1、10FTM10AGMA Technical PaperEvaluation of Methodsfor Calculating Effectsof Tip Relief onTransmission Error,Noise and Stress inLoaded Spur GearsByDr.D.PalmerandDr.M.Fish,Dontyne Systems Ltd.Evaluation of Methods for Calculating Effects of Tip Relief onTransmission Error, Noise and Stress in Loaded Spu
2、r GearsDr. David Palmer and Dr. Michael Fish, Dontyne Systems 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.AbstractThe connection between transmission error and nois
3、e and vibration during operation has long beenestablished. Calculation methods have developed to describe the influence such that it is possible to evaluatethe relative effect of applying a specific modification at the design stage. The calculations can allow thedesigner to minimize the excitation f
4、rom the gear pair engagement at a specific load. This paper explains thetheory behind transmission error and the reasoning behind the method of applying the modifications throughmapping the surface profiles and deducing the load sharing. It can be used to explain the results of laterexperimental val
5、idation on various types of tip relief in low contact ratio (LCR) gears, from very long to veryshort. The paper will also demonstrate that though the effects of modification in any specific case can bemodeled with some certainty, the same modifying strategy can not be applied universally but must co
6、nsiderthe required operating conditions. It illustrates that the effect of tip relief on transmission error and load sharingis not a black art but can be fully explained by applying existing theory.A study of high contact ratio (HCR) gears will be presented to demonstrate why it is often necessary t
7、o applydifferent amounts and extents of tip relief in such designs, and how these modifications affect load sharing andhighest point of tooth loading. Specific attention will be paid to the phenomenon of extended contact, where ifno modification or insufficient tip relief is applied, contact does no
8、t stop at the end of active profile but continuesbeyond this point as the gear rotates resulting in contact on the tip. This effectively increases contact ratio andhas implications for the tooth load and in particular how this may affect the loading position, highest point ofsingle tooth contact (HP
9、STC), which is relevant to both ISO and AGMA standard rating. The paper willconsider 3 methods commonly employed in the industry; a simple 2D mapping procedure carried out on graphpaper, a 3D linear tooth stiffness computation method, and a 3D finite element analysis (FEA) calculation. Thepaper will
10、 demonstrate that though in some cases these methods can produce similar results, albeit withvarying degrees of accuracy, further examples will be presented which demonstrate behavior which can onlybe detected using some of the more complex analysis methods. The commercial viability of implementing
11、abetter quality models against the time constraints in the development process will be discussed andconclusions drawn.Copyright 2010American Gear Manufacturers Association1001 N. Fairfax Street, Suite 500Alexandria, Virginia, 22314October 2010ISBN: 978-1-55589-985-13Evaluation of Methods for Calcula
12、ting Effects of Tip Relief on Transmission Error,Noise and Stress in Loaded Spur GearsD. Palmer and M. Fish, Dontyne Systems Ltd.IntroductionTransmission Error (T.E.) occurs when the drivengear is often momentarily ahead or behind itstheoretical position in respect to the constant speedposition. Gea
13、r design methods assume perfectgeometric conditions and alignment betweencomponents to maintain constant angular velocityoften referred to as conjugate action. This conjug-ate action is usually achieved in spur gears by usinginvolute profiles on the teeth. However due to theirability to transmit lar
14、ge loads, the elastic deflectionof the material from which the gears are made be-comes significant. These small deflections of theteeth cause transmission error (the driven gear isoften momentarily ahead or behind its theoreticalposition), and also the possibility of extended tipcontact, that can le
15、ad to scuffing of the teeth andexcessive noise and vibration. Other causes oftransmission error are the manufacturing pro-cesses often resulting in deviations from the trueinvolute profile, and tooth spacing (pitch) errors.For convenience the transmission error is ex-pressed as a linear value measur
16、ed at the base ra-dius. This eliminates the need to specify on whichgear it is measured as is the case with angularmeasurements.To compensate for transmission errors it is wellestablished practice to apply small profile correc-tions to the gear teeth, often termed tip/root reliefs.The amount of reli
17、ef (material removed from theflank) is generally agreed upon, that is enough toallow for tooth deflections expected at a given loadand also errors due to manufacturing tolerances.The extent of relief (how far down the tooth materialis removed) is not so clear, and in spur gears isknown to have a sig
