 cd gear 26 cd Centre distance allowances DIN 3964: I S 0 tolerance zone js7 Gear tooth quality DIN 3962: 6 For further data see Section A.26) ~ Ai.8969" it mh= .1 Determiningthe theoretical badclash 1.`--- 4.`.```.10.(.7 Pm AjFl =hjF2 19 pm = AiB = . = .I .230) = .`.130) = .1 Minimum backlash 1. A.9 A.. The sum of the tooth thickness allowances is A.2 Maximum backlash 1.(170) + (.2.8969' tan 20' 2 .9. acceptance backlash is j t .`. .. 7.7.`..`.``..1 Theoretical position of the tolerance zones B.l B..3 Actual allowances of normal chordal tooth thickness B. B.9 Symbols and designations as in Appendix A Additionally: Two-flank working distance Facewidth Module Tooth thickness Normal chordal tooth thickness on the y cylinder Addendum modification coefficient Number of teeth Allowance factor of the two-flank working distance Allowance factor of diametral two-ball or tworoller dimension Allowance factor of radial single-ball or singleroller dimension Total tooth trace deviation Diametral two-ball or two-roller dimension Diametral two-ball dimension Diametral two-roller dimension Radial single-ball or singleroller dimension Tooth thickness fluctuation Tooth thickness fluctuation from two-flank working distance Tooth thickness fluctuation from chordal measurement on t h e y cylinder Tooth thickness fluctuation from two-ball measurement Tooth thickness fluctuation from two-roller measurement Tooth thickness fluctuation from base tangent length over 4 measured teeth Tolerance of two-flank working distance Tooth thickness tolerance Tooth thickness tolerance for two-flank working distance Tooth thickness tolerance for the chordal measurement on the y cylinder Tooth thickness tolerance for two-ball measurement Tooth thickness tolerance for two-roller measurement Tooth thickness tolerance for the base tangent length Tolerance of normal chordal tooth thickness on the y cylinder Tolerance of t h e diametral two-ball dimension Tolerance of the diametral two-roller dimension Base tangent length tolerance Base tangent length over 4 measured teeth Transverse pressure angle Helix angle Tooth thickness half angle .`.3 Determining the tooth thickness from the tooth thickness angle Measurement of the normal chordal tooth thickness Measurement of the working distance with the master gear Measurement of the base tangent length Measurement of the dimension over two rollers or balk B.7..7.6 Actual allowances of two-ball measurement B.```.7 Actual allowances of two-roller measurement B.`..2 Actually determined position of allowances B.4 Actual allowances of working distance B.5 B 6 Singleball measurement and singleroller measurement ..8 Tooth thickness fluctuations Reliability of results B.2 8..7.`.Page 18 DIN 3967 Appendix B --``.```.`.7.4 B.`--- Conversion of allowances for the different measuring methods Contents B.`.7 Determining the correction values B.5 Actual allowances of base tangent length measurement B.7..`-`-`. 1 Determining the tooth thickness from the tooth thickness angle The tooth thickness can be measured by mechanically tracing the two flanks of a tooth in the V circle by means of a measuring pick-up (as null indication) in conjunction with an angle measuring instrument. Page 19 over k teeth however enter into the measurement. so that the tooth thickness is determined in accordance with the definition. since t h e measurement is referenced to t h e mounting axis of the gear. The criterion for the lower allowance is Fp (Appendix A.3 Measurement of the working distance with the master gear Apart from the concentricity deviation of the gear teeth. All the measured values were converted to tooth thickness values and presented as sudi.l. This illustration shows (right) how different in position and size are the allowances and tolerances of the measured values although on an ideal-geometry basis they express the same tooth thickness production tolerance. the eccentricity of the gear teeth is not covered. The tooth trace deviation however has already been taken into account in calculating the tooth thickness allowances. Of all the measured values of each measuring method the weighted average was determined and presented each time... Section 5.`.`. October 1976 edition. The tooth thickness is found by converting the measured tooth thickness angle 2 $ into radians.DIN 3967 8.4).1 Theoretical position of the tolerance zones As an example. similarly the measurement over balls or rollers Md.also yields negligible differences. It i s therefore recommended that measurements be made in the various production areas and the correction values determined from the actual values obtained by the different measuring methods.``.October 1976 edition..`. this method i s not suitable for practical application and i s only of significance for scientific investigations.. the normal chordal tooth thickness i s measured in the normal section a t a given depth.This shows both the fluctuations of the individual measured values in one and the same gear and also the differences of the gears one from another. Fig. B. shaved. Where large numbers of teeth or virtual numbers of teeth are involved the conversion of the allowances from the reference circle to t h e V circle according to DIN 3960.4.7. for the tooth to be measured. the tip circle radius with reference to the gear mounting.