1、05FTM04Tooth Meshing Stiffness Optimisation Based onGear Tooth Form Determination for a ProductionProcess Using Different Toolsby: Dr. Ing. U. Kissling, Dipl.- Ing. M. Raabe, Dr. M. Fish,KISSsoft AGTECHNICAL PAPERAmerican Gear Manufacturers AssociationTooth Meshing Stiffness Optimisation Based on Ge
2、arTooth Form Determination for a Production ProcessUsing Different ToolsDr. Ing. Ulrich Kissling, Dipl. Ing. Markus Raabe, Dr. Michael Fish,KISSsoft AGThe statements and opinions contained herein are those of the author and should not be construed as anofficial action or opinion of the American Gear
3、 Manufacturers Association.AbstractThe variation of the tooth meshing stiffness is a source of noise and the exact calculation of tooth form isimportant for the stiffness determination. For this purpose, software was written with the concept of anunlimited number of tools such as hobs, grinding disk
4、, and honing defining a manufacturing sequence. Thetooth shapes after each step to show the material removal, sliding and rolling vectors to optimize the tool lifeare determined. Additionally, meshing stiffness variation can be improved by optimization of final geargeometry with a calculation of the
5、 contact path under load. From this information the meshing stiffness isderived making it possible to study the effect of a proposed profile correction of a gear under different loads.Calculations with AGMA2001 or ISO6336 check the point with the highest root stress. Effect of a grindingnotch is als
6、o included.Copyright 2005American Gear Manufacturers Association500 Montgomery Street, Suite 350Alexandria, Virginia, 22314October, 2005ISBN: 1-55589-852-11Tooth Meshing Stiffness Optimisation based on Gear Tooth Form Determinationfor a Production Process Using Different ToolsDr. Ing. Ulrich Kisslin
7、g, Dipl.-Ing. Markus Raabe, Dr. Michael Fish, KISSsoft AGIntroductionThe methods used in gear production are inconstant development. In recent years form grind-ing (an alternative to the classic meshing grinding)has become the trend. Another example is a meth-od used mostly in automotive industry: t
8、o improvethe working life of tools and in order to get tooth formwith higher root strength, gears are produced usingup to three different pre-cutters plus a final honingor grinding process. One of the latest tendencies inthe development of optimized gears is to apply aspecial wave-form-like profile
9、modification duringthe finishing process for the reduction of transmis-sion error.These production methods require the develop-ment of appropriate calculation methods. In this pa-per the calculation of the resulting tooth form is de-scribed when several tools are used. Then, basedon this tooth form,
10、 the effective meshing stiffness(under load) and the stress calculation are dis-cussed.The variation of the tooth meshing stiffness duringoperation induces a deviation in the rotation-angleof the output gear from the nominal transmission ra-tio (transmission error) causing vibrations andnoise. The m
11、eshing stiffness variation can be im-proved through optimization of the gear geometry(transverse contact ratio and overlap ratio?),but the type of profile modification is also very im-portant for the stiffness under load.The current calculation method for the tooth resist-ance following either AGMA2
12、001 1 or ISO6336 2is based on the assumption of a tooth form pro-duced by one tool in a meshing process. The meth-od includes, when using a tool with protuberance,also a production process with a pre-cutter (withstock allowance for finishing) and final grinding orhoning process.This implies that the
13、 formulas in AGMA or ISO resis-tance calculation methods can not be applied withgears produced by form grinding or other non-con-ventional methods. The problem is that for the cal-culation of the tooth root stresses some values suchas tooth thickness and root rounding must beknown. The calculation m
14、ethod assumes that thetooth form is not exactly known, and therefore pres-ents formulas which permit calculation of the toothform just in the considered section of the tooth.These formulas assume production through ameshing process. But in principle, if the tooth form isgiven, the tooth can be calcu
15、lated by directly usingthe formulas proposed by the standards. Therefore,if the tooth form calculation is integrated into the re-sistance calculation software, AGMA or ISO stan-dards can be used for any production method.Tooth form calculation with different toolsIn most of the available gear calcul
16、ation software itis possible to calculate the tooth form when using astandard tool (hob, generating cutter or gear typecutter). Normally it is also possible to introduce agrinding allowance and so simulate a 2-step pro-duction process (cutting, then grinding). Answeringfrequent requests of users of
17、the widely recognizedgear calculation software KISSsoft 3, it was de-cided to implement a new approach for tooth formcalculation in the software.An unlimited number of tools such as hob, cuttingtool, gear-type cutter, grinding disk (generating orform grinding), and honing wheel can be defined inany
