Probabilistic Simulation Approach to Evaluate the Tooth-Root Strength of Spur Gears with FEM-Based Verification

Transcription

1 Engineering, 2011, 3, doi: /eng Published Online Deember 2011 ( Abstrat Probabilisti Simulation Approah to Evaluate the Tooth-Root Strength o Spur Gears with FEM-Based Veriiation El-Sayed S. Aziz 1*, Constantin Chassapis 2 1 Prodution Engineering and Mehanial Design Department, Faulty o Engineering, Mansoura University, Mansoura, Egypt 2 Department o Mehanial Engineering, Stevens Institute o Tehnology, Hoboken, USA * eaziz@stevens.edu, hassap@stevens.edu Reeived Otober 13, 2011; revised November 1, 2011; aepted November 15, 2011 Dependeny on deterministi design tehniques without attention to inherent proess variations and unertainties in gear design and manuaturing proesses an lead to unreliable results and aet the perormane o a gearing system. A better understanding o the impat o unertainty assoiated with the system input on the system output an be ahieved by inluding reliability tehniques to aomplish a reliable design methodology. This emerged the need to onsider the probabilisti behavior o the stress distribution on the gear tooth during the design phase. The present eort reports on the appliation o the Stress-Strength Intererene (SSI) theory within the ontext o a Design or Reliability approah in support a detailed gear design methodology or the evaluation the tooth-root strength with FEM-based veriiation. The SSI theory is ormulated to predit the eet o the root illet generated by a rak or hob tool with and without protuberane on the gear system reliability. The results obtained rom the probabilisti analysis strongly agreed with the FEM results aross a range o dierent gear tooth illet proiles. A quantitative assessment o the investigated gear sets showed the highest tooth-root stress was assoiated with the lowest tip radius o the generating tool. This approah helps with making the deision by quantiying the impat o stress and strength variations during the gear design stage. Keywords: Gear Design, Design or Unertainties, Probabilisti Design, Reliability, FEM 1. Introdution Traditional deterministi design approahes ompensate or unertainties through the use o empirial saety ators, whih do not provide suiient inormation to ahieve optimal use o available resoures in terms o material, manuaturing and operational osts [1]. For example, omponents resulting rom suh onventional design methods are either over-sae leading to wastage o resoures or unsae leading to unexpeted ailures. They ail to provide the neessary understanding o the variability assoiated with the properties o materials, manuaturing toleranes and in-servie loading. To design a produt that will perorm a untion reliably, the reliability must be onsidered as an important untional requirement all the way through the design proess, rom the ustomer s need to the inal produt [2]. Addressing these issues omprehensively at an early design stage is neessary to produe ompetitive produt that untions onsistently during its intended servie lie [3]. Indeed, onsistent levels o saety and reliability an be ahieved based on the probabilisti design methods. Probabilisti design, suh as reliability-based design and robust design, oers tools or making reliable deisions with the onsideration o unertainty assoiated with design variables/parameters and simulation models [4,5]. It allows the designer to assess the reliability o the mehanial system. This is impossible with the ator o saety approah. One important task o a probabilisti design is unertainty analysis, through whih we understand how muh the impat o the unertainty assoiated with the system input is on the system output by identiying the probabilisti harateristis o system output. The unertainty in a design perormane is desribed Copyright 2011 SiRes.

