A Methodology to Predict Surface Wear of Planetary Gears Under Dynamic Conditions #

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<ul><li><p>This article was downloaded by: [University of Windsor]On: 10 November 2014, At: 20:05Publisher: Taylor &amp; FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK</p><p>Mechanics Based Design of Structures and Machines: AnInternational JournalPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/lmbd20</p><p>A Methodology to Predict Surface Wear of PlanetaryGears Under Dynamic Conditions#</p><p>A. Kahraman a &amp; H. Ding aa Department of Mechanical Engineering , The Ohio State University , Columbus, OhioPublished online: 09 Nov 2010.</p><p>To cite this article: A. Kahraman &amp; H. Ding (2010) A Methodology to Predict Surface Wear of Planetary Gears UnderDynamic Conditions# , Mechanics Based Design of Structures and Machines: An International Journal, 38:4, 493-515, DOI:10.1080/15397734.2010.501312</p><p>To link to this article: http://dx.doi.org/10.1080/15397734.2010.501312</p><p>PLEASE SCROLL DOWN FOR ARTICLE</p><p>Taylor &amp; Francis makes every effort to ensure the accuracy of all the information (the Content) containedin the publications on our platform. However, Taylor &amp; Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of theContent. Any opinions and views expressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor &amp; Francis. The accuracy of the Content should not be relied upon andshould be independently verified with primary sources of information. 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Terms &amp; Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions</p><p>http://www.tandfonline.com/loi/lmbd20http://www.tandfonline.com/action/showCitFormats?doi=10.1080/15397734.2010.501312http://dx.doi.org/10.1080/15397734.2010.501312http://www.tandfonline.com/page/terms-and-conditionshttp://www.tandfonline.com/page/terms-and-conditions</p></li><li><p>Mechanics Based Design of Structures and Machines, 38: 493515, 2010Copyright Taylor &amp; Francis Group, LLCISSN: 1539-7734 print/1539-7742 onlineDOI: 10.1080/15397734.2010.501312</p><p>A METHODOLOGY TO PREDICT SURFACE WEAR OFPLANETARY GEARS UNDER DYNAMIC CONDITIONS#</p><p>Ahmet Kahraman and Huali DingDepartment of Mechanical Engineering, The Ohio State University,Columbus, Ohio</p><p>In this study, a torsional dynamic model and a surface wear model are combined tostudy the interaction between the surface wear and the dynamic response of planetarygear sets. The proposed dynamic planetary gear wear model includes the influence ofworn surface profiles on the dynamic tooth forces and the motion transmission erroras well as the influence of dynamic tooth forces on wear profiles. The dynamic modelincludes the gear backlash and the periodic time variation of gear mesh stiffnesses. Themodel is used to investigate the interactions between the surface wear and the dynamicbehavior within both linear and nonlinear response regimes. Several sets of simulationresults are used to demonstrate the two-way relationship between nonlinear planetarygear dynamics and tooth surface wear.</p><p>Keywords: Gear dynamics; Gear wear; Planetary gear sets.</p><p>INTRODUCTION</p><p>Planetary gear sets commonly are used in power transmission applicationswhere a large speed reduction (or increase) and a higher power density (transmittedpower to weight ratio) are required. Their most common example applications canbe found in automotive automatic transmissions and transfer cases, gas turbines,rotorcraft drive trains, and jet engine turbofans. A simple planetary gear set hasthree central members: a sun gear, an internal (ring) gear, and a rigid structure calledcarrier that holds N number of planet gears (typically N = 37). Each planet is inmesh with the sun and ring gears in an idler configuration. The parallel power flowbranches formed by each planet reduce the individual gear mesh forces acting onthe sun and ring gears, hence increasing power density of the gearbox significantly.In addition, planets are typically positioned in an axisymmetrical orientation suchthat resultant radial forces on the central members are theoretically zero, eliminatingthe need for bearing supports for some of these central members. Another uniquefeature of a planetary gear set is its capability as a mechanism to provide differentpower flows and speed ratios as a function of particular assignments of input,output, and reaction (fixed) member duties to its central members.