18、nificant effect on gear perform-ance. In the past designers tended to use empiricalvalues from previous experience, which may nothave been the optimum but due to lack of informa-tion in the design standards made them cautiousabout change.The theory of profile relief to allow for tooth deflec-tions u
19、nder load was first proposed by Walker 1.The suggested amount of relief was equal to thecombined tooth pair deflection under load and thesuggested extent what we now know as long relief.Harris 2 extended work in this area and covereddifferent types of relief. He also introduced theconcept of T.E. Us
20、ing what have become known as”Harris Maps” he suggested that the T.E. curves fordifferent loads could be used to describe the staticand predict the dynamic behavior of a gear pair.Gregory, Harris and Munro 3 confirmed Harrisspredictions experimentally.Munro 4 later explained the fundamental mechan-i
21、sm behind profile relief and established a soundtheoretical basis for design. He examined the ef-fects of long and short relief and allowed the extentof relief to be varied at an intermediate position toobtain a low variation of T.E. at the desired designload.Mapping 2D tooth profiles to calculatetr
22、ansmission errorThe deviations from the involute profile of the pinionand wheel are be combined from start of active pro-file (SAP) to the end of active profile (EAP). SeeFigure 1.If the combined deviations of all the pairs of teeth fora pair of gears in mesh are superimposed, thepattern of T.E. as
23、the gears are rotated can be cre-ated. This is achieved by spacing the tooth pairprofile deviations one base pitch apart. SeeFigure 2.The uppermost point on the curve from all the over-lapped tooth pairs gives the transmission error atzero load. This is similar to that obtained from thesingle flank
24、tester. When the load is applied fromthe torque acting on the gear, the analysis of thetransmission error in one full tooth length regionfrom SAP to EAP allows us to form a model of thecontact in 2 dimensions. See Figure 3.4Assuming constant tooth pair stiffness:At position aTotal load = (stiffness
25、of pair 1)(x1+x2) + (stiffnessof pair 2)(x2)At position bTotal load = (stiffness of pair 1)(x4) + (stiffness ofpair 2)(x3+x4)When a whole series of these loaded curves isplotted we have produced the Harris map. Itdisplays the quasi-static transmission error forgears under a range of loads. Using thi
26、s methodshows the regions of single and dual pair contact,and allows the effect of different amounts and ex-tents of tip relief to be examined. Each curve underload shows a different deflection from the nominalzero load position and changes in form due tochanges in tooth load share during engagement
27、.The changes in quasi-static curve form representchanges in displacement in a dynamic systemwhich will ultimately be the source of excitation fornoise and vibration in the system. A designershould look for a reduction in the amplitude of thiscurve form to reduce excitation.Figure 1. Combining the to
28、oth profilesFigure 2. Tooth pair profiles offset one base pitch apartFigure 3. Calculating the loaded tooth deflection5Calculating the amount and extent of tipreliefThe 2D mapping technique was used by Munro 4to establish the theoretical basis for spur gear profilerelief design. See Figure 4 and equ
29、ations 1 and 2.Figure 4. Calculating the amount and extentof tip relief in low contact ratio gearsExtent e =EAP SAP ptb2 Porc(1)Amount r =Pmaxc+ fp(2)wherePmaxis maximum load per unit face width(N/mm);Pois design load per unit face width (N/mm);SAP is start of active profile roll distance (mm);EAP i
30、s end of active profile roll distance (mm);c is tooth pair stiffness (N/mm/m);fpis adjacent pitch error (m);r is extent of profile relief from tip;ptbis transverse base pitch.Where the extent occurs at one base pitch from thestart of active profile is termed long relief, and wherethe extent occurs a
31、t half the remaining distancefrom the long position to the end of active profile istermed short relief. The loaded transmission errorsof these 2 types of relief have very differentcharacteristics as will be described in the followingpages. See Figure 5.The effect of any linear tip relief can be show
32、n ongraph paper. The examples for the case of an inter-mediate relief at varying loads using linear tip reliefare shown in the following diagrams. What thetheory allows is the adjustment of the extent of reliefto obtain low variation in transmission error at aspecific design load.The charts in Figur
33、e 6 show that the amount andextent can also be adjusted to account for adjacentpitch errors. Additional relief is applied to com-pensate for the pitch error and the extent of reliefmade slightly shorter to maintain the design loadlow T.E., in this case load 2.Experimental validation of Munrostheoret
