1、90 FTM14A Closed and Fast Solution Formulationfor Practice Oriented Optimization of RealSpiral Bevel and Hypoid Gear Flank Geometryby: H.J. Stadtfeld, The Oerlikon Machine Tool Works, Ltd.American Gear Manufacturers AssociationIIIIIITECHNICAL PAPERA Closed and Fast Solution Formulation for Practice
2、OrientedOptimization of Real Spiral Bevel and Hypoid Gear Flank GeometryH.J. Stadtfeld,The Oerlikon Machine Tool Works, Ltd.The Statements and opinions contained herein are those of the author and should not be construed as anofficial action or opinion of the American Gear Manufacturers Association.
3、ABSTRACT:If a very specific and systematic method is applied to spiral bevel and hypoid gear correction, a newpossibility exists to accurately and quickly design and manufacture high quality gearsets. All generateddata can be archived on a diskette, saved in machine cor,trol memory or stored in a ce
4、ntral hostcomputer. The described algorithm is based on a differential geometry calculation which is to activateabout simple and intuitive input graphics. Gear experts are able to use the newly developed computerprograms for optimization of the theoretical analysis results just as for correction of
5、the real gearset in theworkshop.Copyright 1990American Gear Manufacturers Association1500 King Street, Suite 201Alexandria, Virginia, 22314October, 1990ISBN: 1-55589-566-2A Closed and Fast Solution Formulation forPractice Oriented Optimization of Real SpiralBevel and Hypoid Gear Flank GeometryDr. H.
6、J. Stadtfeld, Chief of Spirornatic DivisionThe Oerlikon Machine Tool Works Ltd.Zurichj Switzerland1. Introduction were developed - each of which, in turn, is backed by so-phisticated mathematical formulas.Previously known calculation methods permit the compen- In addition to nominal - actual compens
7、ation (according to 3-sation of nominal- actual deviations on flank surfaces of spi- D measuring), flank topography modification and toothral bevel and hypoid gear sets on the basis of correction bearing offset, the most interesting (for the gear manufac-matrixes. These are expanded forms of well kn
8、own pro- turer) and simultaneously the most demanding correctionportional changes, e.g., manual ABC corrections, method from the point of view of gear theory is that oftester compensation. Based on two differeqt settings ofThe newly developed method described in this paper func- the bevel gear teste
9、r for the best coast :“d drive toothtions on the basis of data obtained through a one-time theo- bearings, the vectorial calculation method can determine aretical flank generation using a bevel gear simulation pro- gear machine setting with which the two initially contradic-gram. Due to its universa
10、l validity, this vectorial computa- tory tooth bearing positions shown on on the tester appeartion method is applicable in all presently known spiral bevel exactly and simultaneously in the theoretical mounting posi-and hypoid gear manufacturing processes. Yet the method tion.is based on a highly vi
11、sual and transparent model whichmay be described as three spatial triangular vectors in a All algorithms discussed are available as modern, modularlyCartesian coordinate system, constructed FORTRAN programs in both main line com-puter and PC versions. Figure 1 shows a program flow di-Although calcul
12、ations are referenced to the design point P, agram of the bevel gear optimization software. Design be-they also cover the environment of P through an information gins with the SPIROA program which yields the basic gearseries: data, design dimensions, cutterhead and blade geometry aswell as the setti
13、ng of a virtual basic machine.spiral angle gradient along the flank linespiral angle gradient along the profile A data package (Fite.AAA) produced from the foregoing datapressure angle gradient along the profile serves as input for the KEGSIM0 program for zero calcula-Ease-Off Topography as a second
14、 order surface tion of the flank surfaces and for tooth bearing analysis.Should gear characteristic modifications be desired subse-Realizable modifications are divided into corrections of the quent to an initial tooth bearing analysis, the required modi-first order with double flank compatiblity and
15、 single flank fications can be executed with the ABCKORR correctioncorrections of the second order, program - this being the actual subject matter of the presentpaper. To determine whether the modifications lead to theThe true value of a calculation formula as a tool for the gear desired result, the
16、 KEGSIM analysis program is now runmanufacturer is best demonstrated by its usefulness in once again. Since the correction values are actualized dur-practical application. Hence, four different practice oriented ing each KEGSIM run, the vectorial calculation method con-“start-up possibilities for fl