2 1138 EL-S. S. AZIZ probabilistially (e.g., Normal Distribution) by its mean (μ), variane (σ 2 ), the probability density untion (PDF), or the umulative distribution untion (CDF), et. By understanding the probability distributions o the design parameters, the designer an design or a speii reliability or quality level by produing designs that are robust to variations. Figure 1 shows the dierene between deterministi method and the SSI method as a probabilisti design tool or estimating reliability. The area representing the unreliability is a part o the area o the intererene. The unreliability o the mehanial omponent is determined by the probability that the ailure governing stress exeeds the ailure governing strength. One the ailure governing stress and strength distributions are available, the unreliability or reliability o the omponent an be obtained by using the stress-strength distribution intererene approah. The lie and reliability o mehanial systems suh as the gearboxes depend on the lie/reliability harateristis o their omponents at a ertain deined load [6]. Gears, as the most important gearbox omponents, are relatively omplex mahine elements and there are many design parameters involved in their design. Their design methodology involves seleting appropriate attributes rom a large amount o engineering/standards data available in atalogues and design handbooks [7]. Gears are generally designed based on endurane (atigue harateristis) design standards. They should be seleted and shaped to either endure a nominal i.e. rated load ondition or unlimited load yles. Stresses suh as the bending stress at the tooth root under the nominal load must be below the endurane limit. The endurane limit values themselves are not exat, but statistial. For this reason, design standards suh as AGMA, ISO, DIN, et. inlude a number o ators suh as size, surae, lie, load, veloity, et. to adjust the endurane limit to be on the sae side. However, the gear tooth strengths are obtained using deterministi models without suiient knowledge o their statistial distribution. Furthermore, the variation in the gear tooth strength may also be aeted by a ombination o variations in the metallurgial and abriation proesses involved with its manuature, dimensional inauray, surae damage, et., whereas lutuations in the load imposed upon it may originate rom variations in its duty and environment. Thereore, design o gears or speiied design reliability may beneit rom detailed distribution data o gear tooth strength. There are two ailure modes that are important auses o gear tooth ailures. Bending stress (leading to tooth breakage) whih is a maximum at the tooth root, and ompressive stress (leading to pitting), whih reahes a maximum value at the tooth ae. Beause the tooth loading is yli, both o these stress mehanisms impose a atigue loading ondition on this mahine omponent element. Tooth breakage due to the stress onentration and rak initiation at the non-involute portion o gear tooth, when stresses signiiantly exeeds the maximum allowable stress in gear transmission units an be extremely dangerous in automotive, aerospae, or spae industry appliations. The non-involute portion o the tooth proile, the illet, has a diret eet on the motion/ore transmission. The detailed geometry o the illet inluenes the maximum bending stresses developed at the root o the gear tooth and is determined by the gear utting manuaturing proess and utting tool dimensions [8]. Hene, it is neessary to have a detailed knowledge o the root illet geometry and its eet on the tooth bending stresses. In the ollowing setions, a speial attention is given to the analysis o the impat o the gear tooth illet portion on the maximum bending stress value and its distribution along the illet portion. The SSI theory as a probabilisti design tool will be used to examine the inluene o root illet ontour on the gear tooth strength or our gear sets with dierent illet proiles. 2. Investigate the Eet o the Root Fillet Portion on the Gear Tooth Strength Aording to the theory o gearing, the proedure or generating the exat geometry o the tooth proile must be modelled to ollow the atual generation proess whih is used in prodution o real gears [9-11]. The generation o spur gear tooth shape during hobbing proess is modelled to be equivalent to pure rolling o the pith line o rak against the pith irle o gear blank. The illet urve o the gear belongs to the trohoid amily and it is onjugate to the tip o the generating rak, as shown in Figure 2. The radius o urvature o the trohoid illet varies along the illet urve. In the ollowing setion, the mathematial model o the trohoid illet is developed to investigate the eet o the trohoid illet onditions on the root strength o the gear tooth. Figure 2 shows an arbitrary position during a yle o ontat in a transverse plane between the generating hob tool and the gear tooth. Aording to AGMA 930-A05, the enter o the rounded orner at the tip o the rak is loated on the basi rak by its oordinates g, h Figure 1. Comparison between deterministi method and the SSI method or estimating reliability. (a) Deterministi design; (b) Stress/strength intererene. Copyright 2011 SiRes.

