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tooth is once differ for single schematically again reduced tooth illustrated Time due to the double and double tooth in Figures Step Tooth contact. tooth The contact. The change in area contact 2.8 and 2.9. Tooth 1 2 2 4 5 6 Figure 2.9: Schematic NAS A/CR--2000-210062 of load progression 13 on adjacent pinion area of the contact teeth. will is In Figure2.9, tooth 1 and2 aretwo adjacentteethof a spiral bevelpinion. The ellipsesrepresent"snap shot" areasof contact betweena gear and a pinion's tooth. The darkenedellipseis theareathatis currentlyin contactwith the gearat a particular instantin time. Similar to Figure2.8,the larger ellipsesrepresentsingletooth contact, andthe smaller areareasof doubletooth contact. The first row in Figure2.9 begins with tooth 1 at the last momentof singletooth contact. After a discretetime step,the load on tooth 1 hasprogressedup the tooth andtooth 2 hascomeinto contactnearthe root, as depictedin row two. In the final row, or time step,tooth 1 losescontact and tooth 2 advancesinto the stageof singletoothcontact. It is seenin Figures2.8 and 2.9 that the contact areabetweenmating spiral bevel gearteethmovesin threespatialdimensionsduring oneload cycle. Most of the previous researchinto numerically calculating crack trajectoriesin gears has been performedon spurgearswith two dimensionalanalysesandhasnot incorporatedthe moving load discussedabove. Instead,a singleload location on the spur geartooth that producesthe maximumstressesin the tooth root during the load cycle hasbeen usedto analyzethe gear. This load positioncorrespondsto the highestpoint of single tooth contact (HPSTC). Contact between spur gear teeth only moves in two directions, and,therefore,this simplification to investigatea spur gearunder a fixed load at the HPSTC has proven successful[Lewicki 1995] [Lewicki et al. 1997a]. However, since dimensions, the contact the crack dimensional effect. first and compared This approach It has force over assumption that the area between front trajectories As a result, to trajectories is detailed been is the case with gears of the gear teeth be significantly moves influenced in three by this three 5 and 7. assumed since in the tooth where the maximum tensile bending stresses is at the fillet above discussion that The The in the numerical simulations. Dike also asserts that there are two the traction, or lubricant normal main will make forces. areas This in a gear stresses may cause damage. The first is the location of at the fillet of the tooth on the same side as the load. The on the side opposite by drawing an analogy the load, between where the maximum a cantilever beam Basic beam theory predicts that the maximum tensile connection on the outer most fibers on the same side maximum compressive stress occurs at the same vertical and a stress as the location, the load. Similarly, as a gear tooth is loaded, it creates tensile root of the loaded side. In the root of the side opposite the load, there are compressive stresses. fillet and root of the next tooth. NASA/CR--2000-2 small used. to the of the tooth gear tooth, Figure 2.10. occurs at the beam/wall load. forces is always compared compressive stresses occur. This can be visualized on the side opposite stresses in the tooth a lubricant frictional assumption will be utilized In the same paper, applied bevel the contact area, is normal to the surface. Dike [1978] points out that this is valid if there are no frictional forces in the contact area. He also states magnitude second could spiral trajectories under the moving load should be predicted considering only a fixed loading location at HPSTC. in Chapters implicitly mating ! 0062 These compressive 14 stresses might also extend into the Applied Load Maximum Maximum tensile \ stress _ / _ a) , 2.10: The stress IIII Maximum l 1 Maximum tensiZe \ l 1/ compressive stress b) Figure compressive Stresses compressive J in cantilever stresses beam are (a) are analogous noteworthy because to gear tooth root (b). Lewicki al. et [1997b] showed that the magnitude of the compressive stress increases as a gear's rim thickness decreases. The compressive stress could affect the crack propagation trajectories and crack growth large compressive influence have rates. However, stresses it is demonstrated compared on crack predictions. Up to this point, only frictional been discussed. The normal considered Some residual always hydraulic in this thesis. of these stresses used additional However, loads loads loads gears are in operation, sources dynamic do not 15 do produce effects, of the gear. lubricant pressure. NASA/CR--2000-210062 4 that low stress stresses, have ratios, forces get inside since surface to be on the gear. centrifugal In addition, could i.e. a significant and traction normal to the tooth's are the only loading conditions additional include due to the case hardening when in Chapter to tensile forces, a lubricant a crack and is and create 2.4 Gear Materials As discussed in Section transmission systems. performance of the gear. 2.2, spiral bevel In this application, Most often Super variables most Nitroalloy, such and EX-53. as temperature, important for gears the loads, lubricant, fatigue 2.1" Chemical C Mn Minimum 0.07 0.40 .... 