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representedby a singlepoint, the cracktip. At a crack tip thereareonly two modesof displacement;in threedimensionalmodels,however,there is a distribution of three modesof displacementalongthe crack front. Propagatinga crack in two dimensions is completelydefinedby a singleangleandextensionlength. On the otherhand,along the crackfront thereis a distributionof anglesandlengths. Codesdevelopedby the Comell FractureGroup at CornellUniversity, suchas Object Solid Modeler (OSM) andFRactureANalysis Code- 3D (FRANC3D), have beendevelopedto handlethreedimensionalfractureproblems. FRANC3D explicitly models cracks and predictscrack trajectoriesunder static loads. The crack growth models are based on acceptedfatigue crack growth and linear elastic fracture mechanics(LEFM) mixed modetheories. Becausegearsoperateat high loading frequencies,the actualtime from crack initiation to failure is limited. As a result, crack trajectories and preventing catastrophicfailure modesaretheprimary concernin geardesign. Crack growth rates are not as important. predicting three The goal dimensional of this fatigue research crack is to investigate growth in spiral bevel issues gears. related to A simulation that allows for arbitrarily shaped curved crack fronts and crack trajectories will be most accurate. In addition, the loading on a tooth as a function of time, position, and magnitude should 1.2 Numerical be considered. Analyses Computational research area. analyses. As a result, In three of Gears fracture mechanics the majority dimensions, The [1988] complexity used the transmission noise. to little dimensional investigate Individual has been has gear modeled and weight function technique to evaluate has in numerical evolved. gear in two stress trajectories in applying AGMA Blarasin novel to two dimensional crack stresses, increase in accuracy when using the FEM over calculating gear tooth root stresses was demonstrated. FEM limited analyses tooth were is a relatively predicted developments gear gear teeth to gear design work some pertinent to gear design. of two FEM of work very gears. This section summarizes methods and fracture mechanics applied Albrecht resonance, dimensions and and the standard indices for et al. [1997] used the intensity factors (SIFs) in specimens similar to spur gear teeth. Cracks with varying depths were introduced in two dimensional models and a constant single point load was applied. The SIFs were determined of the as a function crack dimensional analyses loading thermal considered Lewicki trajectories never fatigue highest point most often NAS A/CR---2000-210062 Flasker in a gear of a car gearbox. growth from load 1997b] spur to experiments. dimensional crack tooth FEM of gears and loading 3 al. [1993] (HPSTC), used but two The variable and core were simulated with the crack was incrementally The work analyses geometries contact the case history, combined gears. et but predictions performed. single simple lives were calculated, crack of et al. [1997a, concerns Fatigue stresses a given in thin rimmed trajectory predictions Limited three work were to analyze at that point. Residual loading. Based on propagated. crack trajectory FEM of crack depth. and LEFM successfully to investigate matched crack been achieved. The conditions. Pehan et al. have [ 1997] used the FEM stresses sized due to look at two and three to case models were hardening were analyzed: dimensional modeled one tooth spur gear models. as nodal including thermal loads. the arc length Residual Two different of the gear rim directly below the tooth and three teeth with the corresponding gear rim arc length. To determine the new crack front, they used a criterion such that the SIFs along the new front should be constant. the SIFS near magnitude the midpoint and opening simple (mode The and BEM The analyses Their and et al. [1998] the FEM configuration. method for toad three the and characteristics explored with new crack fronts are of a split propagation studies tooth gear to a spur gear. paths for various locations. crack trajectories were performed in addition to calculating fatigue crack growth rates. Very little work, in addition to Lewicki et al.'s research, has used the BEM to gears. Sfakiotakis teeth considering predictions, they conditions bevel can locations. the BEM of crack be created Fatigue for stress gears. The progression investigation performed two dimensional from loading analysis of research growth related BEM analyses related in spiral Handschuh was not considered. to optimizing to computer bevel et al,'s gears. [1991] of gears models computer die of gear program error curves were generated that gave of spiral has led to the of spiral that bevel gears models the surface coordinates in three in conjunction with tooth analysis [Litvin et al. 1991], to determine how bearing (contact gear teeth) changes with different spiral bevel gear tooth surface Transmission of Fu et al. [1995] the forging analysis FEM cutting process of spiral bevel gears to determine tooth dimensions. Litvin et al. [1996] utilized this pro_am, contact mating simulations mechanical and thermal loads. Rather