1 BALANCING THE ROTOR OF TURBOCHARGERS BY REORIENTING THE ROTOR COUPLES

A PROJECT REPORT Submitted by RUTURAJ BARGAL DEPARTMENT OF ENGINEERING DESIGN INDIAN INSTITUTE OF TECHNOLOGY MADRAS CHENNAI 600 036 JULY 2016 2 ABSTRACT Aturbochargerisaturbine-drivenforcedinductiondevicethatincreasesan engine'sefficiencyandpowerbyforcingextraairintothecombustionchamber. Anidealturbochargeroperatesquietlyandefficientlyonlyifitisproperly balanced.Turbochargerrotatinggroupsaremadeupofseveralcomponents,of thesecomponents,onlytheturbineandcompressorwheelsarecomponent balancedpriortoassembly.Whenallthepartsarematedacertainamountof stack up unbalance is introduced into the completed turbo. The symptoms of an unbalanced turbo are oil leakage and screaming, an unbalance induced vibration oftherotatingassembly.Thecarturbochargerrequiresbalancingtomuchfiner limits and this cannot be achieved by balancing of individual components alone. A more precise state of balance can be attained by balancing the complete assembly over a speed range close to the maximum operating speed of the unit.Thepurposeoftheprojectistoreducetheinitialimbalancepresentinthe turbocharger core. This is done by finding the optimum orientation of assembling therotatingcomponentsoftheturbochargerwithrespecttoeachother.Atthis orientation, the imbalance in the turbocharger core would be the least. As a result, thecycletimerequiredforbalancing,theamountofmaterialtoberemoved,the number of correction runs and the rejection rate is reduced.3 ACKNOWLEDGEMENT Iamverymuchthankful to ourinternshipcoordinatorDr.M Ramanathanforhis permission to proceed with this project and encouragement on my project work. I sincerely thank Dr. Srikanth Vedantam, Head of the Department, Department of EngineeringDesign,andIndianInstituteofTechnology,Madrasfortheiractive encouragement and guidance that helped me see this project completed. IwouldliketothankmyIndustrialguide,Dr.AGopalakrishnan,VicePresident R&D, Turbo Energy Limited, for his expertise. I also thank Turbo Energy Limited for their assistance and support and allowing me touse their valuable resources. Finally,I would like toexpress special thanks toTurboEnergyLimitedstaffandworkers,teachingandnon-teachingstaff membersofEngineeringDesignDepartment,IITMadrasfortheirhelpand encouragement during the project. 4 TABLE OF CONTENTS ABSTRACT iii ACKNOWLDEGEMENT iv LIST OF TABLESvii LIST OF FIGURES viii LIST OF SYMBOLS, ABBREVIATIONS AND NOMENCLATURESix 1.INTRODUCTION ................................................................................................................................ 9 1.1 ABOUT THE COMPANY ................................................................................................................. 9 1.1.1 TURBO ENERGY LIMITED ..................................................................................................... 9 1.1.2 HERITAGE: .............................................................................................................................. 10 1.1.3 CUSTOMERS: .......................................................................................................................... 10 1.2 WHAT IS A TURBOCHARGER? ................................................................................................... 11 1.3 WHY DO WE NEED TO USE TURBOCHARGERS ..................................................................... 12 1.4 PARTS OF A TURBOCHARGER................................................................................................... 15 1.5 BALANCING ................................................................................................................................... 21 1.6 TYPES OF UNBALANCE ............................................................................................................... 22 1.6.1 STATIC UNBALANCE ............................................................................................................ 22 1.6.2 COUPLE UNBALANCE .......................................................................................................... 23 1.6.3 DYNAMIC UNBALANCE ....................................................................................................... 24 1.6.4 QUASI-STATIC UNBALANCE .............................................................................................. 24 1.7 SPECIFYING IMBALANCE: .......................................................................................................... 25 1.8 BALANCING LIMITS FOR TURBOCHARGERS ........................................................................ 25 2.MAIN PROJECT ................................................................................................................................ 28 2.1 OBJECTIVE OF THE PROJECT..................................................................................................... 