0.13 0.70 0.015 of material impacts the life is used. and cost. is dependent The composition corn 9osition of AISI P S Si Maximum material in helicopter iron or steel alloy material life, hardenability, yield strength. Table 2.1 shows the chemical Table 2.2 contains relevant material properties. Table gear's used and The bevel gear is AISI 9310 steel (AMS 6265 or gear steels are VASCO X-2, CBS 600, CBS The choice are surface are commonly a high hardenable traditional material for the OH-58 spiral AMS 6260). Some other aircraft quality 1000, gears characteristics fracture of AISI 9310 b Cu Ni 0.15 -- 3.00 1.00 0.35 0.35 3.50 1.40 on operating toughness, 9310 JAMS and 1996]. 9ercent [AMS 1996]. Cr B Mo Fe -- 0.08 95.30 0.15 93.39 = Most In general, 0.015 gears are case hardened. Case hardening increases the gears are vacuum carburized to an effective 0.040 in (0.813 mm - 1.016 mm). The case hardness C (RC), and the core hardness is 31 - 41 RC [AGMA Table Tensile Yield pitting, failures. crushing. Strength Stren£th 2 Toughness Average Grain can be is 60 - 63 Rockwell 185 x 10 3 psi 30 x 106 psi 0.3 Fracture failures specification 1983]. 160 x 10 3 psi Modulus Ratio to Model the wear life of the gear. case depth _ of 0.032 in - pr c)perties of AISI 9310. _ Young's Poisson's 2.5 Motivation Gear 2.2: Material 0.001 85 ksi*in °5 3 ASTM Size 4 No. 5 or finer 0.00244 in) Gear Failures categorized into several failure modes. Tooth bending, spalling, and thermal fatigue can all be placed in the category of fatigue Examples of impact type of failures are tooth shear, tooth chipping, and case Wear and stress [Dudley 1986], bending impact, the three rupture most and abrasive from tooth bending bevel gears, fatigue are two additional common tooth to spalling wear. failures He gives to rolling modes of failure. are tooth bending examples contact The effective case depth is defined as the depth to reach 50 RC. x[Coy et al. 1995] 2 [Townsend et al. 1991 ] 3 [AMS 1996] NAS A]CR--2000-210062 16 fatigue According fatigue, of a variety in both spur to tooth of failures and spiral The focus of this thesisis on tooth bendingfatiguefailure becausethis is one of the mostcommonfailures. In general,tooth bendingfatiguecrackgrowth canlead to two typesof failures. In rotorcraftapplications,the type of failure could be either benign or catastrophic. Crack propagationthat leads to the loss of one or more individual teethwill mostlikely be a benigntype of failure. The remaininggearteeth will still be able to sustainload, and the failure shouldbe detecteddue to excessive vibration andnoise. On the other hand,a crack that propagatesinto andthrough the rim of the gearleavesthe gearinoperable. The gearwill no longer be able to carry any load,andwill mostlikely leadto lossof aircraft andlife. Alban [1985, 1986]proposesa "classic tooth-bendingfatigue" scenario. He suggestsfive conditionsthatcharacterizethe"classic" failure: 1. The origin of the fractureis on theconcavesidein the root. 2. The origin is midwaybetweenthe heel andthetoe. 3. The crack propagatesfirst slowly toward the zero-stresspoint in the root. As the crack grows,the location of the zero-stresspoint movestoward a point underthe root of the convexside. The crackthenprogressesoutward throughthe remainingligamenttowardthe convexside'sroot. 4. As the crack propagates,the tooth deflection increasesonly up to a point when the deflection is large enough that the load is picked up simultaneouslyby the next tooth. Since the load on the first tooth is relieved,the rateof increasein the crackgrowthratedecreases. 5. No materialflaws arepresent. Alban presentsresultsfrom a photoelasticstudyof matingspurgearteeth. The studydemonstratesthe shift in thezero-stresspoint. The zero-stresspoint is wherethe tensile stressesin the root of loadedsideof the tooth shift to compressivestresseson the load free side. Figure 2.11 showsstresscontoursfor two matingspur gearteeth. In the bottom gear,oneof the teethis crackedandanothertooth hasalreadyfractured off. The teeth of the top gearare not flawed. By comparingcontoursbetweenthe matingcrackedanduncrackedteeth,it is easyto pick out the zero-stresslocationshift towardtheroot of the load free side. The shift of the zero-stresslocationdemonstrates the changingstressstatein the tooth. This changingstressstatedrives the crack to turn. The point in the two dimensionalcrosssectionwherethe crack turns is actually a ridge when the third spatialdimension,the length of the tooth, is considered.This classic tooth failure scenario will be used as a guideline when evaluating the predictionandexperimentalresultsin thefollowing chapters. NASA/CR--2000-210062 17 Compressive stress Zero-stress point Tensile stress _....