18、sequence desired. The tooth shape can be vi-sualised after every step, and different shapes canbe superimposed to show the material removal fromone step to another. The manufacturing processfrom tool to gear is also visualised, and sliding androlling velocity vectors are indicated (to optimise theto
19、ol life).Fig. 1 shows the different stages in getting to the de-finitive tooth form when using two cutters and a finalrectifying process. For the gear with a pressureangle n20, a pre-cutter with n25 is used. Thisleads to an increase of the tool service life andcreates a better rounding of the tooth
20、root. As theflank form should be identical to the final design, thebase diameter db of the gear has to remain ident-ical. Therefore the module of the tool must be in-creased by COS(20)/COS(25) for the pre-cutter.2Figure 1. Tooth form generated with a 3-tool-production cycle1: Pre-cutter mn=1.0368, n
21、=25 (green)2: Protuberance-Cutter mn=1.0, n = 20.0 (brown)3: Final rectifying process (blue)Some additional features of the tooth form calcula-tion are:- Tool service life: For the improvement of the toolservice life (no. of gears cut until the tool has tobe re-sharpened or replaced) the display of
22、thespecific sliding on the tool cutting flank is veryimportant (fig. 2). There are many factors in-fluencing the tool service life. One of them is thelocal specific sliding on the tool. A high negativespecific sliding on the tool implies that a shortsection on the cutting edge of the tool producesa
23、large section on the gear. This means that thispart of the tool is highly utilized and consequent-ly subject to high wear. As shown in fig.2, a pre-cutting-tool with higher pressure angle (as infig.1) has a significantly reduced specificsliding.- Grinding notch: For the root strength it is import-an
24、t to know if through the grinding process a socalled “grinding notch” (ISO6336-3, factor Ysg)results. Therefore during the calculation of thegrinding process, the notch has to be recog-nized. Such a notch can reduce considerablythe safety factor for bending stress. In KISS-soft, the corresponding da
25、ta is automaticallytransferred to the resistance calculation and thenotch factor is included in the results.3Sliding (Pre-Cutter 25)- 1 . 5- 1- 0 . 500.511.5200.511.52Specific SlidingTooth FormSliding (Final Cutter, Protuberance, 20)- 1 . 5- 1- 0 . 500.511.5- 0 . 500.511.52Specific SlidingTooth Form
26、Figure 2. Drawing of a cutter-tooth for pre-cutting (pressure angle 25) and finish-cutting (20)process with indication of the specific sliding (between cutter and gear) during manufacture. Highpositive specific sliding (in this case on the dedendum) indicates higher wear risk. The sliding on the 20c
27、utteristwiceashighasonthe25 per-cutter.4Vibrations caused by tooth meshstiffness variationFor many applications today, the noise level is verycritical, and should be as low as possible. Noise isgenerated by transmission errors, which producesan incremental change of the velocity on the gearabout the
28、 nominal value. This effect induces aninstantaneous acceleration / deceleration into thetransmission chain and the result is vibrations. Thetransmission error is produced through the variationof the stiffness during a mesh cycle. It is also wellknown that fabrication errors generate a certaintransmi
29、ssion error. Improvement of the gear qualityhelps to improve the situation, but even with a gearset of highest quality, we have transmission errorsdue to the stiffness variation.For this purpose, the software calculates the singletooth stiffness of any tooth form and determines thereal contact path
30、under load. From this informationthe real transmission error of the gear pair is de-rived. It is possible to study very quickly the effect ofa proposed profile modification on the behaviour ofa gear stage under different loads.The calculation of the tooth pair stiffness under loadpermits a compariso
31、n of different gear geometriesto find the best solution. Fig. 4 shows a typical caseof a standard gear set (tooth form in fig. 3) withtransverse contact ratio () = 1.67 (z: 25:76, DP4.2333, module 6.0). The transmission error is low(stiffness is high) during 67% of the cycle, when 2pairs of teeth ar
32、e in contact, and otherwise high,when only 1 tooth pair is in contact. (Remark: Atheoretical transverse value of 1.67 signifies thatduring 67% of the time 2 pairs of teeth are in contact,and during 33% only 1 pair.) The gear without profilemodification experiences a sudden “jump” from lowto high tra
33、nsmission deviation as shown in fig. 4. Awell designed profile modification would reduce therate (or steepness) of this jump producing asmoother change in the stiffness, but would notchange the level of the low and high stiffness phase.So profile modification is useful but not the solutionto the pro
