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gears and wheel gears are helical. The effect of the design parameters on rattle noise isKeywords:OptimizationringnMechanism and Machine Theory 45 (2010) 15991612Contents lists available at ScienceDirectMechanism and Machine Theoryreducing rattling and clattering noise in the gearbox is important in the automotive transmission for a comfortable car design.Gearsarewidelyusedinautomotivetransmissionstotransmitmechanicalpowerfromoneshafttoanother.Thepurposeofthegears is to couple two shafts together such that the rotation of the output, or driven, shaft is a function of the rotation of the input,or driving, shaft 4.Rattlingandclatteringnoiseinautomotivetransmissionsarecausedbytorsionalvibrationthatistransmittedfromtheinternalcombustion engine to the transmission input shaft. This noise is known as rattling when the transmission is in neutral, and asclattering when the gear is engaged under power or in overrun 914.Rattling and clattering are caused by torsional vibration of loose parts, i.e. parts, such as idler gears, synchronizer rings andsliding sleeves, which are not under load and therefore can move within their functional clearances 9,14.empirical model approach is studied.GearmotioncausesrattlingandclatteGear rattling noise is one of the major problemsmuch time idling under no load or very lightTheparametersthatareresponsibleforraThegeometricparametersincludethemodulThe operational parameters include the angu Corresponding author. Tel.: +90 212 383 27 93.E-mail address: .tr (M. Bozca).0094-114X/$ see front matter 2010 Elsevier Ltd.doi:10.1016/j.mechmachtheory.2010.06.013oise,andnoiseslevelisconsideredtobeacomfortfactorinautomotiveindustry.Therefore,1. IntroductionThe optimization of gearbox geometricanalyzed. The observed rattle noise proles are obtained depending on the design parameters.During the optimization, an empirical average rattle noise level is considered as the objectivefunctions and design parameters are optimized under several constraints that include bendingstress, contact stress and a constant distance between gear centers. Therefore, by optimizingthe geometric parameters of the gearbox such as, the module, number of teeth, axial clearance,andbacklash,itispossibletoobtainalight-weight-gearboxstructureandminimizetherattlingnoise. It is concluded that the optimized geometric design parameters lower the rattle noise by14%comparedtothecalculatedrattlenoiseforsamplegearbox.Alloptimizedgeometricdesignparameters also satisfy all constraints. 2010 Elsevier Ltd. All rights reserved.design parameters to reduce rattle noise in automotive transmissions based on anGearboxRattle noiseAutomotive transmissionAvailable online 8 August 2010Empirical model based optimization of gearbox geometric designparameters to reduce rattle noise in an automotive transmissionMehmet Bozcaa, Peter FietkaubaYildiz Technical University, Mechanical Engineering Faculty, Machine Design Division, 34349, Yildiz, Istanbul, TurkeybUniversity of Stuttgart, Institute of Machine Element (IMA), D-70569, Pfaffenwaldring 9, Stuttgart, Germanyarticle info abstractArticle history:Received 24 November 2009Received in revised form 8 June 2010Accepted 22 June 2010The optimization of gearbox geometric design parameters to reduce rattle noise in anautomotive transmission based on an empirical model approach is studied. Rattle noise iscalculated and simulated based on the design parameters of a 5-speed gearbox, and all pinionjournal homepage: /locate/mechmtfacing the industry, and the car industry in particular, because cars spend soloads 1.ttleandclatterareclassiedasgeometricparametersandoperationalparameter914em,numberofteethz,helixangle,axialclearancesaandbacklashsv,asshowninFig.3.lar acceleration1and excitation frequency an914.All rights reserved.Transmission uid as an engineering design parameter also has a considerable inuence on rattling and clattering noises. Theimportantfactorsincludethetypeofoil,theadditivesusedandtheviscosityandthelevelofoilinthetransmissionwhichtogetheract on a gear pair as drag torque, resulting in a reduction in rattling and clattering noises, especially at low speeds and when cold914.The analysis suggests that in order to reduce the gear rattle noise, rather than to increase the oil level in a gearbox and