3 EL-S. S. AZIZ 1139 relative to the nominal pith line and the tooth enter line. Aording to the geometri relationship depited in Figure 2, it is ound that: t t r br 0.5 Nm d r (1) tan π m rt g br rttan (2) 4.0 os t tr h br rt (3) 2tan Eah generated point on the trohoid is assoiated with a value o the pith point polar angle. At the start o the trohoid, the pith polar angle or the trohoid point on the root irle is 90. At the tangent point r with the involute proile, the pith polar angle is 20. n is the design parameter used to determine the trohoid illet region and limited by n, r whih an be represented by the ollowing equation: n 1 k 1 r n n 1 where k 1 n, n 10 when the generating pith line o the rak utter rolls over the pith irle o gear blank, the generating roll angle rom a pith point at tooth enterline to a pith point at whih (k) trohoid point is generated, as shown in Figure 2, an be deined by: 2g 2h (5) d d tan The Cartesian oordinates o any point x, y on the root illet relative to gear tooth enterline an be obtained as ollows: h rt (6) sin 2 d 2 d sin 2 1 os sin x y (4) (7) (8) sin, os (9) Thereore, Equation (9) an be used to alulate the Cartesian oordinates o the envelope o trohoidal root illets or our dierent utter tip radius having 0.0, 0.25, 0.4 and 0.5 times the module. Figure 3 shows the generating hob tool and the trohoidal root illet in an arbitrary position during a yle o ontat in a transverse plane or a gear with the number o teeth 20 (without under- ut), module o 3 mm and pressure angle o 20. The enter o the rounded orner at the tip traes out a trohoid and then, the rounded tip will envelope another urve deining the root illet. Based on the numerial alulation results, the tooth proile in the transverse plane is produed and the teeth o the whole gear model are ormed to build the omplete 3D gear model. During the meshing o gear pair, large loalized stresses [12] our in the illets o gear teeth due to the sudden hange in the ross-setion o the tooth. Thereore, there is a need to examine the eet o these variations on gear tooth root strength. AGMA [13,14] has established an elasti behavior model or the gear tooth, whih uses the Lewis parabola to determine the ritial setion by the tangeny point o the parabola and the root trohoid x, y. In Figure 4, it is apparent that the bending load omponent o the translated ritial tooth Figure 2. Kinemati relationship between the rak o hobbing utter and generating gear tooth. Figure 3. Fillet proile generated by dierent tip radius. Copyright 2011 SiRes.

4 1140 EL-S. S. AZIZ load W will produe the greatest bending stress at x this ritial setion. The radial omponent W auses x a small ompressive stress aross the root o the gear tooth. This auses the tensile stress to derease by a small amount, and the ompressive stress on the opposite side o the tooth to inrease by a slight amount. In most materials, a tensile stress is more damaging than a slightly higher ompressive stress. The stresses are analyzed on the tooth side loaded in tension beause the irst raks are expeted to appear there [15]. The ombined tensile stress is the dierene between the bending tensile and the ompressive stresses, whih an be obtained by the ollowing equation: h sin W s W W (10) w t 6.0 os W 2 w The ritial tooth root ross setion an be determined based either on the ore appliation on the tip or the ore appliation at the single point o ation. In Equation (10), h is the height o the translated ritial load above the ritial setion w in the tooth root, it is al ulated rom the ollowing equation: dw h os sin tan W W W 2.0 W (11) d b t where, W os, W InvInv W d W d Based on the atual tooth orm, the tooth orm ator ( Y ) and the stress orretion ator ( K ) are alulated at the ritial setion in the tooth root area and subsequently the bending strength geometry ator ( Y K ) is determined. Tooth dimensions are shown in Figure 4 used in the alulation o the tooth orm ator (Y ). This ator onsiders the inluene o shape o tooth at the ritial tooth-root stress. This shape is deined in omputational proedure by the parameters o the ritial setion: 1 Y (12) os W 6h tan W 2 osa w w where the operating pressure angle ollows: a an be ound as d d 1 b 1 b 2 a os 2C a (13) The diretion angle o the ritial tooth ore be ound by the ollowing equation: W W W W an (14) The stress orretion ator K onsiders the omplex stress state at tooth root and stress onentration aused by the illet. It is alulated based on the work o Dolan and Broghamer [16]. They developed the ollowing stress orretion ator urve it relation: K 1 K2 K3 w w r h K (15) From a urve it o the experimental data o Dolan and Broghamer in their initial photoelasti experiments, AGMA gives the ollowing values or the onstants, (K 1, K 2, K 3 ) in terms o the pith irle pressure angle, : K K (16) K where r 3 is the illet radius o urvature, and is given by r b y r 2 r r t rt 0.5d b y r r r t (17) The bending strength geometry ator J is alulated rom the ollowing equation: Y J (18) K This ator relets the inrease in the loal stresses due to sharp hanges in geometry at the ritial setion and diretly will aet the bending strength under repeated loading. I the utter parameters are hosen to generate Figure 4. Data or ombined tensile stress alulation at the ritial illet setion. Copyright 2011 SiRes.

