1、90 FTM 2An Industrial Approach for Load CapacityCalculation of Worm Gears(Verifying and Design)by: Michel Octrue, CETIMAmerican Gear Manufacturers AssociationTECHNICAL PAPERAn Industrial Approach for Load Capacity Calculation of WormGears (Verifying and Design)Michel Octrue, CETIMThe Statements and
2、opinions contained herein are those of the author and should not be construed as anofficial action or opinion of the American Gear Manufacturers Association.ABSTRACT:The method proposed in this paper is based on an analytical rating method which has been developedsome years ago by the author (see AG
3、MA paper 88 FTM 6).The calculation is based on the determination of the maximum of pressure between the mating teethwhich is made by using a specific criteria for worm gears. This method can be used to verify the loadcapability of a worm gear but also to design a new gear covering several types of t
4、ooth profile.Calculations are provided as examples and comparisons have been made with results obtained by theinitial analytical method, and by standardized methods (AGMA, BS).At the end of the paper a comparison is made with experimental results.Copyright 1990American Gear Manufacturers Association
5、1500 King Street, Suite 201Alexandria, Virginia, 22314October, 1990ISBN: 1-55589-554-9AN INDUSTRIAL APPROACH TO LOAD CAPACITY CALCULATION OF WORM GEARSVERIFYING AND DESIGNMichel OCTRUEGear DepartmentCETIM Senlis FRANCE1 INTRODUCTION We have indicated in 2 that this type of calculation leads to expre
6、ssingthe admitted torque to the wormwheel COn re.N) using a rather simpleIf the different rating methods usually used for designing worm gears formula:(1,8 h 10) consider their limited life because of TEETH BREA- ( ZN_ 2c r foKAGE and CONTACT PRESSURES, experience shows that in most = _2 ._ u,m Z-_r
7、) Z_eases, it is the SURFACE DEGRADATIONS OF THE WORMWHEELwhich limit the gearing life. with:The main cause for SURFACE DEGRADATION is the fatigue r_a: operating pitch radius of the wormwheel (in ram)phenomenon on the material layer submitted to CONTACT PRESSU-RES. This has brought us to develop a m
8、ethod of calculation allowing ohm,: admissible pressure of the material (in MPa)to evaluate the level and distribution of contact pressures along the flanks Z g life factorof the teeth. Z_: elasticity factor (in MPa to power I/2)As opposed to existing methods of calculations, the distribution of con
9、tact Z R:pressure distribution factor of the worm gearpressures is no longer supposed to be uniform. (in square nun)The development that foUows presents the method basis used as well as “ZR is composed of the global stiffness factor RDGp which takes intoa simplified method which can easily be applie
10、d industrially either for account the global stiffness of the gear and the (K_. :_, ) o factor whichverifying or design, combines the mean equivalent stiffness (RD _q,_)L.j and the equivalentcurvature radius (/? _q)t. j calculated at the maximum pressure amplitude2 REMINDER OF ANALYTIC METHOD contac
11、t point (index i, j) for the relative position of the worm and thewormwheel (index p):For a worm gear, the cartographic calculation of contact pressures is Z_ = RDGv.(K _ ,_e_) obtained from a detailed analysis of meshing of a cylindrical worm with “the mating conjugate wormwheel. In fact for this t
12、ype ofgear the gearing ( XR_Nq_)conditions evolve continuously and differ from one point to another in (K_,/,-_ )p = R_q.the same contact line; thus the name of analytical method (2 to 4). t.j. pThe analysis of the meshing is performed at 2 steps: with:- STEP 1: it aimed to determine for a relative
