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30000 2500O 20000 --_ 150% Load •_ 125% Load 15000 -- 100% Load 10000 Load Step 1 50OO 1 i 0 l 5 a) 30000 i r 10 15 Crack front position (Orientation: heel to toe) i 20 25 I J 25000 20000 .E 15000 / 10000 Load Step 11 --*-150% Load --_-125%Load 5000 --100% I i 0 -5000 i q T i 5 10 15 20 i 7.10: NAS A/CR--2000-210062 KI distribution Load 25 Crack front position (Orientation: heel to toe) b) Figure f for load step 93 one (a) and load step eleven (b). 15oo Load Step 1 --*--150% Load 125% Load 1000 -- 100% Loa_ 1'7 (2; 0 E. .... -500 _ , \\_ t _ -1000 _'! _ -1500 J j. >.--, _, / , _Xe=_j bier/ Crack front position _ (Orientation: heel to toe) I a) Load Step 11 --,- 150% Load 9000 ' 8000 - --_ 125% Load ad 7000 6000 ¢5 •- 5000 .*.., 4000 3000 2000 1 I000 ti i 0 NASA/CR 5 I0 i i 15 20 25 Crack front position (Orientation: heel to toe) b) Figure t 7.11" Ku distribution 2000-210062 for load step 94 one (a) and load step eleven (b). 0.5 7 Load Step 1 _ 150% Load 0.4 125% Load 0.3 , -- 100% Load /1_ / 0.2 _ 0.i _=_,,. 0 \ 5 ............. i "_ -0.1 ! -0.2 r I//_ f V // _ \ / _/" 25 _Crack front position _ (Onentation: heel to toe) a) 0.5 Load Step 11 0.4 - J ./_ ,/" ./ / _,,,, --'-150% \,,, Load ---125_Load 0.3 0.2 1 0.1 -i i o_ 5 i I i i 10 15 2O 25 Crack front position (Orientation: heel to toe) -0.1 -0.2 - b) Figure 7.12: NASA/CR--2000-210062 KI1/KI distribution for load step 95 one (a) and load step eleven (b). 0 25 Load Step 1 --_ 150% Load [degrees] 20 _ 125% Load -- 100% Load 15 10 5 ,_ 0 -5 -10 -15 -20 -25 -30 -35 a) i i 5 10 i _ 15 i 20 -5 Crack 25 front position (Orientation: heel to toe) -10 -15 -20 -25 X,_ / _i" -_ / f -30 i50% Load ---125% -35 -- 100% Load I -40 I 0 -45 b) Figure [degrees] 7.13: Kink ,_ angle distribution based on the maximum load step one (a) and load step eleven 7.4 Highest Point of Single Comparison the gear's These operating results NASA/CR--2000-210062 were studies conditions reported Tooth Contact to determine were performed 5.2. 96 model when Similar stress theory for (b). (HPSTC) the smallest in Section principal Analysis that accurately developing comparisons the represents BEM are now model. made between the moving load method and a simplified loading assumption is that the moving load method is most accurate. assumes a cyclic load at the HPSTC on the pinion tooth. contact final load ellipse number eleven from the discretized moving step of single tooth contact in the discretized is defined as 100% design load. The model method. load; load step eleven HPSTC the The method to propagate the crack The method assumes proportional produces maximum K1,,,a, and that R is zero. principal stress theory The using length of the initial / / from the moving The cross section Fixed crack is the the moving under the HPSTC is described loading. It is assumed that direction the from ratio of growth of K, is determined to K1 from the loading. The extensions for the discrete crack front points are calculated model modified to account for crack closure. Figure 7.14 is a comparison trajectories occurred. the load data. The magnitude of the parameters and material properties from the moving load analyses are used in the HPSTC predictions. The initial crack location and geometry are the same as those load analyses. Section 3.2.3. Again, The simplified method The HPSTC is taken as in the by HPSTC with Paris' of the crack and HPSTC load methods. Roughly 190,000 cycles have view is taken at the approximate location along the tooth front's midpoint. \E_ Moving,,_,, /' __E/x_Serimental a) Tooth surface \. / ,,-.. / ",Moving ',, _', b) Figure Cross initial section crack ,," ," Experimental /ff of tooth at midpoint 7.14: Comparison of crack trajectories from moving load and HPSTC (fixed location) methods after N 190,000 cycles. NASA/CR--2000-210062 97 load of The midpoint of the crack front is deeperin the HPSTCanalysesafter 190,000 thousand cycles. From Figure 7.14b, it appears that the moving load analysis trajectory will produce rim failure. when the predictions method predict tooth load prediction This comparison matches more closely is purely qualitative. Several obvious differences methods can be observed. larger kink at theheel end; end. Considering shifted 5.14, however, of Section may conclude the observed between the shows that the crack turns the static and moving load into the rim in the moving trajectories trajectories in the tested predicted by pinion. the two As seen in Figure 7.14a, the HPSTC method predicts a the moving load method predicts a larger kink at the toe the location load analyses One Figure are continued. Therefore, both failure. The slope of the trajectory from crack face area since the cross the cracks on the tooth surface this conclusion. of the HPSTC load, this result is consistent with the 7.3.3. Figure 7.14b section view are roughly that the HPSTC method predicts a larger of the crack is deeper, yet the lengths of equal. Figure 7.15, in general, supports 0.25 ,-.., 0.2! ing Load •-_ 0.15 j 0.05 0 _ 0 50000 100000 150000 200000 250000 300000 350000 N [cycles] Figure 7.15: Crack area versus number prediction In summary, load analyses. The the HPSTC crack analyses trajectory of load cycles for HPSTC and moving