5 EL-S. S. AZIZ 1141 the involute lank proile to satisy ertain operation onditions, the illet proile is an outome o the utter motion that will also aet the tooth root strength. Thereore, bending stress redution ahieved by suh illet proile improvement diers and greatly depends on the utter tip radius or gear tooth parameters. The alulation proedure desribed above is integrated into the developed gear program [17-19]. It untions as a omplete gear design tool aording to the relevant standard (AGMA), presenting a omprehensive set o alulations assoiated with geometry o involute gears and design o the internal and external o the gear sets. It provides a 3D export untion used with modules that perorm sizing alulations or the gear models. At the touh o a button, it then transers those alulations to the CAD system and generates a 3D model o sized gears. Then, the strength alulation aording to AGMA an be arried out very easily and quikly or omparative alulations. The developed gear program allows or parametri and automati generation o well-shaped inite element meshes or 2D and 3D gear models with onsideration to proper node and element numbering and required tooth illet proile, as shown in Figure 5. Consequently, the developed gear program an alulate and provide proper design parameters to establish Probabilisti Design Analysis (PDA) to predit the eet o the trohoid illet onditions on the reliability o the gear system. With PDA, muh more an be done than just prediting reliability ater a design is ormulated. PDA will allow us to see how variations in material properties, variations in dimensions and variations in the operating environment aet the mean lie o the gear pair. 3. Development o a Probabilisti Gear Design Approah The design o the mehanial system requires the assurane o its reliability and quality. Variations and unertainties in the inputs and properties o the mehanial system (e.g. dimensions, material properties, errors assoiated with the simulation tools, et.) may lead to large variations in the perormane harateristis o the system and a high hane o ailure. Unertainties in the system harateristis prevent suh assuranes rom being given with absolute ertainty. One ommon approah is to quantiy the reliability o perormane, or risk o ailure, in probabilisti terms [20]. A probabilisti approah, whih requires knowledge o the nominal loads and strengths as well as the statistial variations in eah, allows the designer to assess the reliability o survival o the mehanial system [21,22] Stress-Strength Intererene (SSI) Theory The SSI theory has been used to take into aount variations in the operating load and strength o the mehanial system [23] that are distributed about a mean value and whih may hange with time in servie operation owing to degradation. I the probability density untion urves or both strength and load are shown to interere, as shown in Figure 6, then the shaded area where the urves interere is an indiation that the omponent will ail, beause the stress exeeds its strength. Mathematially, the SSI theory presents the ailure probability o the system P as the probability that the stress exeeds the strength. P P Stress Strength (19) The reliability (R) is the probability that the stress is less than the allowable strength: Figure 5. Integrated gear design analysis tools. Figure 6. Stress and strength intererene theory diagram. Copyright 2011 SiRes.