13、position of the :c_: directive cosine of the normal to plan of contactworm and the wormwheel the instantaneous load distribution The power dissipated in the meshing under friction form is calculatedalong the meshing teeth. This distribution depends on the teeth from the determined load distribution
14、and the sliding velocity whichgeometry, the contact geometry and the materials making up allow to calculate the friction coefficient in all contact point: from whichthe teeth. It is thus possible to calculate a pressure level for we deduce instantaneous efficeney rl. The torque transmitted by the ge
15、areach contact point, is then expressed by:- STEP 2: by varying the relative position of the worm and C_ = rI.Cthe wormwheel it is then possible to study the contact pressurefields on the whole contact zone of the gear. In this type of calculation the difficult point is the determination of theThe p
16、rinciple on which is based the analytical rating method rests on pressure distribution factor of the gear Z Rfor this requires impoffantdetermining the contact point for which the contact pressure field attains calculation means. To obtain a calculation method much easier to use,its maximum. The max
17、imum load capacity admissible by the gear is a simulated analytic method has been undertaken to establish a variationthus limited by the pressure allowed by the most weaker material role fortheZRfaetorin funetionofthebasiegeometricalparameteraof(generally that of the wormwheel) for the number of req
18、uest cycles, the wormgear.13 ELABORATION OF THE SIMPLIFIED METHODPassing from the analytic method to the simplified method is performed Iin 2 phases: IL b2maxi .l- PHASE 1: We have to determine the basic geometrical _parameters affecting Z_. To do so a data base of 375 gears Iwas created, bZl_tandax
19、l- PHASE 2: We have to improve the variation rule for Z_factor in function of the only infiuent geometrical parameters.To this purpose a second data base of 1064 gears was created, fThe fixed parameters for the 2 phases were asfo,lowS:rnxlAxial module 1 _ “-_ ,- _ _ 1:,1C_x Thread thickness distribu
20、tion eoef- 0,45ficientCj_ Worm clearance coefficient 0,15 RG 1h=l Addendum and dedendum According FIGURE 1- Definition of wormwheel face width and external adden-to BS721 dumcoefficientx z Wormwheel addendum modifica- 0 Z l Number of threads in worm 1-2-3tion 4-5-6C j2 Wormwheel clearance coefficien
21、t 0,15 q_ Diameter factor 6 to 20 witha pitch of 1PROFILE Type of thread profile I PROFILE ao_ iNormal pressure angle 20(INVOLUTE)MATERIAL Z 2 Number of teeth in wormwheel 25 to 90 withWORM 0,4 % carbon STEEL a pitch of 5WHEEL Chilled cast phosphor BRONZE b z Wormwheel face width b z_sCHART 1 Consta
22、nt parameters chosen to generate the data bases Ch_2 Wormwheel external addendum 0,7coefficientThe variable _arameters were:CHART 2b Variable parameters to generate the data base of 1064Z l Number of threads in worm 1-2-3 gears4-5 -6q j Diameter factor 6-8-10 3.1 INFLUENCE OF GEOMETRICAL PARAMETERS1
23、2-14-16A previous study 4 has shown that the pressure distribution factor ofa0, Normal pressure angle 15-17,.5-20 the worm gear is proportional to the square of the axial module of the22,5-25 worm. That is why we have set rrtxt = 1.Z z Number of teeth in wormwheel 30-40-50-60 The instantaneous lengt
24、h of the contact lines between the worm threadand the wormwheel teeth influences the ZR factor. It depends on thewormwheel face width bz and the wormwheel external diameter d2b z Wormwheel face width K.b2ss with:K=0,6-0,8-1 defined by the wormwheel external addendum coefficient Ch.zK= 1,2-1,4et The
25、reference wormwheel face width b 28s is given by the formula usuallyb2 ra,x described in the standards (6 p.45). It may be increased if the contactlines continua to exist, that is to say as long as there is no interferenceCh 2 Wormwheel external addendum 0,7-1-1,3 Its maximum theoritical limit b2m_x