load methods. predict and fatigue the same failure life calculations mode vary as the moving between the two methods. Since no experimental fatigue life data exists, the accuracy of one methods fatigue life prediction over the other methods can not be evaluated. The moving load NAS A/CR--2000-210062 98 predictions match the experimental trajectory into the section of the tooth better than the HPSTC prediction. through the rim is what determines tooth failure rim and through the cross Since the trajectory into or or rim failure, it is concluded that the moving load method is necessary to capture that result trajectories on the tooth surface at the heel end, however, most accurately. are in reasonable All of the agreement. Nonetheless, is the significant decrease a distinct computational case needs advantage time to perform of the HPSTC the crack method propagation predictions since only one in load to be analyzed. 7.5 Chapter The Summary results from a fatigue were compared to crack summarized as follows: growth The simulations • The original trajectory predictions observed fracture surfaces in the test. contact probably information growth observations • on the tooth predicted crack in a tested a reasonable modeled used simulation fatigue pinion. failed to capture It was determined the tooth contact in the predictions Change became in contact location more flexible. 2. Differences 3. Crack 4. Misalignment in the magnitude growth under load test bevel of the loading The alignment crack tooth between Some explanations determined to be: as the are to the test data. detailed aspects that the simulated perfect pinion comparisons in the test incorrectly. assumed in the The life with respect pinion and the gear and that the gears were not flawed. differences in contact between the test and theory were 1. in a spiral grew the for the and the tooth of loading. control (simulation) versus displacement control (test). • Additional between simulations observed in the test. location of the crack crack demonstrated through studies were crack growth method's determined that: • The fatigue yielded crack conservative • The crack • A reasonable geometry growth to determine would model how modify the parameters is best described approximation when • The trajectory observed toward the toe end. • The increased on the fracture changes trajectory in some predictions. used as plane in the The initial of the studies prediction strain. of the dimensionless calculating in the SIFs, tested is a value pinion quantity 99 fl, which incorporates of 1. would result torque levels might explain the significant surfaces of the tested pinion. NASA/CR--2000-210062 crack results. front condition effects the of steps. conducted rate to predict crack, which was assumed to approximate the to the formation of the ridge, was used and the a series assumptions in the test. the capability A large initial front just prior was propagated Sensitivity the gear and pinion from amounts a contact of rubbing biased seen A simplified loading methodthat assumesa cyclic load at the HPSTCon the pinion tooth during meshingwas investigated. The failure modepredicted by this methodwas the sameas the moving load predictions. However,the crack trajectory andfatiguelife calculationsvaried betweenthe two methods. The HPSTC methodis advantageous becauseit significantlyreducesthe computationaltime. However,upon comparison of the results from the two methods to experimental results, it is concludedthat themoving load method'strajectoriesaremoreaccurate. In summary,insightsinto the intricaciesof modeling fatiguecrack growth in three dimensionswere gained. Preliminary stepstoward accuratelymodeling crack growth in complicatedthree dimensionalobjects such as spiral bevel gears were completedsuccessfully. To improve the accuracyof the simulations,the changein contactbetweenspiral bevelgearteethduring operationasa crackevolvesis needed. NASA/CR--2000-210062 100 CHAPTER CONCLUDING 8.1 Accomplishments This thesis and investigated spiral bevel gears. because a crack's catastrophic. Having to prevent limited gear Predicting trajectory catastrophic geometry Significance computationally crack growth determines the capability to modeling Prior in gears of Thesis modeling crack to this growth thesis, with simpler model of complicated few predictions Accurately pinion required fatigue crack growth The geometry Army's model of NASA/GRC and three Helicopter the pinion was developed that calculates the surface contact analysis program of contact represented by discrete LEFM theories predictions. The determined fatigue crack calculate In Only loads However, of freedom, dimensional bevel crack used also for traction patches were combined used pinion gears had and spiral been theories demonstrative bevel performed. bevel for predicting system of the U.S. purposes. A BEM computer program developed by of a spiral bevel gear tooth. Their to determine and based been Spur in a spiral the its mating location, gear. on computational orientation, The on the gear tooth. with the BEM to accomplish were had and no closed form cracks. Prior to this trajectories capabilities using a coordinates the by a Paris model, growth This methods contact the crack fracture was growth mechanics by the Cornell