28 2.2 ASSEMBLY PROCEDURE AT TEL .............................................................................................. 29 2.3 BALANCING PROCEDURE FOLLOWED AT TEL ..................................................................... 31 5 2.3.1 COMPRESSOR WHEEL BALANCING .................................................................................. 32 2.3.2 TURBINE WHEEL BALANCING ........................................................................................... 33 2.3.3 TURBOCHARGER CORE BALANCING: .............................................................................. 34 2.4 BENCHMARK DATA: .................................................................................................................... 35 2.5 REASONS FOR REJECTION: ........................................................................................................ 37 2.6 SOLUTION: DYNAMIC BALANCING OF ROTATING MASSES ............................................. 38 2.6.1 BALANCING OF SEVERAL MASSES ROTATING IN DIFFERENT PLANES ................. 38 2.6.2 APPLYING THE ABOVE METHOD TO TURBOCHARGER KP35 .................................... 42 2.7 PROGRAM ON PYTHON ............................................................................................................... 47 2.7.1 FORMULAE USED IN PYTHON ............................................................................................ 48 2.7.2 PROGRAM ................................................................................................................................ 49 2.8.1 INDIVIDUAL COMPRESSOR WHEEL IMBALANCE VALUES ........................................ 53 2.8.2 INDIVIDUAL TURBINE WHEEL IMBALANCE VALUES ................................................. 54 2.8.3 OPTIMUM ORIENTATION OF THE COMPRESSOR WHEEL ............................................ 55 2.8.4 RESULTS FROM OPTIMAL ORIENTATION ....................................................................... 56 2.8.5 ASSEMBLING AT WRONG ORIENTATION ........................................................................ 57 3. CONCLUSION ....................................................................................................................................... 59 3.1 FUTURE SCOPE .......................................................................................................................... 60 4. REFERENCES ....................................................................................................................................... 61 6 LIST OF TABLES Table 1: Balancing quality grades ............................................................................27 Table 2: Tabulation for Dynamic balancing method ...............................................40 Table 3: Dynamic Balancing of Masses applied to Turbocharger KP35 ................44 Table 4: Balancing measurement for Golden CoreError! Bookmark not defined. Table 5: Individual Compressor wheel imbalance values (5Times) ............... Error! Bookmark not defined. Table 6: Individual Turbine wheel imbalance values (5 Times) .. Error! Bookmark not defined. Table 7: Golden core assembled at 90 degree orientation ..... Error! Bookmark not defined. Table 8: Golden core assembled at 180 degree orientation ... Error! Bookmark not defined. Table 9: Golden core assembled at 270 degree orientation ... Error! Bookmark not defined. Table 10:Compressor wheel imbalance for test turbochargers .............................54 Table 11:Turbine wheel imbalance for test turbochargers ....................................55 Table 12: Optimum Orientation table for test turbochargers ...................................55 Table 13: Readings for wrong orientation of turbocharger .....................................57 Table 14: Comparison between methods .................................................................59 7 LIST OF FIGURES Figure 1: Turbo Energy Limited - Customers .........................................................10 Figure 2: Parts of a Turbocharger ............................................................................16 Figure 3: Turbine Housing .......................................................................................17 Figure 4: Turbine Wheel ..........................................................................................18 Figure 5: Compressor Housing ................................................................................18 Figure 6: Compressor wheel ....................................................................................19 Figure 7: Bearing Housing .......................................................................................19 Figure 8: Rotor Assembly of Turbocharger .............................................................20 Figure 9: Rotary components taken into consideration ...........................................29 Figure 10: Compressor wheel balancing .................................................................33 Figure 11: Turbine wheel balancing ........................................................................34 Figure 12: Turbocharger Core balancing .................................................................35 Figure 13: Benchmark readings ...............................................................................36 Figure 14: Dynamic Balancing of rotating masses ..................................................39 Figure 15: Dynamic Balancing of rotating masses method applied to KP35 Turbocharger ............................................................................................................43 Figure 16: Graphs for Golden Core ........................ Error! Bookmark not defined. Figure 17: Reference line on Golden core .......... Error! Bookmark not defined. Figure 18: Imbalance graph for 90 degrees compressor wheel assembly ....... Error! Bookmark not defined. 8 Figure 19: Imbalance graph for 180 degrees compressor wheel assembly ..... Error! Bookmark not defined. Figure 20: Imbalance graph for 270 degrees compressor wheel assembly ..... Error! Bookmark not defined. Figure 21: Graph of optimally oriented turbocharger ..............................................57 LIST OF SYMBOLS, ABBREVIATIONS AND NOMENCLATURES M, m: Mass of the body (Subscript represents the plane) R, r: Radius of the rotating mass (Subscript represents the plane) ,:Angleoftherotatingmasswithrespecttoareference(Subscriptrepresents the plane).t: Angle used in the python program(Subscript represent the plane) L, l: Length F: Force (Subscript represent the plane). 9 1.INTRODUCTION 1.1 ABOUT THE COMPANY 1.1.1 TURBO ENERGY LIMITED Turbo Energy Private Limited (TEL) was incorporated on 3rd May 1982, as a joint VenturebetweenBrakesIndiaPrivateLimited,SundramFinancePrivateLimited andBorgWarnerTurboSystem(formerlyknownasKKK-Germany).The companyisengagedinmanufactureofturbochargersaswellaspartsof turbocharger.Theproductsofthecompanyfindapplicationinautomobile, industrialandmarinesegments.TheCompanyisaPartofTVSGroup.TELhas achievedasalesturnoverofRs.7.8billioninFY2013-14.TELhasbeenableto achievecustomersatisfactionbybeingabletoprovideproductsandservicesof high quality at globally competitive prices. In line with the corporate values of the TVSGroup,TELhasafirmcommitmenttowardsitsstakeholders,thereby ensuring sustained growth of the organization. TEL also recognizes that its vendors are partners in progress. Emphasizing team work, trust and care among the employees, TEL always strives forhigherstandardsofperformance.Thecompanycontrolsitsactivitiesfromits 10 head office at Chennai and two manufacturing facilities located in Tamilnadu and oneinUttarakhand.ThecompanyhassetupanexclusiveResearchand Development(R&D)centerformanufacturinganddevelopingturbochargersof internalcombustionengine.The(R&D)centerislocatedinChennaiandis equippedwiththerequisitetestfacilities.Italsohasauthorizedservicescenters and distributors for spare parts to the requirements of its customers throughout the country.TheCompanycatersbothtothedomesticandtheinternationalmarkets. ThecompanyhasbeenaccreditedwithISO14001:2004andISO/TS16949 Certification from Bureau Veritas Certification (India) Private Limitedfor quality management System.1.1.2 HERITAGE: TheTVSGrouptracesitsoriginstoaruraltransportservice,foundedin1911in TamilNadu,India.Today,thisrenownedbusinessconglomerateremainsfaithful toitscoreidealsoftrust,values,serviceandethics.TheTVSGroupisIndias leadingsupplierofautomotivecomponentsandoneofthecountrysmost respected business groups. With a combined turnover of more than US$ 4 billion, theTVSGroupemploysatotalworkforceofcloseto25,000.Chartingasteady growthpathofexpansionanddiversification,itcurrentlycomprisesaround30 companies.Theseoperateindiversefieldsthatrangefromtwo-wheelerand automotivecomponentmanufacturingtoautomotivedealerships,financeand electronics.Unitingthesemultiplebusinessesisacommonethosofquality, customer service and social responsibility.11 1.1.3 CUSTOMERS: Figure 1: Turbo Energy Limited - Customers 1.2 WHAT IS A TURBOCHARGER? Anengineisdesignedtoburnafuel-airmixturetoproducemechanicalenergy. Themechanicalenergythenmovespistonsupanddowntocreatetherotary motion that turns the wheels of a vehicle. Themoremechanical energy, the more powertheenginecanproduce.Initssimplestformaturbochargerisanexhaust drivencentrifugalcompressorthatfeedsyourenginewithmoreairthanitcan normally ingest allowing a greater amount of power to be produced than the same engine normally aspirated. Since all internal combustion engines rely on oxygen to burntheirfuel,feedingmoreair,oroxygen,bytheuseofaturbocharger,more power can be produced. In most cases, power increases of 50-75% will be achieved with turbo charging.Normally,theexhaust gas byproductof the combustion processis expelledfrom theengineoutintotheatmosphere.Aturbochargerutilizesthisnormallywasted