6 1142 EL-S. S. AZIZ a B R P (20) Reliability (R) o a omponent an be alulated, i the density untions or the allowable strength a ( a ) and the atual stress B ( B ) are known. The random variable (U) is a measurement or the distane between the atual stress and the allowable strength: U a B with U a B (21) PR P U 0 is the probability that: U 0 the reliability. PF P U 0 is the probability that U < 0: the ailure probability. I the random variables stress B and allowable strength a are assumed to be normally distributed due to their numerable random inluenes, then along with the parameters mean and statistial spread, B, SB, a, Sa the density untion o a normal distributed stress an be determined as ollows: B B S B 1 e 2π 2 B B 2 2S B (22) The density untion o the allowable strength an be determined in the same way. The random variable (U) is, likewise, normally distributed. The unreliability o ailure or the intererene area o the two distributions an be alulated using the ollowing equation: Z U U S S 2 2 a B (23) Then, the reliability an be simply alulated rom the normal distribution untion ( ): a B R S S 2 2 a B 3.2. Appliation o the SSI Theory (24) In this setion, we disuss an appliation o probabilisti modelling and analysis during the gear design proess. Current gear design analysis methods do not diretly aount or the probabilisti nature o most input parameters. The result o treating parameters suh as material properties, geometry, environment, and loads as solely determined (deterministi) values is a design o unknown reliability, or onversely, unknown risk. The minimization o risk o the gear tooth ailure has been handled in the design phase by the appliation o ators o unertainty and the use o judiious material properties. Probabilisti design analysis an be applied at the various phases o the gear design as long as inormation is available on the strength (material properties, manuaturing proess, et.) and stress (loads, environments, et.) parameters. Figure 7 illustrates this proess. The let-hand side shows the input data to determine the applied stress distribution at the ritial illet loation that an be ound rom the applied torque and geometry multiplied by dierent ators related to the speii appliation. The result o the gear design is to yield as a tensile bending stress distribution at the ritial illet loation, while the other three omponents in the right-hand side suh as material strength, manuaturing proess and operations and environment will work together to yield as an adjusting gear material strength distribution. The middle part in the igure shows the output o the proess, that being an applied stress and resistive gear strength distribution, with an assoiated probability o ailure. Thus, random design variables aeting both stress and strength are statistially deined. Thereore, the appliation o this approah allows or alulation o reliability and probability o ailure o the gearing system in terms o statistially varying load and strength harateristis, whih more nearly model the true situation. The previous setions indiated that the shape o the tooth illet is known to be ritially important to the tooth bending strength and is the ous o muh urrent gear researh. In pratie, there are many other parameters that ould also aet the bending stress, inluding gear material properties, applied torque, misalignment, et. However, these parameters are by nature unertain due to variability o material properties, toleranes in manuaturing, and the random nature o applied torques. In suh ase, it is more appropriate to perorm unertainty analysis and evaluate the reliability o gearing system in terms o probability o ailure, rather than the maximum value o the bending stress. Among many input parameters that an aet the unertainty o the tooth bending stress, the shape o the tooth root and the applied Figure 7. Development o a probabilisti gear design approah. Copyright 2011 SiRes.

7 EL-S. S. AZIZ 1143 torque are hosen or the present investigation. As a omplementary requirement in the investigation o the unertainties enountered in the design o gear tooth or bending stress based the SSI theory, the unertainty o gear material strength is onsidered. The unertainty in gear material strength omes rom variability in the material struture and mahinability. The gear material is hosen to be Alloy Steel and a standard deviation based a variation oeiient o ±10% was applied to be able to relet variability in gear material strength. The apability and simpliity o the normal distribution make it the best hoie or representing the variability in gear material strength. For this proposed analysis, a probabilisti analysis algorithm is developed to interae with the CAD-Integrated analysis tools (see Figure 5) so that the analysis output an be diretly ed to the probabilisti program (Minitab Statistial Sotware [23]) and vie-versa. The main idea is to integrate the probability o applied stress and material strength over the root region where stress exeeds the strength. The result o this integration is to determine the gear reliability and probability o ailure, as shown in Figure 8. Thereby, the output distributions will be alulated and the realisti system behaviour an be extrated rom these output distributions. To get aurate output distributions, Monte Carlo simulation (MCS) was used to generate a large number o sample points in order to estimate the statistial probability o gear tooth bending stress and strength. To examine how the tooth-root geometry inluenes the gear tooth strength, our gear sets with dierent root geometries (see Figure 3) that are generated by dierent tip radius o the hobbing utter have been onsidered as the test gears. Eah test gear has module 3 mm, pressure angle 20 degrees, number o teeth 20 and the same material. The analysis was onduted at a onstant gear rotational speed o 1500 rpm and at dierent applied torque levels ranging between N-m. Nominal tooth-root stress is produed at the tooth root when the gear pair is loaded by the applied torque. Figure 9 shows the histogram o the distribution density untion and the probability ourrene o the tooth bending stress when Figure 8. Integrate probabilisti and deterministi gear design analysis. Figure 9. Histograms o the distribution density untions and probability ourrenes o the tooth bending stresses or the our ase. Copyright 2011 SiRes.