26、 is given by the intersection of thecoefficient 1,6-1,9 wormwheel external circle with the worm addendum circle. Thesimulation showed that it was not useful to increase the wormwheel faceCHART 2a Variable parameters to generate the data base of 375 width beyond a value equal to:gears b z= 1,2.b2_ sN
27、OTE: bz_a x = _/dZ=t _(2.a _d,z)Z b2, s - 2.rrt,_t-_l + 1with: As for the wormwheel external addendum coefficient C n_z its optimumda _:worm tip diameter value is:a,.: centre distance Ca, 2= 1.6d,2: wormwheel external diameterTo improve the influence parameters we have tried to study separatelyGEAR
28、RATIO: Taking into account the values imposed hereabove, the the global stiffness factor R DC v and the mean equivalent curvatu-gear ratio is included between 1/60 and 1/5. re/stiffness factor (K, i_,= )1,which compose Zn. The torque effects ofIn all cases the existence of a worm gear is tied to the
29、 existence of the each geometrical parameter is such that it is impossible to deduce simplethread profile: the worm root diameter must absolutely be more than rnles.We have established the qualitatlve rulesofthese effectsby settingthe worm base diameter. Cn,2 = 0,7and be = b2Bs.The chart here-below
30、resumes these variations: 3.2 EQUATION OF Z_ FACTORZ _ cLo_ q _ Z a A program of curve smoothing in the data base created from 1064 gearsstudied has lead to represent the Z Rfactor under the form:z. - /up,o ,o,.:if a,_ 17,_ = _- 17,_ with:c(z,) r,_RDGp A(qt)= a(Zt)+b(Zl)q: ql J- “ xlKtJ_n_ _ifJif 2(
31、a_2Ca“ 2( “RD_e_ _r upto Zl=l zl=2 Zl3 Zl=4 Zl=5 Zl=6constant after a .509487 .480134 .312491 .124563 -.246016 -.1895216.10:-1.82184 .174084 6.50529 12.0703 24.4142 21.1091CHART 3- Directional influence of main basic geometrical parameterson the value of ZR and its components c -2.31131 -2.82418 -2.
32、31135 -1.35173 .923392 .459922- - -4_ light influence a“ -57.0706 -57.9277 -66.5854 -72.6328 -107.574 -87.2934mean influencehigh influence 6 -361.889 -333.547 -49.5023 297.434 1397.2 945.126c 2798.43 3715.18 2758.29 526.426 -6748.53-3669.39The number of threads on the worm Z i has very little influe
33、nce on the CHART 5- Coefficients for the calculation of Z 8 factorZ Rfactor. VALIDATION OF RESULTS: the differences in percentage betweenThe diameter factor q t has a great influence on the Z _factor as long as the value of Z a originating from the data base and obtained from theq _ 60thus C_o_= 20)
34、 with chromium-molybdenum 1235 867Finally the Z Rfactor varies almost linearly according to the number of CHART 6- Admissible pressure in surface as function of the twoteeth on the wheel Za. materials used3Note that these values should be corrected by a multiplying factor as where:function of the wh
35、eel dimension to take into account the increasing E=. E2: YOUNGs moduli of materials (in MPa)heterogeneity of the material regarding the mass of the part. This factor v _, v _ POISSONs ratio of materialsis expressed by :As indicating information we give hereafter a chart of practical values_/1,9097
36、.(r w2.sin % )-0.2 for the usual materials:with:rwz: operating wormwheel pitch radius YOUNGs POISSONsmoduli ratio%: base lead angle of the worm (in MPa)Of the two parameters participating in the calculation of this factor, it Structural steel 210000 0.285is mainly the operating pitch radius of the w
37、heel which is important. Itis near 1.0 for pitch radius of 50 mm wormwheel and it drops to 0,74 Nickel steel 203000 0.290for pitch radius of 500 mm wormwheel. Gray east iron 110000 0.290FonteGS 160000 0.290Note that these stress values are not affected by any corrective factor Bronze 110000 0.320tak
