----------------------------------------------------------------------------------.. The European Rack Shift Coefficient ··x" for Americans Don McViittie Ge.ar Englinee.rs, Inc .• Sea.ltle. WA Basic Rack Introduction and equations in this article The definition The ue of dimensionless factors to describe seems to have a strong are ba ed on a "basic rack" ill which addendum appeal 10 gear engineers ..The stress factors I and and dedendum are mea ured from a reference J, for instance. are well established in AGMA line located where the tooth thieknes literature. The use of the rack shift coefficient space width on the reference line are equal, The gear 'loath geometry and the "x" to describe nonstandard gear proportions i . basic rack repre ent the tooth form in the nor- common in Europe, but is not as commonly used ma] plane of a gear with an infinite number of in the United States. When it is encountered in teeth .. The normal module of the basic rack is the Eu ropean literature or in the operating manu- equal to thenormal circular pitch divided by 7t. als for imported machine tools, it can be a source The normal diametral pitch of the basic rack is of confusion to the American engineer. equal to 7t divided by the normal circular pitch. The basic rack repre ents the theoretical. gear What follows is intended to provide a source "1{" tooth form, not the form of the cutting 11001. No European standards and pa- allowance i. made for backlash, fini hing stock, for the background factor as II edin pers. The addendum and derivations modification, of the rack shift, or manufacturing method. Q The or profile shift factor bas several mathematical definitions documents in the U.S. Most European use a specific definition, a theoretical "zero backlash" right me h at the nominal (See McVittie, based on gear pair in center distance. tandard 20 norma] pres ure angle ba ic rack of ISO S3 is commonly document used. This is va]id for that basic rack and for any other basic rack which meets the criteria of Fig. I. 86 FTM 1 for discussion.) Addendum Modification The addendum Factor modification factor "x" (Profitverscniebungsfaktor, "profile shift fac- tor" in German) represent the distance, in tight mesh, from the reference line of the basic rack to the reference circle of the gear (rack shift or profile sHift) for normal module diametral pitch = e 1.0 or normal LO. Sum of X Faetees The European practice is to define the sum of Hg. 1 - 34 GEI\.R Basic rack definition - normal plane. TECfffllOI.O(lV· Xl + "'2' (Ix) for a theoretical gear pair which operates in tight mesh (has. no backlash) on the nominal. operating center distance. Table of Symbols The basic equation can be derived from the basic tooth thickness involute geometry equa- ISO tions and the requirement. that the sum. of the c transverse at the operating d pitch diameter is equal to the transverse circular IE pitch at that diameter .. hal tooth thicknesses Definition Genter Distance Gliearan,ee (ha - hfl Diameter Tooth Thinning for Backlash Addendum Dedendurn Tip Shortening Factor Backlash Module Circular Pi,tch Tooth Thickness Addendum Modification Factor d a hf k iny Ctwl. + illY ut tan ,un (]) DIN 3992 Eq 9 B m m p p t s x tan un p cos (2) DIN 3992 Eq 6 z 1 +z AGMA C a, a Pressure Angle Helix Angle III p. 'I' Table of Subscripts 2 Subscript (nonel 2 (3) DIN 3992 Eq 5 At Reference a Diameter At Addendum (Tipl Diameter At Base Cylinder Diameter b f At Root Diameter n NormallPllane o ToollDimensions t w Transverse Plane At Working: Diameter Y At Any (Undefilledl D'iameter 3992, and lSOrrR 4467 for further information 1 on choice ofx factors) according to the operating 2 Pinion Gear or iRack (4) X Factor for Each Gear Th.e values of x for gear and pinion are chosen somewhat arbitrarily (See Maag. DlN conditions and gear ratio, so that their total is equal to LX. The theoretical addendum (tip) diameters and tooth thicknesses of the two gears in the gear pair are defined by their X factors, tion is to be measured must be specified, since there is no recognized convention. Working Group (WG)2 ofISorrC60 is con- sidering a draft technical report. DTRI006412, (5) containing tables which recommend that the tooth thinning for backlash, called "upper allow1t - -0 )o'm " ance of size", Essn' be a function of the pitch diameter of each part. The values are measured (6) ISO DTR 1006412 Eq 6 ..4 normal to the helix angle in the reference cylin- - Sn =(·-+2·oxotanu 2·· .. der. The values can be converted as follows: The actual. addendum. diameters and tooth The transverse circular allowance, Esst' is: thicknesses ar,e then adjusted (usually reduced) (7) to control backlash and! tip to root clearance, Backlash Allowance A common convention facturersis among gear manu- to reduce the normal tooth thick- ness of each member by the same amount, which may be a value in um or a function of module, such as ..024· mn. This maintains the same cutting maximizes depth for both members contact ratio, The direction mal, transverse, The normal allowance in the base tangent reference Don IMcVitti,e plane, Ebsn' (normal to the tooth surface) is ~sn = E SI.