thusdecreasingthemechanicalefciency,itwouldbeopportunetoguaranteetheoilpresencebetweenthemeshingteethbymeansofa suitable lubrication device feeding the lubricant only in the meshing zone 15.Sliding friction between meshing teeth is one of the primary excitations for noise and vibration in geared systems. Among thedifferent kinds of non linearities in gear system, such as clearance, spatial variations and sliding friction, the effect of friction is theleastunderstood.Certainuniquecharacteristicsofgeartoothslidingmakeitapotentiallydominantfactor.Forinstance,duetothereversal in direction during meshing action, friction is associated with a large oscillatory component, which causes a higherbandwidthinthesystemresponse.Furthermore,itbecomesmoresignicantathighvaluesoftorqueandlowerspeeds,duetothetribological characteristics as well as due to higher force transmissibility in the sliding direction 17.Optimization of the gears macro-geometry i.e. the use of high contact ratio gears that has lead to minor noise emissions withhigher transmitted power levels. Optimization of the gears micro-geometry i.e. trying to balance load-induced teeth deectionswith prole corrections that has generally lead to less noisy transmission effects. This is not a suitable solution for an overallworkingrange;thereforeprolecorrectionsmustbedeterminedstatisticallytotakeintoaccountmanufacturingdeviationswhichwill overlap their effect 16.1600 M. Bozca, P. Fietkau / Mechanism and Machine Theory 45 (2010) 159916121.1. Gearbox mechanismThe gearboxmechanismincludespiniongears,wheelgears, an inputshaft,an output shaft,a layshaft,a bearingsupport,and asynchronizers, as shown in Fig. . Pinion gears and gear wheelsAll pinion gears and wheel gears are helical, and all gears are made of 16MnCr. Input shaftThe constant pinion gear and rear wheel gear are engaged on the input shaft. The rear wheel gear is the idler gear and runs inthe needle bearing on the input shaft. The synchronizer of the rear wheel gear is connected to the input shaft.1.1.3. Output shaftThe 1st, 2nd and 3rd wheel gears and 4th pinion gear are engaged on the output shaft. The 1st, 2nd and 3rd wheel gears areidler gearsandrunin theneedlebearingsontheoutput shaft.Thesynchronizer of the1stand2ndwheelgears isconnectedtotheoutput shaft.1.1.4. Lay shaftThe1st,2nd,3rdpiniongears,therearpiniongearandthe4thwheelgearandtheconstantwheelgears,areengagedonthelayshaft. The 4th wheel gear is the idler gear and runs in the needle bearing on the lay shaft. The synchronizer of the 3rd and 4thwheel gears is connected to the lay shaft.Fig. 1. 5-speed gearbox for automotive transmission.1.1.5. SynchronizerAll synchronizers run in the needle bearings to maximize smoothness.2. Calculation of rattle noiseRattle noise is calculated and simulated based on the gearbox design parameters for a 5-speed gearbox for an automotivetransmission, which is shown in Figs. 1 and 2.The rattle noise level of a complete automotive transmission is calculated as follows 913:LpComp=10log ni=1100;1Lp;iC16C171The rattle noise Lpis calculated as follows by correlating the computed noise value and the measured noise level 913:Lp=10logkIm+100;1Lbasic 2where k is the calibration factor and Imis theaverageimpactintensity N. Theaverageimpactintensity Imis writtenas follows913:Im= m21rb1CIm3where2averagewherefollowswheredimensiowheredB.1601M. Bozca, P. Fietkau / Mechanism and Machine Theory 45 (2010) 15991612Fig. 2. View of 5-speed gearbox for automotive transmission.m2is a loose part kg, 1is the angular acceleration rad/s , rb1is the pitch circle radius mm and CImis the relatedimpact intensity . The average impact intensity,CImis written as follows 913:CIm=Csvp1;4620;714CfaCsa0;016Cfa+0;12Csv!4Csvis the non-dimensional circumferential backlash . The non-dimensional circumferential backlash Csvisdened as913:Csv=sv2anrb115svis backlash mm, anis the excitation frequency rad/s and Csais the non-dimensional axial clearance . The non-nal axial clearance, Csaisdened as follows 913:Csa=sa2antanrb116saistheaxialclearancemm,isthehelixangle0, Cfais therelatedaxialfrictionforceand Lbasicis thebasicnoiselevel3.4. Axial clearance (sa)Increasing of axial clearance causes the rattle noise to increase until axial clearance reaches its maximum value and decreasesafterwards. The axial clearancerattle noise relationship is shown in Fig. 