8 1144 EL-S. S. AZIZ generating random values (the sample size is 10,000) o the tooth bending stresses or the our ases, as well as the urve orresponding to the analytial desription o the same density untion. Case (1) is showing the data o the generated illet by utter with the sharp orner tips. It symmetrially distributed with a entral tendeny about the mean stress o 255 MPa. A normal urve is superimposed over the histogram showing the distribution symmetry and good it. The dispersion o the histogram data bins are also indiating the variability in stress values experiened at this partiular root radius. The run hart on the let side o the graph o ase (1) is showing data dispersion o the sample about the mean, it has been notied that the highest stress values an reah to 480 MPa. In ase (2), the mean stress value redued to 200 MPa and the highest stress value is about 380 MPa. For the tip with protuberane, ase (3), an improvement in data lustering about the mean (185 MPa) with high stress around (320 MPa) is notied ompared to the two previous ases. In the ully rounded orner tip radius o the utter, ase (4), the stress results showed less variability about the mean stress (175 MPa) with the highest stress around 300 MPa. The tooth bending stress distribution or eah ase is represented graphially against the strength o the gear material distribution based the prinipal o the SSI theory, as shown in Figure 10. SSI diagram or the stress/strength distributions. Aording to the intererene area between the stress and strength distribution urves, ase (1) is showing the highest intererene area whih indiates more stress points are exeeding the allowable strength distribution, hene reduing reliability and inreasing ailure probability. The probability o ailure in ase (1) is ound to be about 8.23%. On the other hand, ase (4) is showing less intererene between the two distributions whih enhanes the gear reliability and redues the hanes o sudden ailing due to high bending stress. The probability o ailure in ase (4) is ound to be about 0.1%. Aording to Equation (23), the area o intererene between eah normally distributed stress ase and the normally distributed strength an be alulated to deine the unreliability o ailure (Z) or eah ase, as shown in Table 1. Then, the reliability o eah ase an be determined by subtrating the ailure area rom the total area under the urve whih is 1 (see Equation (24)). Figure 11 shows the umulative density untions o the tooth bending stress distributions alulated or the dierent our ases that desribed above. From the igure, we see that the probability o getting higher stresses inreases as the dereasing in the value o the tip radius o the utter. The igure also shows the tooth bending stress range or the ase (4) is less than the other ases whih indiates less stress variability will be experiened at this partiular ase. Aording to Figure 11, at 80% sample perentile o the stress ourrene, ase (4) provides higher resistane to the bending breakage ompared with the rest three ases. Figure 10. SSI diagram or the stress/strength distributions. Copyright 2011 SiRes.

9 EL-S. S. AZIZ 1145 Figure 11. Cumulative density untions o the tooth bending stress distributions. Table 1. Reliability assessment based graphially representtation o statistial data or the our ases. Tip Radius Z Intererene % (Unreliability) Reliability 0.00 m n m n m n m n FEM-Based Veriiation Approah Finite Element Method (FEM) is used to veriy the auray o the deterministi and probabilisti stress results at the root illet. The advantage o using FEM is that it an aurately onsider the eet o detailed geometry, as well as omplex loading onditions. In order to examine preisely the inluene o root illet ontour in the gear tooth strength, as shown in Figure 12, the FEM ode that was developed or this analysis is split into three untions, eah o whih serves its own purpose in deining parameters, reating the model, perorming analysis, and extrating analysis results. The irst untion deines all variables that are required in onstruting the gear geometries, material properties and applied loading onditions. Sine all onstrution proesses are parameterized, our dierent gear sets an be easily onstruted by modiying parameters in this untion. The seond untion onsists o the ommands that build a three-dimensional solid model, inluding only three teeth and generate a inite element mesh by using the automati mesh generation untionality. Only three teeth are modeled beause o periodi symmetry, whih will signiiantly redue the omputational analysis times required to perorm inite element analysis o the model. This proess is repeated or the pinion and gear to build a gear set assembly. During the real operation, the tip-load ondition is not being the worst load beause in this ondition another pair o teeth will be in ontat. Examination o run-in teeth reveals that the heaviest loads our near the middle o the teeth arrying the ull load, that is, at a point where another pair o teeth is on the edge o oming into ontat. When a single tooth arries the ull load, it has been generally aepted that the maximum tensile bending stress ours at the highest point o the single-tooth ontat (HPSTC). Contat between the gear and pinion is established by deining ontat elements on the lank o the gear teeth and target elements on the lank o the pinion teeth. A tangential distributed load is applied uniormly on the pinion surae. Both the gear and pinion are pinned at the enter, but is allowed to pinion to rotate and the gear is ixed. The ixed displaement onstraint is applied on the boundary surae o the gear and the ra- Copyright 2011 SiRes.