</p><p>Received January 15, 2010; Accepted March 24, 2010#Communicated by S. Velinsky.Correspondence: Ahmet Kahraman, Department of Mechanical Engineering, The Ohio State</p><p>University, 201 W. 19th Avenue, Columbus, OH 43210, USA; E-mail: kahraman.1@osu.edu</p><p>493</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Uni</p><p>vers</p><p>ity o</p><p>f W</p><p>inds</p><p>or] </p><p>at 2</p><p>0:05</p><p> 10 </p><p>Nov</p><p>embe</p><p>r 20</p><p>14 </p></li><li><p>494 KAHRAMAN AND DING</p><p>In many of these applications, planetary gear sets are expected to operateunder complex duty cycles defined by wide ranges of speed and torque. This bringsthe dynamic response of gears to the forefront as a major concern due to severalfactors. One factor is that excessive vibratory motions exhibited by the gear settypically generate higher levels of noise. In addition, under dynamic conditions,each gear mesh and bearings in the gear train experience dynamic loads that exhibitsizable fluctuations about the static load at characteristic gear mesh frequencies.Such dynamic loads can be expected to accelerate the occurrence of contact (pittingand micro-pitting) and tooth bending fatigue failures while increasing the surfacewear rate, hence impairing the functionality of the gear system. In summary, anyattempt to improve durability and reduce noise in gear systems requires a betterunderstanding of the system behavior under dynamic conditions.</p><p>Excessive tooth surface wear is characterized by loss of material on thetooth profiles, which might result in higher dynamic gear mesh and tooth forces,as a consequence, shorter contact and tooth bending fatigue lives. Surface wearchanges the contact pattern and the load distribution, as well as the vibrationand noise characteristics of the gear system significantly. It is well documentedthat the dynamic response of a geared system is very sensitive to deviations ofthe tooth surface profiles from a perfect involute (Kahraman and Blankenship,1999). Intentional tooth modifications such as tip and root relieves are commonlyused to reduce the dynamic forces at certain design torque values. Unavoidablemanufacturing errors influence the dynamic response as well, since they act as amotion transmission error excitation at the gear mesh interface. As surface wear isessentially a material removal process that results in a deviation from the intendedtooth profiles and alters the gear mesh excitations, gear systems with worn surfacesshould have dynamic behavior that is quite different from their counterparts with nowear. This seems to be the primary reason for many real-life gear systems to becomenoisier after years of operation. On the other hand, dynamic gear tooth forces aredifferent from the quasi-static forces in both magnitude and shape. These toothforces also reflect various nonlinear phenomena such as backlash-induced toothseparations, jump discontinuities, subharmonic (parametric), and superharmonicresonances (Kahraman and Blankenship, 1997). Therefore, surface wear outcomethat is strongly related to the contact stresses should be highly dependent on thedynamic behavior. These arguments suggest that gear dynamics and gear wear aremutually dependent on each other. Therefore, design of planetary gear systems withan acceptable dynamic (and hence noise) behavior throughout their entire life cyclemust include wear in the design process. Also, any attempt to reduce surface wearmust take into account the dynamic loads that the gears experience.</p><p>In a recent study, these authors investigated the two-way interactions betweenthe vibrations and the wear behavior of a spur gear pair (Ding and Kahraman,2007). They proposed a dynamic gear pair wear model that included the influenceof worn surface profiles on the dynamic tooth forces and the transmission error aswell as the influence of the dynamic tooth forces on the wear profiles. The dynamicmodel was combined with a gear wear model to study the interaction of surface wearand dynamic behavior in both linear and nonlinear response regimes. Several sets ofsimulation results were used to demonstrate this two-way relationship between thenonlinear gear dynamics and the surface wear of a single gear pair.</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Uni</p><p>vers</p><p>ity o</p><p>f W</p><p>inds</p><p>or] </p><p>at 2</p><p>0:05</p><p> 10 </p><p>Nov</p><p>embe</p><p>r 20</p><p>14 </p></li><li><p>SURFACE WEAR OF PLANETARY GEARS 495</p><p>Apart from the single pair dynamic models that contain only a single gearmesh, there are 2N gear meshes (N external sun-planet meshes and N internal ring-planet meshes) in a planetary gear set, where N is the total number of planets.Opposite tooth flanks of a planet tooth come to contact with the sun and ring gearteeth, hence making the wear of the internal and external meshes separate events,while they are still coupled through unique dynamic behavior exhibited by the entireplanetary gear set. Several published studies (e.g., Al-Shyyab and Kahraman, 2007;Botman, 1976; Cunliffe et al., 1974; Kahraman, 1994a,b,c; Kahraman et al., 2003;Lin and Parker, 2000; Saada and Velex, 1995; Sun and Hu, 2003) proposed modelsof varying complexity to describe the dynamic behavior of a planetary gear set andsuggested that nonlinear phenomena such as tooth separations might occur in aplanetary gear set having spur gears as well. This paper