7.3.5. Backlash (sv)Increasing the backlash result decreases the rattle noise until backlash reaches its maximum value; the rattle noise thenincreases. The backlashrattle noise relationship is shown in Fig. 8.3.6. Angular acceleration (1)Increasing the angular acceleration also increases the rattle noise. The angular accelerationrattle noise relationship is shownin Fig. 9.3.7. Excitation frequency (an)Increasing the excitation frequency also increases the rattle noise. The excitation frequencyrattle noise relationship is shownin Fig. 10.1602 M. Bozca, P. Fietkau / Mechanism and Machine Theory 45 (2010) 159916123. Effect of design parameters on rattle noiseThe rattle noise is calculated and simulated based on the gearbox design parameters. Thus, the effect of design parameters onrattle noise is shown in Figs. 410 graphically. The gearbox design parameters are shown in Fig. 3. Rattle noise simulationparameters are shown in Table 1.3.1. Module (m)As the module increases, rattle noise also increases. The modulerattle noise relationship is shown in Fig. 4.3.2. Number of teeth (z)Increasing of number of teeth increases the rattle noise. Their relationship is shown in Fig. 5.3.3. Helix angle ()Changing the helix angle results in different levels of rattle noise. The relationship is shown in Fig. 6.Fig. 3. Design parameters for a gearbox.Fig. 4. Modulerattle noise relationship for the 1st gear.Fig. 5. Number of teethrattle noise relationship for the 1st gear.Fig. 6. Helix anglerattle noise relationship for the 1st gear.Fig. 7. Axial clearancerattle noise relationship for the 1st gear.Fig. 8. Backlashrattle noise relationship for the 1st gear.Table 1Rattle noise simulation parameters.Parameters Unit 1st pinion 2nd pinion 3rd pinion 4th pinion Constant pinion Rear pinionCalibration factor k 1 1 1 1 1 1Loose part m2kg 1 1 1 1 1 1Angular acceleration1 rad/s2 500 500 500 500 500 500Backlash svmm 0.1 0.1 0.1 0.1 0.1 0.1Excitation frequency anrad/s 220 220 220 220 220 220Axial clearance samm 0.3 0.3 0.3 0.3 0.3 0.3Helix angle mm 26 30 30 30 30 30Related axial friction force Cfa 0.5 0.5 0.5 0.5 0.5 0.5Basic noise level LbasicdB 65 65 65 65 65 651603M. Bozca, P. Fietkau / Mechanism and Machine Theory 45 (2010) 159916124. Calculating the load capacity of helical gearsDetermining the service life and/or determining the strength of gears are crucial for gear manufacturers. Gear strength isdened by the bending and contact strength 20.4.1. Tooth bending stressThe tooth bendingstress is calculated as follows, Fig. 11 2,3,5,7. Accordingto the ISO 6336, shear stresses due to lateral forcesFig. 9. Angular accelerationrattle noise relationship for the 1st gear.Fig. 10. Excitation frequencyrattle noise relationship for the 1st gear.Fig. 11. Bending stress at the tooth-root.1604 M. Bozca, P. Fietkau / Mechanism and Machine Theory 45 (2010) 15991612werenottakenintoaccountwhendeterminingtheloadingcapacityofgear3,20,21.Atooth-rootbendingfatiguefractureusuallystarts at the 300tangent of the root 3,8,21,22.The real tooth-root stress F, is calculated as follows 2,3,5,7:F=FtbmnYFYSYYKAKVKFKF7where Ftis the nominal tangential load N, b is the face width mm, mnis the normal module mm, YFis the form factor , YSisthe stress correctionfactor , Yis the contact ratio factor , KAis the applicationfactor , KVis the internal dynamic factor,KFis the face load factor for tooth-root stress and KFis the transverse load factor for tooth-root stress .The permissible bending stress Fpis calculated as follows 2,3,5,7:Fp= FlimYSTYNYYRYX8where Flimis the nominal stress number (bending) N/mm2, YSTis the stress correction factor , YNis the life factor for thetooth-root stress , Yis the relative notch sensitivity factor , YRis the relative surface factor and YXis the size factor thatrepresents the tooth-root strength .The safety factor for bending stress SFis calculated as follows 2,3,5,7:SF=FpF94.2. Tooth contact stressThe tooth contact stress HCis calculated as follows, as seen in Fig. 12 2,3,5,6:The real contact stress His calculated as follows 2,3,5,6:the transverseThe1605M. Bozca, P. Fietkau / Mechanism and Machine Theory 45 (2010) 15991612Fig. 12. Contact stress at the tooth ank.load factor for contact stress .permissible contact stress Hpis calculated as follows 2,3,5,6:Hp= HlimZNZLZVZRZWZX11where d1is the reference diameter