`--- B.7. the allowance factor A& may alter to such an extent that it can no longer be disregarded. The mean measured value u" and the allowance factor A:H are to be calculated as in DIN 3960. it i s necessary in this case also to add a correction value to the conversion factor. The results of the measurements on all five gears were combined (right-handside of illustration) the offset of --``. It is therefore appro priate to calculate it for the mean value of the tooth thickness allowances and not for the nominal dimension of the tooth thickness (see DIN 3960.`. If this i s larger than or equal to the width of t h e working gear all tooth trace deviations will be covered. the upper and lower actual allowances of the normal chordal tooth thickness are found.. It must be borne in mind however that where relatively large allowances amounting practically to an additional addendum modification are involved. Section 4. if it is smaller an allowance in proportion to the two widths is sufficiently accurate. and the allowance factor AM. Therefore in order to be sure that the tooth thickness i s not exceeded a t any point on the gear. Since angle measuring facilities of the necessary accuracy are usually not available in industry. B. B. Nevertheless to avoid obscurity the reference should always be to chordal measurement.2 Actually determined position of allowances The results of measurements on five gears (milled.2 Measurement of the normal chordal tooth thickness On t h e basis of a reference diameter.4 Measurement of base tangent length This measurement does not cover the eccentricity of the gear teeth relative to the gear mounting. Section 5. they also cover eccentricities and are thus equatable with the chordal measurement.2.5 Measurement of dimension over two rollers or baIIs Here.`-`-`. t h e expression tooth thickness measurement should beavoided in thiscase. The same applies to the base tangent length fluctuation. Conwersion of the too* thickness allowances from the arc to the chord can normally be dispensed with. too.```.. Section A. The mean measured value M.. With the concentricity deviation of t h e tip circle taken into account and a sufficient number of measurements made on the gear circumference.`. B. are to be calculated as in DIN 3960. Hence although theoretically the tooth thickness tolerance could be converted with a factor for the base tangent length tolerance.`.`. The upper allowance is displaced according to the master gear width. normally on the V circle. it is not possible to make any general statements. which can conveniently be the tip diameter. 8. B. Pitch deviations . this measurement also covers the tooth trace deviation up to the width of the master gear. Section 51. The eccentricity of the gear teeth can be found. 6.```. October 1976 edition. . For this purpose it is necessary to determine beforehand. B. case hardened) on all teeth are compared in Fig.. Consequently before converting it is necessary t o apply a plus correction to the permissible allowances by t h e amount of the tooth trace deviation.l. it is necessary here to add a correction value. 8. 1 shows the tolerance zones for a given gear.7 Determining the correction values Although the influences which make correction values necessary are known. October 1976 edition.6 Single-ball measurement and single-roller measurement If these measurements are made radially from a reference diameter or from centring elements which can be equated with the gear mounting. neither the minimum backlash nor the maximum backlash would be guaranteed any more.7.2. since the causes of this have already been taken into account in the calculation of the allowances.) Fig. I f the maximum value of the working distance is within the theoretically calculated tolerance the minimum backlash is guaranteed.7. B. If the allowances of the chordal tooth thickness measurement were to be converted purely theoretically as allowances for the base tangent length measure- . 6. These are not taken into account in the calculationof the a!lowances of tooth thickness.Page 20 DIN 3967 ment (Fig. This security of compliance with t h e minimum and maximum backlash however is bought a t the cost of making the tolerance small. gears 3 and 5 also the limits mown by dashed lines in . This however is taken into account in the calculation of the allowances. Hence if an offset occurs the tolerance zone can be displaced accordingly.7. The mean offset relative to the normal chordal tooth thickness measurement is thus in this case + 1 pm. Since the width of bearing on the tooth i s larger than with the chordal measurement a tooth trace form deviation could exert influence.2.7. - - - 6.7 Actual allowances of the two-rolier measurement Here. It is therefore f necessary to alter the site o the tolerance if it i s to be guaranteed that all tooth thicknesses are within tolerance when checking is carried out by way of only individual measured values (and not over the whole circumference). and ought for this reason to be the preferred measurement. The error propagation law can of course also be applied here. The mean value of the measurements is not displaced compared with the chordal measurement. From these the correction value for the tolerance centre of each method of measurement can be determined. i. The lower allowance must however be displaced by the amount 2 cosq' f Fß - 6. This makes the measurement "thicker" than with the two-ball measurement. The cause of t h i s lies in the profile and tooth trace deviations. B. to determine the ball and roller diameters approximately for the measurement points of the base tangent length measurement. Apart from this however the measurements on the individual gears yield fluctuations differing widely in magnitude. 