34、ical basis for the application of tipreliefMunros theory of transmission error was experi-mentally proven during the 1990s ref. 5, where aseries of 6 low contact ratio spur gears with thesame amount of tip relief but different extents weretested. These results are displayed in Figure 7.The differenc
35、e in the overall T.E. level of the curvesin the measured data in Figure 7 is due to the bear-ing deflections that are not considered in the 2Dmapping.The peak to peak T.E. and the measured soundpressure level for the long, short and intermediatetip relief cases are shown in Figure 8.Figure 5. The ex
36、tent of long and short tip relief6Intermediate tip relief Intermediate tip relief with adjacent pitch errorTip contactIntermediate tip relief with additional pitch errorallowance and extentIntermediate tip relief with additional pitch errorallowance and extent with pitch errorLow T.E.No tip contactI
37、ntermediate tip relief with additional pitch errorallowance only no extent adjustmentT.E. variationFigure 6. Mapping the 2D transmission error7Figure 7. Measured and predicted quasi-static transmission errorFigure 8. Measured quasi-static transmission error and sound pressure level83D Effects and us
38、ing a simple strip theoryto calculate T.E.Although the 2D method produces reasonable T.E.predictions there are effects from sources such aslead modifications or mesh misalignment acrossthe tooth surface that cannot be taken into accountusing this method. A simple strip model can be usedto approximat
39、e the 3D effects, where the gear isdivided into a series of strips, or narrow spur gearsall acting in parallel but independent to each other.They can even be incremented rotationally torepresent a helical gear. See Figure 9.Figure 9. Dividing the gear into stripsEach strip has its own stiffness whic
40、h can be asingle value or vary from SAP to EAP. The examplein Figure 10 shows a case of long relief where thetooth stiffness is reduced to 70% in a parabolic man-ner from the pitch point to the SAP and EAP. Invest-igations into measuring the tooth stiffness have pre-viously been undertaken. Ref. 6Th
41、e example in Figure 11 shows that misalignmentor lead modification, (lead crown in this example)can change the effective tooth stiffness. This couldchange the optimum load level and also the contactmay extend to the EAP even though enough tiprelief was applied to prevent this in the 2D modelsince th
42、e tooth will deflect more.The strip method is reliant upon the tooth stiffnessdata being representative of the tooth geometry.Improvements to the model employ a tooth stiffnesscalculation and also link the deflections of each stripto each other across the face width.The stiffness is made a function
43、of height position onthe tooth profile and lateral position across the toothface Ref. 7 8. The 2D and improved stripmethods represent a quick and relatively easy cal-culation, especially when converted to a computerprogram. They have the limitation of being aninaccurate representation of the behavio
44、r of thegear as there is in fact a complex relationshipbetween the force applied and the deformation.They will give a good approximation in general oper-ation, but may be found to be inaccurate in limitingcases or extreme geometry. This can have an im-pact when designing for safety critical, special
45、ized,or high-cost applications.Figure 10. Long relief with variable tooth pairstiffnessFigure 11. Long relief with lead crowncalculated using the strip method9More advanced 3D tooth contact analysisusing FEA for tooth bending stiffnessA Finite Element (FE) method to calculate toothstiffness provides
46、 an improved level of accuracy asit represents the relationships between all neighbor-ing points on a surface and sub-surface regardlessof geometry. Previously an FE calculation used tobe highly specialized requiring hours or days tocomplete. Due to improvements in computer powerthe calculation can
47、be reduced to minutes enablingthe development of a practical design tool. The restof the examples in this paper are produced using theGATES (Gear Analysis for Transmission Error andStress) software originally developed at the DesignUnit, Newcastle UK. The software calculates thetooth stiffness using
48、 a 3D FE model. It also includeseffects such as extended contact at the tip of thegear if insufficient tip relief is applied. There areother programs commercially available that use ad-vanced methods for the 3D stiffness effects. Thetransmission error results are shown in Figure 12for the zero, very
49、 short, and very long tip relief case.For the zero tip relief case the results show theeffect of the extended contact as a rounded effectrather than a step change in the T.E. This effect willbe covered later in the paper.The GATES tooth contact analysis is used to revieweffects of tip relief in high contact ratio gears andtheir potential to produce lower transmission errorlevels. Some of this material is covered by Yildirim9, who extended Munros theory to high contactratio spur gears.In high contact ratio gears the long and shortdefinitions of tip re