10 1146 EL-S. S. AZIZ dial onstraint is deined on the boundary surae o the pinion. The third untion perorms postproessing, whih extrats output results suh as deormed shapes, normal stresses at the root illet, and ontat pressures. Figure 12 shows the distributions o the eetive stresses along the tooth line passing through ritial setions in various FEM models o spur gears and along illet in various transversal setions. To obtain the results o the analysis, maximum prinipal stre- sses were searhed at every alulation ase near the tooth root illet area to ompare with the results alulated by the probabilisti design method. Figure 13 shows the distribution o the eetive stresses along the illet in various transversal setions orresponding to the our ases disussed in Figure 12. Comparison o these ases indiates that bending stress distribution inreases with the root illet that is generated by small utter tip radius as in ase (1) and the eetive stresses or the our ases alulated by FEM are loser Figure 12. FEM model and tensile bending stress ontours or our ases o dierent illet proiles. Figure 13. Comparison o eetive normal stress distributions along tooth ae width and illet proiles or our ases (3D and 2D plots). Copyright 2011 SiRes.

11 EL-S. S. AZIZ 1147 to those obtained by the probabilisti design method. The 2D hart shows the average o the bending stress distribution along the dierent illet proiles or the our ases. The bending stress distribution or the ase (4) is muh more uniorm than the other three ases. Other ases have the signiiantly greater maximum stresses that are sharply onentrated. The hart also indiates that as the illet proile hanges the position o ritial point on the illet ontour or eah ase is being determined by the maximum tensile bending stress. Generally, this study investigated omparatively larger root illet within the ase (4), whih results in notable stress redution. 5. Conlusions A reliable approah was developed to evaluate the gear tooth reliability by the bending strength riterion and integrated into the developed gear program as a probabilisti design tool. The SSI theory as a probabilisti design tool was applied to examine the inluene o root illet ontour on the gear tooth strength or our gear sets with dierent illet proiles. This approah is implemented upront during the gear design phase to assist the designer in making deisions on the best balaned design with respet to several design riteria. Thus, unertainty analysis is perormed to estimate the probabilisti distribution o the tooth root stress. Consideration the SSI ailure probability alulation or bending breakage, has shown that the shortest portion o the illet proile in ase (1) provided the highest intererene area whih indiates more stress points are exeeding the allowable strength distribution, hene reduing reliability with about 9% probability o ailure. The analysis o the mean nominal bending strength revealed that the illet proile generated by the ully rounded tip radius in ase (4) had the highest strength ompared with the rest ases. FEM o tooth-root strength is perormed or eah modeled gear set to veriy the auray o the deterministi and probabilisti stress results at the root illet. There is strong agreement between the probabilisti analysis and FEM s results aross a range o dierent illet proiles; this onirms the illet generated by the ull tip radius in ase (4) has the lowest maximum bending stress, whih is uniormly distributed along the large portion o the illet proile. Other illet proiles have signiiantly greater maximum stresses that are sharply onentrated. Understanding the relationship between proess variables unertainties with the gear perormane and saety is aomplished. This approah helps making the deision regarding the most reliable design by quantiying the impat o stress and strength variations during the gear design stage. 