6.6 apply. and also in the fact that the measurement circle differs and in this way profile deviations become effective.3 Actual allowances of normal chordal tooth thidtness In the example the fluctuation of the measured values on one and the same gear ranges up to 55 pm. The tolerance must therefore be reduced not only by the concentricity tolerance but also by the pitch-span tolerance. The upper allowance of the working distance measurement can thus be displaced relative to the normal chordal tooth thickness measurement by the amount determined. confirms the concentricity deviation. B. The tolerance can be displaced by the offset compared with the chordal measurement. according to measuring method. The reason for t h i s may lie in the fact that only point contact is made with the flanks. which points to a concentricity deviation. Example: With gear 1 the fluctuation of the measured values of base tangent length ranges from + 11 pm to . without this endangering the stipulated minimum backlash.5 Actual allowances o the base tangent f length measurement Although the concentricity deviation does not enter into this measurement the fluctuation of the measured values is larger than with the two-ball and two-roller measure ment This is due to the fact that the pitch-span deviations influence the measurement. the minimum backlash is guaranteed. since eccentricity dÒes not enter into the result. the mean value of a particular type of measurement relative to the mean value of the normal chordal tooth thickness measurement of the gear concerned being shown each time. Compared with the chordal measurement however the mean value of the measurements i s offset by + 10 pm. These however have already been taken into account in the calculation of t h e tooth thickness allowances. 8. This affects in particular shaved gears which may have relatively large pitch deviations which are not always allocatable to the concentricity deviations (see Fig.11. converted to the tooth thickness.4 Actual allowances of working distance The fluctuation of the measured values of the gear.7.e. This shows that the concentricity deviation determined is attributable to eccentricity and not t o out-of-roundness.7. It i s of course possible for tooth trace form deviations to have an effect. For the fluctuation of the values the weighted average was again determined. For t h i s value the fluctuation for this offset is between + 10 pm and 7 pn for all five gears (see Fig. The offset relative to the weighted average of the normal chordal tooth thickness is 7 pm. 6 3 . and to calculate the normal chordal tooth thicknesses also for this. If the theoretically calculated tolerance were to be utilized. as regards t h e functional conditions. it is also the method which involves reducing the theoretical tolerance by the largest amount in order to guarantee the backlash in the same way as with the chordal measurement. . The weighted average of this fluctuation is + 1pm.6 Actual allowances of the two-ball measurement The fluctuation of the measured values converted to the tooth thickness is significantly smaller than i s t h e case with the foregoing measured values. The tolerance has therefore to be reduced by the amount of the permissible concentricity deviation. The weighted average is 10 pm. The mean value of the measurements is slightly displaced compared with the chordal measurement. right). the same remarks as in Section B. Although in general the base tangent length measurement i s subject t o the smallest measuring errors. neither the minimum backlash nor the maximum backlash would be guaranteed if t h i s tolerance were to be utilized to the full. in t h e two-flank working test.27 pm. compared with the other measuring methods. If the maximum value is in tolerance. For comparative measurements it is therefore expedient to carry out all measurements at approximately the same points on the flanks. so that in come cases objections may be made against gears which are found to be serviceable. %- Tolerancezone of specifid tooth thickness Derived tolerance zones o f test dimensions 0. The tolerances stated in DIN 3962 Part 1 apply to the fluctuation of the tooth thicknesses.DIN 3967 Page 2 1 The tolerance must however likewise be reduced by the amount of the permissible concentricity deviation.. 8. For other measurements these values have to be converted. Tolerance zones of test dimensions after ideal-geometry conversion of tooth thickness tolerance zone t = 29. Therefore so long as no firm conversion figures are available it is recommended when using the tolerances to the full t h a t a check should be carried out by measuring the normal chordal tooth thicknesses or the working distance with a master gear in order to avoid difficulties at acceptance. B.`.```. B..051 7. The zero line corresponds ' 0 with zero-backlash design of the gear transmission. Example: m = 4. The mean allowance .l necessary from the evaluation of Fig.8 Tooth thickness fluctuations Tooth thickness fluctuations are often measured a flucs tuations of the base tangent length and in some cases also as fluctuations of the two-ball or two-roller dimension. For this conversion the same applies as for converting t h e tooth thicknesses themselves.. should however be carried out where possible.7.2 (right) are shown in Fig.0.4 to B.2063.the mean tooth thickness is then s = 7. The modifications of Fig.25 mm.`.l. For quantitative statements not only is the number of measurements too small.`-`-`. B. --``. B.`.314 mm.`--- .154 mm corresponding to x = 0. but also it i s not permissible to transfer conclusions from one size of gear to others. I r.7.7.9 Reliability of results The relationships of the different measuring methods shown here by way of an example can only apply qualitatively... tooth thickness for this s = 7..3.