38、ing into account the numbers loading cycles of the material.As for as we are concerned we determine the admissible stress of the CHART 7- Usual values of YOUNGs moduli and POISSONs ratiotwo materials used on a disc-roller simulator. This allows obtaining adirect fatigue curve characterizing the beha
39、vior of the material to For a steel-bronze pair we take Z _- 15 7.461 (en M Pa _ z)superficial pressure according to the number of loading cycles (Fig. 2)This curve is then expressed by the product of a life factor ZNand anadmissible stress limit _ H,_, defined for 5.107 cycles. 4 DESIGN METHODi I I
40、 I 111 torquetobe transmitted)7so I I I I I II_LI I I IIIII - the ROTATING SPEED of the wormwheel and wormI IIIIII I I II II1_b I IlllI - the DURATIONOFLIFEEXPECTEDforthe wormgearI I?_gll I II IIi11 _ I IIIIII (Hinhours)“= I I11111_ I I I IIIII “N,LIIItJ -theGEARRATIO-_ I II IIII _,_ I I I IIIII I I
41、-IIIlJ -the CENTERDISTANCE to achieve6 r_x “ _1_ I“ The designer Engineer deeides the matedals to be used whiehallowsIIIIIIlllllIIIIII1_1111111&“g41111!lIIIIII11111 tocalculatetheelasticityfactorZrandtheadmissiblestresslimitofthetw materials G.IIIIII III I I 1: From the rotating speed known and the
42、reduction ratio it is easy toI IIIII calculate the rotating speed of the wheel n2 from which we obtain the26c IIIIIII I IIIIlll IIIIIInumber of loading cycles N dI IIIIII I I IIIIII I IIIIII IIIIII t I IIIIII 1 IIIII Nc=60“nzHIIIIIII I IIIIIII II1111o I 11t11| I I IIIII! 11 IHI Wethushave2 equations
43、allowingthecalculationof thefactorZ_:1E5 1E6 IE? 1ElN cycle= - one proceeds from the conditions imposed :C2, .Z_FIGURE 2 Superficial pressure endurance curves Z ea(ql) 2a)Bronze UE12P chilled cast r=2 .om,_ .qb)Bronze UE12P centrifugally east - the other proceeds from the simplified method :The resu
44、lts we presently have concern the two following materials : ZRb(q,)=A.exp(_._z )CASE-HARDENED STEEL-UE12P BRONZE CHILLED CAST:(5.lOr_IsZn=/_ ) The principle of the solution is to set a number of threads Z l for theZ s_,a_ = 1.80 worm, from which we calculate the number of theeth on the wheel :Z N,.,
45、o,_= I zz=f(u.Z_.0,5)_m_ = 3SOMPaCASE-HARDENED STEEL-UEI2P BRONZE CENTRIFU- For a diameter factor value q t fixed, it is possible to calculate, fromGALLY CAST: the center distance imposed, the axial module of the worm :/5IOZ_Is 2.awZ“L-_c ) m,_,=Z,+q Zn,ax_= 1.80Z nmn_= I The operatingpitchradiusoft
46、hewheelisgivenby :o#. m= 600MPa r_z = O,S,m xl.zz10,000 hours of tests are required, that is 35 test units to determine the The efficiency of the gear can be evaluated from :behavior of each materials. _ ta n V,The other factor depending on the marerials is the elasticity factor Z r tan(_+ _,)which
47、is expressed by:1 Where V, is the lead angle of threadsZe“ / -f ,._) andisthemeanfrictionanglegivenby:n. ( .=)+( rz j _b=tan“(_)withtt=l(V,)4The friction coefficient it as a function of the mean sliding velocity I/, CASE 1 THE GEAR IS OVERSIZED:is obtained in rrds by : C 2_ran_= 5 0 0 m. N there exi
48、st no intersection point between theV s = 1. O 47.10-4. n _r t curves and we still have Z n=(q 1) ZRo(q 1 )housing (K/factor) The gear never transmits the torque no mattez- the type of lubricant used the diametral quotient values.To optimize the gearing the center distanceExcept for the application
49、factor K Afor which we can keep the customary must be increased or some other more resistan!values used for circular gearings, all other parameters are being to be materials chosen.defined.The final formulation of the simplified method could be of that form : ZRFACTSRE._SC2_“_“ r_2 %“_Z-rE “ .KHy.K t.K_P.OO6 APPLICATION OF DESIGNING METHODg. 90 /Given to design a wormgear with 125 mm center distance u = 40 from ._ _-P,q&which the worm is driven by an electric motor operating at the s