focuses on studying thedynamic wear characteristics of a planetary gear set by coupling a torsional dynamicmodel with the dynamic wear methodology that was used for spur gear pairs earlier.The sun-planet and ring-planet meshes wear in different rates due to their geometryand the kinematics of the gear set. While there are a few published studies thatfocused on the influence of static surface wear from the sun-planet mesh on theoverall dynamic response of a planetary gear set (Yuksel and Kahraman, 2004), thetwo-way relationship between surface wear and dynamic response of a planetarygear set is yet to be established. Work by Yuksel and Kahraman (2004) computedthe sun-planet wear profiles separately by using a quasi-static external gear pair wearmodel and applied these wear profiles to a finite element (FE) model of a planetarygear set to observe the changes in the dynamic behavior as a result of these toothsurface deviations. Wear profiles in that study corresponded to the quasi-static toothloads, not dynamic tooth loads, such that the influence of the dynamic behavioron the accumulation of surface wear was not considered. The model of Yukseland Kahraman did not include the wear accumulated at the ring-planet meshes. Inaddition, since the model of Yuksel and Kahraman used a single gear pair modelfor wear computations, it failed to capture any system-level planetary effects suchas relative phasing among the meshes.</p><p>This study extends the methodology proposed by Ding and Kahraman (2007)to planetary gear sets to investigate the relationship between the surface wearand the dynamic behavior. The proposed model captures the two-way interactionsbetween surface wear and dynamic behavior as actual dynamic tooth forces are usedto compute surface wear. This model carries out wear simulations for all meshessimultaneously. The effects of the disturbances caused by the wear of the sun mesheson ring gear meshes (and visa versa) are also captured in the proposed model. Useof a discrete dynamic model here in place of the FE model of Yuksel and Kahraman(2004) also enhances the computational efficiency significantly, making it potentiallysuitable as a design tool.</p><p>In this paper, first, the static wear model will be applied to an N -planetplanetary gear set. A discrete dynamic model will be proposed whose internaland external gear mesh excitation parameters are predicted by a quasi-static loaddistribution model. The dynamic model will include both the periodically varyingmesh stiffnesses and the tooth contact loss nonlinearity. The dynamic model willthen be combined with the surface wear model to quantify both dynamic response ofgear sets having worn gear surface and wear profiles due to operation under variousdynamic conditions.</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Uni</p><p>vers</p><p>ity o</p><p>f W</p><p>inds</p><p>or] </p><p>at 2</p><p>0:05</p><p> 10 </p><p>Nov</p><p>embe</p><p>r 20</p><p>14 </p></li><li><p>496 KAHRAMAN AND DING</p><p>QUASI-STATIC WEAR BEHAVIOR OF PLANETARY GEAR SETS</p><p>Wear Model for a Gear Pair</p><p>The static wear model used in this study uses the Archards wear equation inthe form</p><p>h =</p><p>kP dS (1)</p><p>where k is a dimensional wear coefficient, P is the amplitude of the contact pressure,and S is the sliding distance between a local point and the corresponding contactpoint on the mating gear. Several earlier studies proposed computational proceduresto apply Eq. (1) to different forms of gear pairs, including spur gears (Flodin andAndersson, 1997), helical gears (Bajpai et al., 2004; Flodin and Andersson, 2000;Kahraman et al., 2005), and spiral bevel and hypoid gears (Park and Kahraman,2009). Here the methodology proposed by Bajpai et al. (2004) will be adapted asshown in Fig. 1.</p><p>The first step in Fig. 1 is to determine the initial geometric descriptions ofthe gear tooth surfaces to serve as the initial state for wear prediction. The toothsurface modifications must be included in quantifying the initial contact of the gearsurfaces. Considering a contact of a pair of gears p and g, the surface deviationfrom a perfect involute at a point i on gear including surface modifications isdefined as Gi ( = p g). By selecting points i at the nodes of a predetermined surfacegrid, a discretized description of both contacting surfaces is obtained. The secondstep is the computation of the contact pressure at each nodal point i at differentrotational positions of the gears in mesh. The geometric data consisting of Gi areinput into a deformable-body contact mechanics model to predict the instantaneouscontact pressure distribution Pi r at each rotational position r 0 R. Here, thetotal number of rotational positions R and the increment of the rotation are suchthat the amount of gear rotation achieved covers a complete wear cycle from theposition where the tooth of interest enters the mesh zone (r = 0) to the positionwhere the tooth ex...</p></li></ul>