gasasanenergysourcetodrivetheturbinewheelbeforebeingreturnedtothe 12 exhaustsystem.Theturbinewheelhasashaftthatconnectsitdirectlytothe compressorwheellocatedontheoppositeendoftheturbocharger.This compressor wheel draws in fresh air from the air filter assembly, then compresses ittoprovideaforcefedchargeofairforyourengineutilizeincombustion.The turbochargerspinsatanextremelyhighspeed,somemodelsapproach160,000 revolutionsperminutetoprovideenoughairflowtofeedahungryengine.The turboreliesonoilsuspendedbearingstoallowitsshafttorotate.Thespeedat whichtheturboturns,combinedwiththeheatgeneratedbytheexhaustsystem equatestoaverysophisticatedpartmadewithveryspecializedmaterials, machined and manufactured to strict tolerances.Asignificantdifferencebetweenaturbochargeddieselengineandatraditional naturallyaspiratedgasolineengineisthattheairenteringadieselengineis compressed before the fuel is injected. This is where the turbocharger is critical to thepoweroutputandefficiencyofthedieselengine.Itisthejobofthe turbocharger to compress more air flowing into the engines cylinder. When air is compressedtheoxygenmoleculesarepackedclosertogether. This increase inair meansthatmorefuelcanbeaddedforthesamesizenaturallyaspiratedengine. This generates increased mechanical power and overall efficiency improvement of thecombustionprocess.Therefore,theenginesizecanbereducedfora turbochargedengineleadingtobetterpackaging,weightsavingbenefitsand overallimprovedfueleconomy.Althoughturbochargingisarelativelysimple concept,theturbochargeriscriticaltotheoperationofthedieselengineand therefore requires a highly engineered component.1.3 WHY DO WE NEED TO USE TURBOCHARGERS 13 In the early 1980s most of the worlds Formula One teams turbocharged their race carsforonemajorreason:morepowerfromasmallerenginesizeandweight. Turbochargedcarsoffermorepower,morespeedandbetterhandlingthantheir rivalswithconventionalnaturallyaspiratedengines.Aturbochargedenginecan providegoodengineresponsewiththereassuranceofreliablereservesofpower whenyouneedit.Eversincetheintroductionofdieselengines,thedemandhas always been for small engines capable of producing maximum power. Add to this thepresentenvironmentalproblemsofleadpollutionetc.,plustheeconomicsof thedieselengineuserandyou haveanareainwhichthe turbochargercan playa major part.We have already given some reasons for turbocharging i.e. the power of an engine is controlled by the ratio of air to fuel. The size and weight of engines has become important;no-onewantstolosealargeamountofthepowerfromtheengine simplytopropeltheexcessweightdownaroad,ortohavehugevehiclesjustto carry normal loads. Under normal circumstances the addition of a turbocharger to a naturally aspirated enginewillincreasethepoweroutputbyapproximately30%.Remember,this meansbasicallythesamesizeenginegivingmorepowertobeusedbythe operator.Intheearlydaysitwascommonpracticetoputaturbochargerontoa naturallyaspiratedengine,nowturbochargedenginesaredesignedtotakeinto account the higher stress thermal and mechanical loading placed on the engine and other ancillary parts by the turbocharger Evermorestringentemissionsregulationsacrosstheworldarechallenging automotive manufacturers to create engines that meet the needs of the environment whilst still satisfying the demands of consumers for vehicles that are fun to drive. 14 The main focus is on engine downsizing, emissions control and fuel economy but notattheexpenseofdriveability.Thesegoalsarecomplementaryandbring togethertheperformancequalitiestomakeanautomobilesafe,cleanandfunto drive. Honeywell turbochargers deliver significant benefits to end users: Safer Aturbochargedenginecangenerateasmuchas7timesmorepowerthana naturallyaspirated(non-turbocharged)engineofequivalentdisplacement.For example,Formula11.5Lturbochargedenginesproducedmorethan1000HP.In morestandardapplications,itisrealistictodoublethepowerofagivenengine throughturbocharging,makingvehiclesmoreresponsiveandsafertodrive. Turbochargersalsopreventthelossofpowerathighaltitudes,thusproviding significant advantages to turbocharged trucks and off-road machinery. Fast reactionInstandardapplicationthepoweroutputofaturbochargedengineisdoubled whichisconsiderable,andfortheresultithasfasterresponseindrive. Turbochargersalsohaveinfluenceonthepoweroutputlosssothatathigher enginespeeds,moreexhaustgasatlowerpressurecanbepassedthroughthe turbine,whichinsuresadvantage totrucks,generators and industrialengineswith turbocharged engines. Economic advantagesTurbochargersrecyclingenergywhichenginesproduced,bytransformationmore ofexhaustgasenergyinpoweroutputwithlessfrictionalandthermallosses. Comparedwithanaturallyaspiratedenginethefuelconsumptionofa turbocharged engine is lower. The ratio power-to-weight of the exhaust gas turbine 15 engine is much better than that of the naturally aspirated engine. The turbocharged engines installation space requirement is smaller than that of a naturally aspirated