``... On the one hand this will lead to the backlash effectively obtained being largely the same with all measuring methods. fi = . I I Z 'y a 'TW' 'd U. or from one method of manufacture t o another.`. Use of the theoretical allowance factors alone can thus lead to errors. as described in Sections B.`.1546. and on the other hand the necessary production effort also. The tolerance modifications of the given tooth thickness tolerances.`. It is therefore necessary to have a large number of measurement series for the widely differing areas before statistically affirmed correction values can be stated for the conversion. O = 22 mm.`. 'MdR Figure B.```. x = 0.160 pm corresponds to Ax = . Page 22 DIN 3967 L O . 200 -300 -LOO . and the allowances of the test dimensions determined therefrom.2 (right).`--- .`-`-`.5) were calculated for position and site according to the evaluation in Fig.`.500 Figure 6.`.. --``.`. Section 8. Altered tolerance zones of tooth thickness and resulting tolerance zones of the test dimensions On the basis of the unmodified tooth thickness allowancesof Fig..`.DIN3967 Page23 Tolerancezones o f specified tooth thicknesses Derived tolerance zones of test dimensions O Pm ...```. cf.`.`.3. see upper illustration (left)..```.`. B.``...7. These tolerance zones result in approximately equal production effort and the same effective backlash. The limits shown by dashed lines of the tooth thickness tolerance for t h e base tangent length (TSw)and base tangent length tolerance (Tw) are needed for shaved gear teeth (see Section 7.l for the chordal measurement the tooth thickness tolerance zones for the other test dimensions (apart from the base tangent length measurement..100 .5 and Explanations). B. The parameters. In any case it is recommendable to determine correction values by measurements. When values obtained from experience for the backlash or tooth thickness allowances are available or when for functional reasons no exact determinationof backlash is necessary.`. Assuming a sufficiently large number of measurement series for the most diverse gears and manufacturing methods there is a possibility that the offset of the tolerances for measurements by way of working distance.`-`-`.`. It is left t o the user t o establish a selection system or reference values for the backlash concerned. The allowance series (Table 1) are based on a progression of certain series near the zero line being omitted. base tangent length or the two-balldistance can be kept the same relative to the normal chordal tooth thickness. vet despite this the gears are within tolerance (see limits shown by dashed lines in Fig. Within an allowance series the upper tooth thickness allowances have a progression of "@ The calculated values have been rounded according to ISO/R 286.Page 24 DIN 3967 Explanations With this Standard the system of fits for securing the minimum backlash and limiting the maximum backlash is placed on a broader footing compared with DIN 3963 and DIN 3967 dating from 1953. It is also intended to stimulate the collection of practical experience. The previously standardized allocation of tooth thickness tolerances to gear tooth quality has thus been dropped.. The progression from one tolerance series to the next is 1.`.```.3).``.`.`--- '"a. To simplify use of this Standard the main text i s followed in Appendix A by information on the calculationof tooth thickness allowances and in Appendix B by information on the conversion of allowances for the different measuring methods. Empirical values for specific systems of fits involving given gear transmission categories could not be laid down in a uniform system by the responsible committee because with differing diameters no uniform minimum ba&lash can be stated. The proposed system of fits is regarded as a general basis. It is also possible that the tolerance for the base tangent length need not be so severely restricted a indicated in s Section 5 of Appendix B. the accurate calculation of the tooth thickness allowances is superfluous. but instead has to be superseded by indirect tooth thickness measurements using different methods.`.. Otherwise.. since in the case of the five gears measured the base tangent length is not always within the calculated reduced tolerance. so that further general pronouncements can be made on the system of gear tooth fits..6. DIN 3998 and DIN 3999. It has also not been considered expedient to adopt coding of the tolerance values with multiples of the individual pitch deviation fpt as contained in the InternationalStandard IS0 1328 1975.`. - --``.. .. symbols and designations of this Standard have been redefined in conformity with DIN 3960.`.```. the calculation can be carried out according to Appendix A. The examples given in Appendix B are not directly transferable to other gears. Ten tolerance series based on the preferred number series R 10 are given and allocated to the reference diameter (see Table 2).. The values of t h e upper tooth thickness allowances and the tooth thickness tolerances should be taken from Tables 1 and 2. B.. The information in Appendix B on the relationships between the values obtained with different measuring methods should be considered additionally because in industry the tooth thickness is not measured as defined. so that the desired backlash is complied with despite the different influence exerted on the measured variables by the individual deviations of the teeth.