6. Reerenes [1] A. Vlahinos, S. Kelkar, S. Reh, R. Seaur and S. Pliz, Reliability Based Optimization within the CAD Environment, miz-bmx-ansys.pd [2] G. Cooper and G. Thompson, Conept Design and Reliability, Ata Polytehnia, Vol. 42, No. 2, 2002, pp [3] O. P. Yadav, S. S. Bhamareb and A. Rathoreb, Reliability-Based Robust Design Optimization: A Multi-Objetive Framework Using Hybrid Quality Loss Funtion, Quality and Reliability Engineering International Journal, Vol. 26, No. 1, 2010, pp doi: /qre.1027 [4] X. He and S. O. Oyadiji, A Study o Pratial Reliability Estimation Method or a Gear Redution Unit, IEEE SMC 99 Conerene Proeedings, Vol. 1, 1999, pp [5] X. Du and W. Chen, Sequential Optimization and Reliability Assessment Method or Eiient Probabilisti Design, Journal o Mehanial Design, Vol. 126, No. 2, 2004, pp doi: / [6] D. P. Townsend, Dudley s Gear Handbook, MGraw- Hill, New York, [7] A. Kawale, J. Wiktor and D. Ceglarek, Comparative Analysis o Tooth-Root Strength Using ISO and AGMA Standards in Spur and Helial Gears with FEM-Based Veriiation, Journal o Mehanial Design, Vol. 128, No. 3, 2006, pp doi: / [8] A. L. Kapelevih and T. M. MNamara, Diret Gear Design or Automotive Appliations, SAE World Congress & Exhibition, Detroit, [9] S. Xiaogen and D. R. Houser, Charateristis o Trohoids and Their Appliation to Determining Gear Teeth Fillet Shapes, Journal o Mehanism and Mahine Theory, Vol. 35, No. 2, 2000, pp doi: /s x(99)00004-x [10] V. B. Math and S. Chand, An Approah to the Determination o Spur Gear Tooth Root Fillet, Journal o Mehanial Design, Vol. 126, No. 2, 2004, pp doi: / [11] J. Brauer, Analytial Geometry o Straight Conial Involute Gears, Journal o Mehanism and Mahine Theory, Vol. 37, No. 1, 2002, pp doi: /s x(01) [12] M. Savage, K. L. Rubadeux and H. H. Coe, Bending Strength Model or Internal Spur Gear Teeth, NASA Lewis Researh Center, Cleveland, [13] AGMA 930-A05, Calulated Bending Load Capaity o Powder Metallurgy (P/M) External Spur Gears, Amerian Gear Manuaturers Assoiation, Alexandria, [14] AGMA Inormation Sheet 908-B89, Geometry Fators or Determining the Pitting Resistane and Bending Strength Copyright 2011 SiRes.

12 1148 EL-S. S. AZIZ o Spur, Helial and Herringbone Gear Teeth, Amerian Gear Manuaturers Assoiation, Alexandria, [15] K. Stoker, A. Chaudhuri and N. H. Kim, Saety o Spur Gear Design under Non-Ideal Conditions with Unertainty, Proeedings o the ASME 2010 IDETC/CIE, Montreal, August 2010, pp [16] T. J. Dolan and E. L. Broghame, A Photoelasti Study o Stresses in Gear Tooth Fillets, University o Illinois Engineering Experiment Station Bulletin 335, [17] E. S. Aziz and C. Chassapis, Knowledge-Based Geometry Generation or Spur and Helial Gears, Conurrent Engineering: Researh & Appliations (CERA) Journal, Vol. 10, No. 3, 2002, pp [18] E. S. Aziz and C. Chassapis, An Intelligent Design System or Agile Design and Manuaturing o Mehanial Transmission Systems, The 9th International ASME Power Transmission and Gearing Conerene, Chiago, 2-6 September [19] E. S. Aziz and C. Chassapis, A Deision-Making Frame- work Model or Design and Manuaturing o Mehanial Transmission System Development, Engineering with Computers Journal, Vol. 21, No. 2, 2005, pp doi: /s z [20] B. Bertshe, Reliability in Automotive and Mehanial Engineering: Determination o Component and System Reliability, VDI-Buh, Springer-Verlag, Berlin, [21] R. C. Kuzera and Z. P. Mourelatos, On Estimating the Reliability o Multiple Failure Region Problems Using Approximate Metamodels, Journal o Mehanial Design, Vol. 131, No. 12, 2009, pp doi: / [22] E. G. Haugen, Probabilisti Mehanial Design, John Wiley and Sons, New York, [23] P. Martin, A Review o Mehanial Reliability, Proeedings o the Institution o Mehanial Engineers, Part E: Journal o Proess Mehanial Engineering, Vol. 212, No. 4, 1998, pp doi: / Copyright 2011 SiRes.