than perform trajectory calculated SIFs for different size initial cracks with various loading and crack also used [1997] dimensional is not crack analyze is that three constant propagation is similar on only crack effects crack configuration of this work location dimensional dimensional fracture lives based et al. to consider determining three locations point load Pehan since of the split tooth strong the fatigue A constant allowed limited to investigate single The to calculate front. performed geometry used was used gear geometry intensive Lewicki model of the crack spur I) effects. computationally accommodated. using Paris' an indication between designs. of the efficiency of the gear, has been Along with Litvin et al.'s work explored by Bibel et al. [1995 et al. [1991]. Bibel et al. successfully deformable contact bevel and analytically gears [1991], tooth contact analysis of mating gears and 1996], Savage et al. [1989], and Bingyuan using the FEM. modeled They modeled, multi-tooth conducted using spiral a stress gap elements bevel analysis from deflections can alter developed analytical methods contact ellipses due to elastic the to predict, deflections under loadsl Savage et al. and however, they did not incorporate NAS A]CR--2000-210062 contact zone between gear purpose finite general teeth. using tooth contact analysis, of a spiral bevel gear's shafts Bibel et al.'s work fracture mechanics. with spiral element codes, the rolling contact between the gear teeth. Bibel et al.'s used to investigate how changes in gear geometry affect tooth deflections. in tooth gears of mating was related to spiral On the other hand, work can be Variations Savage et al. the shift in and bearings bevel gears, Bingyuan et al. approximated the geometry of gears in contact as a pair of disk rollers compressed together. The linear elastic stresses in the disks could be written in closed form. The SIFs were calculated using the closed form expressions. focus was to calculate surface fatigue life and compute trajectory to predictions were Bingyuan et al.'s primary crack growth rates. No made. The majority of the aforementioned research on spiral bevel gears failure, but rather associated with design and efficiency; methods developed Crack to create trajectories represented by have two research to date. in spiral bevel 1.3 numerical been predicted dimensional The next of spiral gears in gears with This thesis models. and predict simpler step is to computationally geometry is a natural model contact areas. that can extension fatigue crack be of the trajectories of Chapters This thesis is divided overview and summary. propose, apply, and into The eight chapters. remaining evaluate The chapters methods each for predicting first and last build fatigue chapters upon one crack growth gears, with are another and in spiral gears. Chapter Two contains background information attention to spiral bevel gears. The objective related to spiral bevel gears that will be used work of the design thesis is further objectives for gears. A focus of this thesis can be used LEFM to analyze and Chapter fatigue Three. to compute crack bevel gears. Overview bevel models is unrelated have been theories by examples is to demonstrate complex that Methods crack motivated gear of gear failures that computational geometries are utilized under realistic to accomplish that are currently trajectories on implemented are demonstrated through and of the rim thickness. compression, concept in AISI then growth If fatigue rates of fatigue crack closure 9310, a common gear discussed. may crack warrant fracture mechanics conditions. this task are presented in two and three rates attention are highly method is presented on calculated tooth root is a sensitive in designing is used to investigate fatigue crack steel. First, the concept of fatigue A material-independent in dimensions examples. growth more the current loading Chapter Four explores the significance of compression loading growth rates. The magnitude of compressive stresses in a gear's function particular is to define vocabulary and concepts throughout the thesis. In addition, the to this gears. The propagation rates crack closure is for obtaining fatigue crack growth rate data that do not vary with stress ratio. The method is demonstrated using data at various stress ratios for pressure vessel steel. Next, the concepts are applied to AISI 9310 steel data used to investigate fatigue to obtain the effect Chapter Five is crack trajectories an intrinsic of low stress an fatigue ratios initial investigation in a spiral bevel crack growth on fatigue pinion into crack predicting under model. growth three a moving This model is in AISI 9310. dimensional load. First, a boundary element model of a pinion is developed. A method to represent the moving contact area on a gear tooth is discussed. Next, studies are conducted to determine the smallest model that still accurately Once the model is defined, a crack NASA/CR--2000-210062 represents the operating conditions of the pinion. is introduced into the model, and the initial stress 5 intensity fatigue applied factor crack history under trajectories to predict Fatigue fatigue