of the pinion mm, u is gear ratio , ZHis the zone factor , ZEis the elasticity factorN= mm2p,Zisthecontactratiofactor,Zisthehelixanglefactor,KHisthefaceloadfactorforcontactstressandKHisH=Ftbmnu +1usZHZEZZKAKVKHKHq10where,ZVstressTheConstrainstress,InoptimizingLetThen,differen6.1. ObjectivesAverageconsiderinoptimizationTheThen, to nd the constrained minimum of the average of rattle noise level Lpaverage(m, z, , b, sa, sv), the following optimizationproblemsv. Iteration1606 M. Bozca, P. Fietkau / Mechanism and Machine Theory 45 (2010) 15991612LB and UB dene, the sets of lower and upper bounds on the design parameters (variables) vector such as m, z, b, sa, andbegins with the initial design parameter vector, which include, e.g., m0, z0, 0, b0, sa0and sv0and a solution vectorsubject to : LBm;z;b;sa;svUB and GX0 18where,is solved:minLpaveragem;z;b;sa;sv 17F = Lpaverage15Average of rattle noise levels Lpaverageare dened as follows:Lpaverage=1nni=1LpComplete;i16AverageofrattlenoiselevelsLpaverage(m,z,b,sa,sv)areconsideredtobetheobjectivefunctionstobeminimized,wheremodulem, the number of teeth z, helix angle , axial clearance saand backlash svare the design parameters (variables) to be determined.t initial value vectors are used to identify the global minimum solution of the objective function Lpaverage(m, z, , b, sa, sv).functionof rattlenoise levels of thegear systemareconsideredasobjectivefunctionsandthe designparameters areoptimizedg bending stress, contact stress and distance between gear center constraints. The owchart of the design parameterprocedure is shown in Fig. 13.following objective function is employed:min FX 13subject to : LBXUB and GX0 14where,LB and UB dene, the sets of lower and upper bounds on the design parameter(variable) X. Iterations start with the initialdesign parameter vector X0and a solution vector X is found that minimizes the objective function F(X) subject to the nonlinearinequalities G(X)0 18,19.6. Numerical exampleConstrainedoptimizationapproachesareappliedtothe5-speedgearboxforautomotivetransmission.Allprogramsaredevelopedusing the MATLAB program. In all optimization studies, the sequential quadratic programming (SQP) method is employed.Tondtheoptimumdesignparameter,theinitialdesignparametersofthe5-speedgearboxforautomotivetransmissionsuchasm,z,b,sa,andsvarevaried.Thirty-sixdesignparametersareoptimizedsimultaneouslyusingthedevelopedprograms.Duringoptimization,F(X) denote the objective function to be minimized, where X is the design parameter (variable) vector to be determined.to nd the constrained minimum of F(X), the following optimization problem is solved 18,19:13.ed optimization is a very useful tool for light-weight-structure design of machine elements with constraints such asdeformation and vibration.optimization, the goal is usually to minimize the cost of a structure while satisfying the design specications 19.Bythe responsible parameters, it is possible to obtain a light-weight-gearbox structure and minimize the rattling noise5. Optimization of gearbox design parametersHlimistheallowablestressnumbers(contact)N/mm2, ZNisthelifefactorforcontactstress, ZListhelubricationfactoristhevelocityfactor,ZRistheroughnessfactor,ZWistheworkhardeningfactorandZXisthesizefactorforcontact.safety factor for contact stress SHis calculated as follows 2,3,5,6:SH=HpH12with m, z, , b, sa,andsvis found that minimizes the objective function Lpaverage(m, z, , b, sa, sv)subjecttothenonlinearinequalities G(X)0.6.2. Constraint functionsTooth bending stress, contact stress and distance between gear centers are considered to be the constraint functions in theoptimization. The tooth bending stress parameters and tooth contact stress parameters are shown in Tables 2 and 3 respectively.The following constraints are considered to be constraint functions1.FFp0 19where Fis the real tooth-root stress N/mm2 and Fpis the permissible bending stress N/mm2.2.HHp0 20where His the real contact stress N/mm2 and Hpis the permissible contact stress N/mm2.3.a1= a2= a3= a4= a5= aR= constant 211607M. Bozca, P. Fietkau / Mechanism and Machine Theory 45 (2010) 15991612Fig. 13. Flow chart to optimize gearbox design parameters.The safety factor for bending stress SFranges between 1.00 and 2.68 during optimization. In addition, the safety factor forcontact stress SHvaries between 1.15 and 1.