· cos ~ • co ilb is President of Gear Engineers, tnc., Seattle, WA. He is a past president oj (8) AGMA and Chairman oj the U, S. Technical Advi- which can aha be expressed as and (nor- (9) circle, or base tan- gent plane) I.D which tile tooth thickness reduc- The resulting transverse circular backlash at sory Group for Intefila· tional Gear Standards. He is a licensed professional engineer in the Stare oj Wa,shingIOIi. JULY/AUGUST 1993 35 the working diameter is at function of allowance, center distance, and tooth accuracy. (See AGMA 2002 for more information.) Tip Sbortening for Clearance When Lx. > 0, the tips of external gear pairs should be shortened to maintain standard tip to root clearance. The reduction :in clearance is skiving, or grinding, a more detailed study is required to estimate the fim hed root diameter. (See Appendix E, Sec. E60f AGMA 2[8.01 for more informatlon.) Convention. for Signs For external gear, the value of x i positive thickne s is increased arid the whenthetooth often ignored for small value. of Ix, but for value of Ix is positive when the center distance larger values the addendum i greater than standard. by k· hould be shertened The same convention can be used for inter- mo' nal gears jf the sign of the center distance is k= considered negative. "Long addendum" inter- aal gears have a negative x. This convention is + Zz [inV a"..,l - inv ~ --.' 2 . tan ~ I ZI ( cos '0; ----··----1 co I~ cos <Xwt - )] (0) Maag Eq 68 but is not. universal. common Intemal 'Gears The equations in thi article are arranged for external gears. With a few exceptions, they can Tip diameters of intern aJ gear pairs should be be u ed for internal gears if the internal. diam- checked for clearance and interference with cut- eters, center distance, and number of teeth are for signs must ters and mates by calculation of actual cutting made negative. The convention and mating conditions. be checked carefully. One trap i divi ion by a Actual Root Diameter and. Cleaeance negative value to. calculate all involute func- The European method doesn't. calculate the tion, which must be positive, h is good pro- actual root diameter of gear which are thinned. gramming practice to take the absolute value of the quotient before calculating the angle from the iavolute function. I. for backlash by feeding the cutler to greater depth. When the actual root diameters are calculated, the addendum diameter required for standard clearance can be calculated more accurately from Eq .. 11. Appendix ~so :: - Derivations Esst • cos ~ • CD of Equation 9 P . b Essn ' = --cos P • cos '0;. • cos I~b The root diameter at maximum tooth thickness can be calculated as follows: Il Po r.t Pbn cos ... =····p·- co "'b=':"- cosa d(",d.-2.(h-x + .• m ao 1I Equation 12 is ba ed Essn 2. tan ~ • cos 011. .)i02) the cutter addendum, hao is measured as shown ill Fig .. 1 for the basic rack, If the gear i to be finished in a second operation, " / ......... t;a: <, I - lRefer,enCIi Diameter i I Fig. 2 - 36 G.EAR Addendum modification TECHNOLOGY as by shaving. ........ ~r...... '" .... Diameter " Pbn n =-Pn Pbl cos a =I PI (9) cos~ fi the assumption that. Pbt t References: American Gear Manufacturers As oclatien AGMA 908. Geometry Factors/or Determining the Pilling Resistance and Bending Teeth. SIre/lgth of Spur and Helical Gear DIN 3992. Profilverschiebung bei Stimrddem mil AUSstnller:;alrnnng. ISO 53. Cylindrical Gears for General and Heavy Engineering - Basic Rack. ISO/DTR 10064/4,. Part 2. Inspection Related to Radial Composite Deviations, Runout, Tooth Thickness, and Backlash. ]SOfTR 4467. Addendum ModiJjcariol10/ the Teetb 0/ Cylindrical Gears /01' Speed·Reducing and SpeedIncreasing Gear Pairs. Lorenz. Gear Curling Tools. 1961. Maag. Maag Gear Book. 1990. McVittie.Don. "De cribing Nonstandard Gears-c-An Alternative to the Rack Shift Coefficient." AGMA 86 fTM L ALso in Gear Technology, Vol. 5, No. I, January! February, J 988, p, lOff.

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