crack the under crack growth moving the load moving is calculated. load growth trajectories results from A method is proposed. and rates a spiral bevel The in a spiral pinion to predict method bevel is then pinion. in operation are necessary to validate the predictions. The sponsor of the research efforts of this thesis, NASA-Glenn Research Center (NASA/GRC), provided a pinion that was tested in their gear test fixture. Notches were fabricated beginning the test. The test data In addition, in an effort to obtain the fracture surfaces are observed are given in the chapter. The crack trajectory pinion the are compared prediction and in Chapter and test, the into several and crack growth crack front shape with a scanning fatigue life results Seven. influence electron microscope, from the simulation To gain insight of model of the teeth's roots prior to results are presented in Chapter 6. and crack growth rate information, and the results and the tested into the discrepancies parameter between assumptions and loading simplifications on crack trajectories and calculated fatigue crack growth rates are studied. Next, the necessity of the moving, non-proportional load crack growth method is evaluated by comparing the results to predictions that assume proportional loading. Finally, previous Chapter chapters. Eight Implications summarizes the accomplishments of the research work are given. NASA/CR----2000-210062 6 conducted of the work and suggestions in the for future CHAPTER GEAR TWO: GEOMETRY AND MODELING 2.1 Introduction Chapter Two This terminology this thesis. covers A gear's fundamentals the basic terms and background design are explained and geometry is essential and geometry aspects to motivate can be quite of a spiral the numerical complex; bevel simulations however, only gear. of the in this chapter. 2.2 Basics of Spiral Bevel Gear Geometry Gears two mating pinion, are used in machinery gears have and the larger similar to transmit shapes. the gear. Motion successively engaging teeth. There are various types of gears. the mating gears are mounted the gear types. Gears with common be used. The motion. smaller shape shape of a bevel gear dimensional difference lies. from are a few of the distinguishing intersecting shafts are called the by at which between The most and the shafts of the gears are parallel. The geometry of a fully illustrated in two dimensions. However, the conical requires a three dimensional illustration. This two and three is where the complexity of the work contained in this thesis Axes of gears run parallel to each other Gear NASA/CR--2000-210062 is called any intersecting angle could as illustrated in Figure 2.1, ® operate The to another characteristics bevel gears. Pinion a) Spur gears gears one gear in pairs. of the teeth and the angle angle to mount bevel gears is 0 = 90 ° , although A bevel gear's form is conical. For comparison, spur gears are cylindrical, spur gear can be almost operate of the mating is transferred The Gears with parallel axes 7 B b) Bevel gears The tangency define operate with intersecting Figure 2.1" Schematics cone with the the pitch mating gears, geometry, defined angle gear is called The o.__/COl,which gear ratio also equals the ratio of the number a) Straight bevel Figure Two common The main difference the straight by the mating cones. bevel axes of spur (a) and bevel between the pitch is the ratio the ratio gear's In Figure of the angular of sin(02) b) Spiral gear drawings axis cone. of gear teeth to the number gear 2.2: Bevel a bevel (b) gears. bevel and the line 2. i b, O_ and frequencies to sin(01), of pinion or, of 82_ of the due to teeth. gear [Coy et al. 1988]. bevel gears are the straight bevel gear and the spiral bevel gear. between these two gears is the shape of their teeth. The teeth of gear are straight, and the teeth of the spiral bevel gear are curved. Figure 2.2 illustrates this difference. When looking along the axis of a spiral bevel gear, the teeth will either curve counterclockwise or clockwise, depending on whether NASA/CR--2000-210062 8 the gear mesh, is left- a spiral or right-handed, bevel gear respectively. and pinion So that the will always have opposite and height of a spiral bevel gear tooth varies along the tooth is the heel, and the smaller the toe. The curvature and convex tooth surfaces on opposite sides teeth of the tooth, can hands. fit together, The or thickness cone. The larger end of the of the tooth creates concave Figure 2.3. Heel Convexside Concav_ Figure The tooth 2.3: Schematic profile, as shown of a single in Figure spiral bevel 2.4, is one gear tooth. side of the cross section of a gear tooth. The fillet curve is at the bottom of the tooth profile where it joins the space between the teeth. The region of the tooth near the fillet is the bottom land, and the area near the top of the profile is the top land. Top Land Tooth Profile Fillet Curve \ Bottom Land 3 Figure 2.4: Schematic of cross The advantage of the spiral bevel gear's one tooth to be in contact at a time. This makes bevel gear of equal size. Consequently, high speed and high force applications. this thesis, is in helicopter transmission NAS A/CR--2000-210062 