Fracture Group, which allow for arbitrarily shaped, curved crack fronts and crack trajectories. The crack trajectories rates kink angles. operation, magnitude. was between simulations software developed three dimensional in spiral was in a designer such as spur gears. analyses. fatigue the state-of-the-art allows rates and crack trajectories. of a spiral bevel pinion from the transmission Kiowa magnitude were growth dimensional expanding OH-58 tooth of crack modeling numerical growth of gear design be benign or the geometry and cracks. Three to create, require greater computing power because of the significant increase in degrees solutions exist to predict the growth of arbitrary three thesis, crack in gears geometry, the use of two dimensional gears require a three dimensional dimensional models are much more fatigue is significant in the context whether the failure will to predict failures. cracking permits EIGHT: REMARKS the moving normal modified in conjunction load on load effect to a gear a tooth's to incorporate crack with the maximum gear was tooth varies incorporated surface were closure, principal over time to calculate stress theory in location into the propagation considered. It was to and method. discovered that the moving normal load produces a non-proportional load history in the tooth root. Proposed prediction methods for fatigue crack growth under non-proportional loads in the literature were determined to be insufficient for the spiral bevel gear model. As a result, a method to predict three dimensional fatigue proportional loading was developed. The method front for a series of discrete load steps throughout cycles number cycle; were then specified, of specified the process NASA/CR--2000-210062 load was and cycles then the and repeated. crack was the calculated Some 101 crack growth under non- incrementally advanced the crack one load cycle. A number of load advanced aspects an amount trajectory of the from final based the crack single on the load trajectory predicted by this moving however, the method experimental data. Other For example, trajectories was confirmed. the root. is directly AISI 9310 focus of current steel compressive was design portion compared simplified approach analyzing crack crack growth The analyses This into Paris' life in a tested pinion; comparable to in the tooth's trajectory loads examination crack is significant a gear's weight. when did not significantly very close to growth rates in because Reducing model to calculate modify fatigue a principal the amount the rates crack were the in a gear that the when growth carried of crack rates. As out ignoring from the moving load crack propagation method were when only HPSTC was considered. HPSTC is a more and has been commonly Crack trajectories A crack crack method lack of experimental are of primary of fatigue crack growth in a tested spiral bevel surfaces. HPSTC when growth suggested This result growth scenario utilized fatigue of the crack growth rates predicted crack trajectory predictions agreed addition, the observations the surface was fatigue. fatigue The The in past research numerically fatigue trajectories. in gears. used loading. fatigue an evaluation load method's crack growth to predict was devised from the of tooth deflections by the NASA]CR--2000-210062 that on the contact the deflections adjacent tooth failure In the majority simulations of to in the gear. fatigue crack growth in spiral limitations can be summarized area between mating not modeled. Capturing this effect will increase the accuracy crack trajectories are Strongly determined by the load locations. It is anticipated the observations. A scarcity of experimental data prohibited validations of calculated rates, fatigue life predictions, and crack front shape evolution. effect data rate data and crack front shape information pinion motivated SEM observations of the that the failure mechanism along supported the use of the numerical trajectories rate by the two methods. The more closely to the tested importance As this thesis was a first attempt at predicting bevel gears, certain limitations were encountered. The as follows: limited the existing The dearth tooth failures The root, will remain on fatigue analyses of a spiral beve ! pinion of the loading history. propagation failures. predict a failure fatigue theories since there was a single load location and proportional in this thesis with the two loading methods predicted different and crack fracture the crack is to minimize of a load cycle predictions to predictions from a may increase the magnitude of the compressive stresses could influence crack growth rates. It was discovered a result, the BEM/LEFM the compressive portions pinion mode. from that has initiated of compressive examined. gear was incorporated hindered moving differed predicting the crack, the effect material in the gear tooth's root, which lives in For a crack above Additionally, The method issues related to modeling crack growth in a gear were also investigated. the effect of shifting the load location along a tooth's length on the crack load location closure load succeeded of a cracked picking up 102 the spiral load. bevel The crack gear growth teeth of the model gear tooth magnitude was since will of be this