34、blem. The best method to reduce the vari-ation of the stiffness is to use a deep tooth profilewith, theoretically, a transverse contact ratio of 2.0.In this case, as 2 pairs of teeth are always in contact,the “jump” is eliminated.Best results will be achieved when the third toothcoming into contact
35、is unloaded. This is possiblewhen an appropriate profile modification is applied.In this case, the theoretical contact ratio has to behigher than 2.0 according to the length of profilecorrection planned.Figure 3. Representation of the gear pair used for the calculation of the transmission error5-0.0
36、1-0.009-0.008-0.007-0.006-0.005-0.004-0.003-0.002-0.0010- 8 - 4048phi1 ()Deltaphi2()T(0%)T (25%)T (50%)T (75%)T (100%)-0.01-0.009-0.008-0.007-0.006-0.005-0.004-0.003-0.002-0.0010- 8 - 4048phi1 ()Deltaphi2()T(0%)T (25%)T (50%)T (75%)T (100%)-0.01-0.009-0.008-0.007-0.006-0.005-0.004-0.003-0.002-0.0010
37、- 8 - 4048phi1 ()Deltaphi2()T(0%)T (25%)T (50%)T (75%)T (100%)Figure 4. Transmission error calculated considering the stiffness under load for0, 25, 50, 75 and 100% of the nominal power.Top: Gear set with unmodified profile.Middle: Gear with small symmetric profile modification (13 m).Bottom: Gear w
38、ith normal symmetric profile modification (26 m).6The calculation of the transmission errordue to tooth mesh stiffnessThe stiffness of a single tooth is composed of fourimportant effects. These are:- Tooth bending- Shear deformation of the tooth- Hertzian compression in the contact- Tilting of the t
39、ooth in the gear bodyEquations for the calculation of these effects weredeveloped by Peterson 4 for involute gears. Themethod can also be adapted to gears with profilemodification and to non-involute gears. The stiff-ness of two teeth in contact is called Combinedtooth stiffness of one pair of teeth
40、 (c ) and the totalstiffness of all teeth in contact is the Mesh stiffness(c).The calculation of the behaviour of the meshing stiff-ness during a contact cycle is important. For un-loaded involute gears with no profile modification,the path of contact is a straight line, but for realgears under load
41、 the effective path of contact iscomplicated to find. Due to bending of the teeth thereal transverse contact ratio increases. There-fore the contact path has to be calculated step bystep; and for every single step, the effective numberof teeth in contact has to be determined. As thetooth stiffness d
42、epends on the load, and the load ona single tooth (with constant torque on the pinion)depends on the number of teeth in contact, the solu-tion must be found through iteration (Fig. 6). Thecalculation is time-consuming because the stiff-ness itself depends on the applied normal force(due to the Hertz
43、ian compression); therefore everysingle step on the contact path has to be found bydouble iteration.The result is quite impressive. Fig 4 shows a typicalexample of a gear set without (top), and with differ-ent degrees of profile modification (middle and bot-tom). The technique of displaying transmis
44、sionerror under varying load such as 25, 50, 75 and100% of the nominal torque load, and defined as farback as 1958 by Harris5, is a very helpful illustra-tion for evaluating the behaviour of a proposed pro-file modification. Most gearboxes do not always runwith a steady torque, so the performance of
45、 the gearset should be optimal within a certain torque range.The transmission error of the unmodified gear in fig.4 also shows an often discussed feature of gears:due to the flexure of the teeth, the transverse con-tact ratio () increases with higher torque form 1.67to approximately 1.78 (50% load)
46、and 1.81 (at 100%load).-0.008-0.006-0.004-0.00200.0020.004- 8 - 4048phi1 ()Deltaphi2()T (25%)T (50%)T (75%)T (100%)T(0%)Figure 5. Gear set with small symmetric profile modification (13mm) as in Fig. 4, but with anadditional sinus-wave form modification in the single contact region of the tooth flank
47、.Left: The modified tooth form (here exaggerated) with additional thickness 10 mm.Right: Resulting transmission error under different loads7Figure 6. Calculation of the transmission error of a gear pair under load8Any small modification on the tooth profile has animportant influence on the transmiss
48、ion error curveas the comparison of the different gear sets in fig. 4shows. A new tendency in the development of opti-mized gears is to apply a special profile modificationduring the finishing process for the reduction oftransmission error. When increasing the thicknessof the tooth between the limit
49、s of the single contactdiameters (the section of the tooth flank with onetooth pair in contact), the value of the transmissionerror in this section can be reduced. To demonstratethis, fig. 5 presents the transmission error showingthe effect of a sinus-wave like modification. It is evi-dent that such a modification produces higher trans-mission errors in low load conditions, but in the nom-inal torque range the change of the transmissionerror is significan
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