spiral section of a gear tooth. curved teeth is to allow it significantly stronger bevel gears for more than than a straight are commonly found in One such application, which is the focus of systems. The mating spiral bevel gears in the 9 transmissionsystemconvertthe power from the horizontalengineshaftto the vertical shaft of the main of 6000 rotor. Many sketched parallel in Figure from a spool while an involute curve. generated tooth's Gears rpm and transmit from profile. in this application on the order axis gears, 2.5, the such coordinates cutting as spur gears, involute curve circle evolute A closed of a spiral form curves. The solution at rotational speeds involute tooth profiles. As by unwrapping thread involute bevel in creating gear tooth. the gear, along of a spiral model. The using coordinates the tooth geometry Involute Figure curve then for the coordinates along becomes a spur the curve exists gear for the tooth's surface coordinates can be calculated geometry of a spiral bevel tooth is much more calculates the coordinates input to a finite element NASA/CR--2000-210062 have can be visualized and there is no closed form solution et al. [1991] developed a program process operate keeping the thread taut. The path traced by the end of the string is The spool is the evolute curve. All involute gear geometries are this type of geometry. As a result, with relative ease. However, the complex, Handschuh typically of 300 hp of power. The to describe the surface to numerically calculate program with the basic models the gear geometry. coordinates. the surface kinetics The program bevel gear tooth in three dimensions numerical models in this thesis were as defined by Handschuh 2.5: Generation of an involute 10 for use as all created et al.' s program. Curve curve. of the Tooth fractures Figure A spiral 2.6:OH-58 bevel gear Army's OH-58 Kiowa fracture during an experiment set, spiral set is used Helicopter. is shown schematically are supported large shaft. in Figure pinion in the with two fractured main An OH-58 The geometry of the OH-58 a 19 tooth spiral bevel pinion pinion's shafts of the pinion's bevel spiral in Figure rotor bevel teeth. transmission pinion U.S. tooth 2.6. gear set will be used throughout meshes with a 71 tooth spiral by ball bearings. The approximate of the that exhibited The input dimensions this thesis. In the bevel gear. The torque is applied at the end of a pinion tooth are given 2.7. 099/ r............. ! : 1016 Figure 2.3 Teeth 2.7: Approximate Contact According and dimensions Loading to the theory of OH-58 of a Gear of gears, there spiral bevel pinion tooth. Tooth is a point of contact between a spiral bevel gear and pinion at any instant in time where their surfaces share a common normal vector. In reality, the tooth surfaces deform elastically under the contact. The deformation spreads traditionally been conventionally the point over using Hertzian approximated idealized NAS A/CR--2000-210062 of contact to spread a larger over an elliptical 11 contact area. The theory. area [Johnson larger This 1985]. area contact has is The center of theellipseis the meancontactpoint, which determinesthe on the tooth defined surface. by the tooth The orientations surface's gear and pinion. The length the ratio of the axes' lengths the equations respectively, for the length is [Johnson of the ellipse's geometry, curvature, contact minor and the ellipse's and major alignment location axes are between the of the axes is a function of the load. It can be shown that is constant and is not a function of the load. The form of of the ellipse's 1985] semi-major [Timoshenko and semi-minor axes, a and b, et al. 1970]: a = f (2.1a) b = gL-_-j where f and exerted g are functions on the tooth The meshing the area of contact of the mating process by the geometry. to the input gear teeth and magnitude time as the gear rotates. contact area along a tooth continuous defined is proportional torque The level iS a continuous of the force exerted magnitude process. between Figure 2.8 illustrates schematically of a left-handed spiral bevel pinion. has been discretized into a series of force, P, and gear geometry. The the teeth position varies of with the progression of the In the schematic, the of elliptical contact patches, or load step increments. The darkened arrow demonstrates the direction the load moves. The actual tooth contact pattern during operation is a function of the alignment of the gear and pinion. Heel Single tooth contact Double tooth Figure 2.8: Schematic Overlap single l tooth one tooth, of tooth contact shape and direction during left-handed spiral bevel pinion tooth. in tooth c0ntaci contact two teeth _ and double between tooth adjacent contact. teetff resuitsin At-the of the i3inion are in contact beginning with the gear. one load cycle two modes of contact: of a meshing As the of a pinion cycle for rotates, the adjoining tooth lose s contactwith the gear and only one pinion tooth receives all of the force. As the pinion continues to rotate, the load moves further up the pinion tooth, and the next pinion tooth comes into contact with the gear; the force on a pinion NAS A/CR--2000-210062 12