17、ank modification calculations verges on the desired solution during each new run evenfor difficult bevel gear geometries. Aachen as comm;ssioned by the ForschungsvereinigungAntriebstechnik /FVA)/1,2,3,4/. Figure 2 illustrates thegraphical output of the KEGSIM geometry calculation.(START) _ The two v
18、ertical sequences show the analysis of the coastflank paring, left, and the analysis of the drive flank paring,_ t at the right in Figure2. The Ease-Off-Topography(top)geardesigncalculation _ showsthe deviation of the present pair from the crowningS P I R O A free conjugated gear. In other words, fl
19、ank line and profile_r _ corrections as well as longitudinal and profile crowning, andalso all higher value flank surface modifications are recog-_ I flankgeeretio. I nizable in the Ease-Off presentation. The ordinate valuesTCA.f,rstcalculat,on _ are applied over the spatially located projection of
20、the gear-_ K E GS IM _ flanks (polar projection of flanks on axis section planes_ anyoptlmrzat,onsteps analog to cross sections as presented in technical draw- leap over correction _ ;: ; “: . “ “ -_DEZ7 ing of coast and drive side reduced by half. Should, for ex-_iii . :,:.,.:-:.: ample, the drive
21、side remain uncorrected and the coast side . _!_-_ be altered with BCX, then AKOR achieves a modification ofr-El I jl DEZ8 the drive flank of -BCX/2 and a modification of the coastDEZ 6 t tflank of +BCX/2. Alternatively, CKOR changes drive andcoast flanks by +BCX/2. Hereby the change to the coastsid
22、e is doubled, i.e., equal to BCX and that of the drive flank_lu_ ,. ,1c.o,_ is cancelled, or equals zero.-*4-move “,_start program E_c Menu rl Help rl0 Hardcopy 0EI_LIKON C0SOne may use additional masks to decide whether the cor-BOX = (DEZ 7 + DEZs) “( DEZ 5 + DEZd); BDX - BOX2.ZABR AKOR=_ rection i
23、s to be performed on the pinion or the gear andwhether the blade geometry may be altered.BDY= (DEZ5 + DEZT) “(DEZ6 + DEZ8; BDX + BCX2“HGER CKOR = 28CX= (DEZ3 + DEZ4 “(DEZ 1 + DEZ2; 8DY + BCY2ZABR BKOR = 2BCY = !DEZ 1 + DEZ3) “ (DEZ2 + DEZ4); BDY 8CY2“HGER DKOR=_ 6. Tooth bearing offsetIf a theoretic
24、al tooth bearing is found to be in an undesirableFigure 7: Data input illustration of the flank topography position, the gear specialist merely needs to express wheremodification he desires to offset either the tooth bearing or the MeanPoint. This also applies for tooth bearing patterns at thezero s
25、etting on the hypoid tester. Here also, a practical in-put mask was developed as is shown in Figure 8. Theses during the design stage or while measuring with test-ing methods used in the shop, without first having to per-form complicated interpretations and conversions. In thedesign phase, the Ease-
26、Off-Topography and the calculated ROOTtooth bearing are to be viewed as starting point for a de- Schubsired correction. In the shop, it is the tooth bearing test or _-Z_CX-_the V-H optimization using a hypoid tester and measure- 1 /x-_e ACY = ements performed on coordinate measuring devices. Inorder
27、 to do justice to most standard expectations, the fol- XGclowing start-up possibilities have been developed to date:EEl. _ f00fH TOP TOE- nominal- actualcompensation _-_z&DX_- flank topography modification Zug xtooth bearing offset “- - testercompensation ADYxGo INominal - actual deviations are obta
28、inable according to 3-Dmeasurements against theoretical nominal coordinates.These deviations are interpolated with a regression poly- = X ReoTnomial of the second order and the polynomial coefficients x : (a)ctu_l p0sid0n e : (n)0=inal p0sig0n CUI_SOR:5 X: 0 Y: 0are compensated to a double flank cor
29、rection. ,+_*,0v_tarl pr08ra=rtcll_nu11tell)rl0Hard:0py 0ERLIi(0NCOS1AKOR = (Z&DX.2“A4 -ADY. A6 -( z_CX2“ B4+ ACY Bd)“_“A similar method is the flank topography modification in thatthe information to the correction program is restricted to the CKOR= (Z_DX.2cA4-ADY. A6 + (Z_CX.2. B4+/*,CY. Bd). 1four
30、 corner points of each flank. A picture with a symbolical 2geartooth was createdas aninputmask,thus allowingthe 1user a very quick orientation (Figure 7). As calculated from BKOR= (z&DY.2.A5 -z&DXA6 + (ACY. 2. B5-Z&CX.Bd)“ 2the input end-point deviations of the correction magnitudes1AKOR, BKOR, CKOR
31、 and DKOR, the formal relationships DKOR = (Z_DY.2-A5-Z_DX-Ad-(z_CY.2.BS-z&CXoBd). -_may be seen in Figure 7. As a rule, all individual effects areactivated and only in special exceptions will an effect be-come zero. Figure 8: Input display for tooth bearing offsets6ZG (XG, YG) = A1 +A2.XG+A3oyG+A4.