____l ' _e_nt str_SrSalture d /tooth Figure 2.11: Photoelastic Crack .... from mating results photograph 2.5.1 Gear from spur gear teeth [Alban (stress contour 1985]). Failures Gears in rotorcrafl applications Therefore, gear failures are not common. result from manufacturing Some of the more thick, grinding Dudley [1996] flaws, gives burns on metallurgical an overview common are currently designed for infinite However, failures do occur primarily case, and misalignment. of the various metallurgical the flaws, flaws core listed hardness life. as a factors affecting are case too low, depth a gear's life. too thin or too inhomogeneities in the material microstructure, composition of the steel not within specification limits, and quenching cracks. In addition, examples of surface durability problems, such as pitting, are presented. failure ............. Pepi [1996] A pitting examined flaw could develop a failed spiral bevel A grinding bum was determined learned that the carburized case crack origin, microstructure cause which contributed inhomogeneity, of a fatigue failures could Albrecht helicopter, crack which to crack introduced as manufacturing [1988] gives were caused et al. [1993] gives an example The excessive misaliglament gear an example growth. during AISI crack in an Army as the origin of the fatigue was deeper than acceptable in a carburized be classified into a starter crack. limits for a fatigue cargo helicopter. In addition, in the area it was of the Roth et al. [1992] determined the remelting process, to be 9310 spiral bevel gear. Both a the of these flaws. of a series by gear resonance of failures with insufficient in the Boeing damping. Chinook Couchon of a gear failure resulting from excessive misalignment. was due to a failed bearing that supported the pinion. The misalignment led to a fatigue crack on the loaded side of the tooth. An analysis of an input spiral bevel pinion fatigue crack failure in a Royal Australian Navy helicopter NAS A/CR--2000-210062 18 is given by McFadden occur in service. Gear changes experts in the weight, strength [1985]. These are researching geometry. noise, and and crack ways However, reliability. trajectory to design spiral bevel gears demonstrate to make at the Geometry characteristics performance of proposed gear designs [1995], would be extremely useful. procedure examples gears same quieter time changes could of the gear. there OH-58 Spiral In rotorcrafl produced vibration result, recent design vibration and noise. using gear tooth design objective Adjusting strength [Coy et al. 1987] for the majority has focused In addition, Lewicki failures lighter between by Lewicki optimization strength changes and contact on the strength of vibration of the gear and box is [Lewicki et al. 1993]. In turn, the of the interior cabin noise. As a on modifying the gear's geometry to reduce the due to the application of the gear, a continuous et al. [1997a] showed jeopardize the that the failure mode gears is closely related to the gear's rim thickness. It was demonstrated initial flaw exists in the root of a tooth, the crack would propagate either rim or through the tooth result, a tool to evaluate do through is a tradeoff bending is to make the gear lighter and more reliable. the geometry of the gear, however, may characteristics. and and changes, such as discussed Savage et al. [1992] used an Bevel Gear Design Objectives applications, a primary source by the spiral bevel gears of the gear box accounts gear have negative effects on the A design tool to predict the parameters as constraints. Including effects of geometry failure modes could contribute greatly to his procedures. 2.5.2 that gear's in spur that if an through the for a thin rimmed and thick rimmed gear respectively. the strength and fatigue life characteristics of proposed designs and would be useful. Albrecht [ 1988] demonstrated life were insufficient. He also As a gear that AGMA standards showed the advantages method, such as the FEM, over the currently accepted The work of this thesis is an extension of the numerical stresses to determine gear stresses of a numerical simulation AGMA standards at that approaches to determine time. gear and life. 2.6 Chapter This Concepts visualize Summary chapter Characteristics of materials properties examples of gear significance covered basic terminology and geometry aspects of gears. related to spiral bevel gears were the primary focus. In addition, methods to and model the contact between mating spiral bevel gears were presented. a common will failures of modeling NASA/CR--2000-210062 gear be used and gear gear failures steel, in the design AISI 9310, numerical objectives numerically. 