32、XG2+AEoyG2+AeXGYGprofilX _k._._ / crowning_(I, J) / back lash lengthwise crowning wound element 2,. / YGEase-Off-Topography_/_ _ _“_._ /._ generation with KEGSIM, /XG (I,J)YG (I,J) JYG ( I, J ) =- in space with = ZG = f (XG, YG )ZG (I,J) I I=l,N second order surfaceJ=I,MFigure 9: Approximation of Ea
33、se-Off-Topography andanalytical presentation as a second order sur-facesymbol illustrates the the finalized surface of the gear tooth. 7. Tester compensationUsing the cursor, an offset vector can be defined for thecoast and drive flanks through input of nominal and actual The idea that led to this m
34、ethod is the attempt to create anMean Point positions for each. optimal procedure for the user who desires to utilize theresults of the hypoid tester for gear machine corrections.The tooth bearing offset is realized through a combination The simple computer input mask in Fiqure 10 shows sym-of linea
35、r effects with double flank cut compatiblity. To bolical gear and pinion in the tester start up position. Axleswitch from an offset input to a spiral or pressure angle cor- displacement and pinion mounting distance in the testerrection, it is necessary to know the sensibility to which the should be
36、so adjusted that, for example, an optimal bearingflank pairs react to tilting. This sensibility can be derived exists for the coast side. The differences AAk and AA1from the Ease-Off characteristic. A typical Ease-Off is illus- (relative to zero setting) are written into the input mask.trated in Fig
37、ure 9. This topography is brought into an ana- Thereafter, an analog graphic for the drive side appears intolytical form through a coordinate system referenced to the which the values for an optimal drive bearing are typed.mid point of the flank. The directional division of the ex- Hereby it is impo
38、rtant that the coast and drive values maypression ZG(XG,YG) according to XG yields the variable be completely different.elements 2 A4 XG + A6 YG for the drive side and 2 B4XG + B6 YG for the coast side. Elements A4 and B4 arethe sensibility in the derived direction. Multiplied with the shaft positio
39、ns (MM,DEG) Mean-Pointpositions(MM)desired offset (&DX respectively ACX), they yield the flank J TTX TTY TTZ AWI XTRA YTRAtilt. Elements A6 and 86 express flank torsion, in other i-38.000-13.4_0 12.839 90.000 i“_.1764.240words, by an Ease-Off tilt in profile direction (ADY respec- 2 -35.295 -13.796
40、13.047 90.000 8.174 6.389tively &CY) one obtains a simultaneous and undesired Iongi- 3 -35.675 -12.660 12.653 90.000 9.404 1.2014 -35.402 -13.191 12.777 90. 000 9.037 3.579t,dinal influence. If one enters the torsion elements with 5 -34.888 -13.221 12.715 90.000 25.206 5.3466 -34.845 -13.426 12.867
41、90.000 22.826 2.439negative pre-sign into the correction formula, then this is 7 -34.771 -13.596 12.849 90.000 23.936 3.8008 -34.919 -13.815 12.976 90.000 16.445 5.825equal to zeroing the disruptive effect through negation. By 9 -35.175 -13.110 12.756 90.000 16.187 1.867entering elements with the pr
42、oper pre-signs and observing 10-34.989. -13.466 12.835 90.000 16.536 3.679the rules for double flank cut uncoupling (see above), thefour formulas shown in Figure 8 are derived. These formu-las allow a Mean Point offset which is both an uncoupleddouble flank cut and free of disrupting effects. Table
43、2: Displacement sensitivity matrix7The subsequent computation run requires, in addition to the TTX are used in the following. The offset characteristic sur-information described above, one more matrix from which face for TTX is generated, for example, in that the Aq-X val-the displacement behavior o
44、f the theoretical gear (separate ues in the projected flank region are imposed over the cor-for coast and drive) can be extracted. This matrix is illus- responding tooth bearing center positions (Mean Point)/7/.trated in Table 2 for a coast flank pair, and is also calculatedby the KEGSIM analysis pr
45、ogram. Values FIX, FLY, T-Z arethe three translatory gegrees of freedom and the value AWlis the gear set axle angle. XTRA and YTRA represent the The same applies for &FLY. In Figure 11, two typical dis-coordinates of the Mean-Point for each of the offsets 1 placement characteristic surfaces are disp
46、layed. The inter-through 10 in the flank referenced coordinate system, section line of the formulation with the supporting gridplane is of special importance if the figure input value 6AKis subtracted from all ordinate values for the &TTX surfaceFIle I I_onflgura_lon I _:alcutat_on I,_.tQ_T_I Machin
47、. I COS-Disk I Pltnt I End and, likewise, from the ordinate values for the &,TTY sur-Tester _Ol_)ertSS._lO_ face, the input value &A1 is subtracted. The possible MeanPoint lies along the intersection line if-AAk, respectively, -_t (_ AA1 is set on the theoreticalzerogear. Onlyafter consider-(_ _ ing
48、 both formulation intersection lines simultaneously is it(_) “ . “ possible to reduce the dual one-dimensional multiplicity of(_ the solution to a specific coordinate. This coordinateequates to the Mean Point Position of the real gear on theAA_ _7_q_K tester at zero setting. The correction setting o
49、f the testerunit has so been converted into a fictitious, erroneous toothbearing position. The difference vector between Pb and Pa(Figure 11), as the tooth bearing offset vector, will now beCO&St S._e va, i, ,_u._,_E_ further modified by the ABCKORR program according to themethod described earlier. The correction, applied to the“_,-_-mov. E=cMenuFlHelpEl0Hardcopu 0E_LXDNCDS theoretical gear, leads to a preliminary Mean Point, Pc. Thenew gear machine setting compensates for the previouslymeasured e