19 were summarized. simulations. were discussed Finally, to motivate These some the CHAPTER THREE: COMPUTATIONAL FRACTURE 3.1 Introduction the This chapter discusses work of this thesis. The BEM is used between the in a fashion methods areas of computational fracture mechanics relevant to areas of focus are LEFM, fatigue, and the BEM. The similar in three analysis how and where are used a crack The analyses to calculate of this work developed the SIFs. a geometry model of the OH-58 Mechanics are conducted The using by the Cornell post-processor to the boundary features to compute SIFs using closed FEM. The problems primary is that to the volume and/or stress SIFs difference with the BEM that is meshed results from a are in turn used to predict may grow. programs Fracture common elasticity are meshed, as opposed LEFM, the displacement mechanics 3.2 to the more dimensional only the surfaces, or boundaries, in the FEM. In computational numerical MECHANICS spiral bevel pinion. element solver the displacement and a suite of computational Fracture Group. OSM FRANC3D fracture is used to create is used as a pre- program, BES. FRANC3D correlation technique. and has built in Fatigue Westergaard [1939], Irwin [1957], and Williams [1957] form solutions for the stress distribution near a flaw. were the first to write Their solutions were limited to very specific geometries and loading conditions. Their results, in the form of a series solution, showed that the stress a distance r from a crack tip varied as r -_/2 . It can be shown for the stress that, under linear elastic near a flaw in any body, K t r and 0 are polar dependent sub- and coordinates the first term of the series loading (01 by: as defined (3.1) in Figure 3.1,fj is a function on the mode of loading, and Kt is the mode I stress super-scripts (/) denote mode I loading. Similarly, but with all of the sub- and super-scripts I replaced 21 with H or IlL of 0 that is intensity factor. The two other modes of loading can be defined as in-plane shear, mode II, and out-of-plane The stress solutions for mode II and III loading are identical in form NASA/CR--2000-210062 solution is given - (z) o,/" -where conditions, under mode I, or opening, shear, mode III. to Equation (3.1), OFF Y O'r0 X Figure A significant approaches the 3.1' Coordinate feature crack system of Equation tip, this first (3.1) term at a crack is that of the tip. as r goes series solution to zero, or as one approaches infinity. However, the higher order terms of the series will remain finite. For this reason, a large portion of LEFM focuses on this first term of the series expansion only. In reality, the stresses the tip where This zone LEFM ......... : do not approach linear elastic is called the infinity conditions plastic tip. do not h01d and plastic zone holds when the p!ast!c zone _e SIF is a convement at the crack and results issmall way There deformfition in blunting in relation to describe is a zone of the to the length the stress takes sharp scale and around place. crack tip. of the crack. displacement d_stributions near a flaw !9 i]nearelastic bodles. The SIF for any mode is a function ge0metryl Crack length, andloading. The general equation for a SIF is K = flo'_/-_ /3 is a dimensionless the far stress field factor stress. be seen on geometry, from Equation the energy supplied 2a is the crack (3.2) that length, the and units cr is of K are * leith. For a crack than or equal energy to the surface new (3.2) that depends It can of to propagate, to the energy necessary ....... system, the energy can formation. surfaces. LEFM assumes As a result, LEFM for new primarily system must energy the material supplied at a crack goes [ASTM 1997]. The tests can be performed tests affects NAS A/CR--2000-210062 the I loading mode will be a combination fracture criterion and crack 22 is self-similar. I loading. growth loading mode to pure toughness tip in Figure 3.1 under only mode I loading will extend along the x-axis. However, it is rare that a crack is subjected to pure mode I loading. loading pure specimen of fracture crack the under a standard values In other of all the modes. trajectory. For The the Kin. Fracture on geometry. in mode I, is crack realistically, direction subject to measure into forming tip will fail when mode I SIF, Kz, reaches a critica! intensity called the fracture toughness, toughness is a material property and by definition is not dependent Therefore, the criterion for fracture, or crack propagation, under LEFM, K t > K m . Standard be greater surface formation. When supplying go into plastic deformation or new that all of the predicts to the words, the More mixed example, The mode Mode I/