engine with the same power output. Turbochargers harness and recycle the energy produced by automobile engines, transformingmore of the fuel energy consumed intopowerbycreatinglessparasiticheatandfriction.Asaresult,turbocharged enginesdeliversignificantfuelcostadvantagesovertheirnaturally-aspirated counterparts.Ecology Whereas, turbochargers supplied more air mass into engines combustion chamber, thecombustioniseasier,plain,andalsoemissionislower.Today,the turbochargeddieselenginesincomparisonwithnaturallyaspiratedengineshave CO2 and NOx emissions about 50 % lower.The high altitude performance The turbine engine performance at high altitude is significantly better. The lower airpressureathighaltitudescanproducethepowerlossofanaturallyaspirated engine.Incontrast,theperformanceoftheturbineimprovesathighaltitudeasa result of the greater pressure difference between the pressure of the turbine and the ambientpressureatoutletwhichislowerwiththehigheraltitude.Thelowerair density at the compressor inlet is largely equalized. Hence, the turbine engine has barely any power loss at higher altitude. Noise levelBecause of reduced overall size, the sound-radiating outer surface of a turbo engine is smaller; it is therefore less noisy than a naturally aspirated engine with identical output. The turbocharger itself acts as an additional silencer. 16 More fun Turbochargersdelivergreatertorquewhich,inturn,translatesintoimproved performance on the road and make driving a real pleasure 1.4 PARTS OF A TURBOCHARGER Aturbochargerismadeup oftwomainsections: the turbineand thecompressor. Theturbineconsistsoftheturbinewheelandtheturbinehousing.It isthejobof theturbinehousingtoguidetheexhaustgasintotheturbinewheel.Theenergy fromtheexhaustgasturnstheturbinewheel,andthegasthenexitstheturbine housing through an exhaust outlet area.Thecompressoralsoconsistsoftwoparts:thecompressorwheelandthe compressorhousing.Thecompressorsmodeofactionisoppositethatofthe turbine.Thecompressorwheelisattachedtotheturbinebyaforgedsteelshaft, and as the turbine turns the compressor wheel, the high-velocity spinning draws in airandcompressesit.Thecompressorhousingthenconvertsthehigh-velocity, low-pressureairstreamintoahigh-pressure,low-velocityairstreamthrougha processcalleddiffusion.Thecompressedairispushedintotheengine,allowing the engine to burn more fuel to produce more power. 17 Figure 2: Parts of a Turbocharger Turbine Housing: Turbine housings are manufactured in various grades of spheroidal graphite iron to dealwiththermalfatigueandwheelburstcontainment.Aswiththeimpeller, profilemachiningtosuitturbinebladeshapeiscarefullycontrolledforoptimum performance. The turbine housing inlet flange acts as the reference point for fixing turbochargerpositionrelativetoitsinstallation.Itisnormallytheloadbearing interface. Figure 3: Turbine Housing (1)The turbine wheel (2)The turbine housing (3)Exhaust gas(4)Exhaust outlet area (5)The compressor wheel (6)The compressor housing (7)Forged steel shaft(8)Compressed air 18 Turbine wheel: The turbine wheel is made from a high nickel superalloy investment casting. This methodproducesaccurateturbinebladesectionsandforms.Largerunitsarecast individually.Forsmallersizesthefoundrywillcastmultiplewheelsusingatree configuration.Duetothestrengthofthematerialandthetemperaturesitcan handle,materialcannotberemovedfromtheturbinewheelsideduring turbochargercorebalancing.Thereforematerialisremovedonlyfromthe compressor wheel during the core balancing procedure. Figure 4: Turbine Wheel Compressor Housing: Compressor housings are also made in castaluminum. Various grades are used to suit the application. Both gravity die and sand casting techniques are used. Profile machining to match the developed compressor blade shape is important to achieve performance consistency. 19 Figure 5: Compressor Housing Compressor Wheel (Impellor): Compressorimpellersareproducedusingavariantofthealuminuminvestment casting process. A rubber former is made to replicate the impeller around which a casting mould is created. The rubber former can then be extracted from the mould into which the metal is poured. Accurate blade sections and profiles are important inachievingcompressorperformance.Backfaceprofilemachiningoptimizes impeller stress conditions. Boring to tight tolerance and burnishing assist balancing andfatigueresistance.Theimpellerislocatedontheshaftassemblyusinga threaded nut. Figure 6: Compressor wheel Bearing Housing: 20 Agreycastironbearinghousingprovideslocationsforafullyfloatingbearing systemfortheshaft,turbineandcompressorwhichcanrotateatspeedsupto 170,000 rev/min. Shell moldings is used to provide positional accuracy of critical featuresofthehousingsuchastheshaftbearingandseallocations.CNC machinerymills, turns, drills and taps housing faces and connections. The bore is finishhonedtomeetstringentroundness,straightnessandsurfacefinish specifications. Figure 7: Bearing Housing Bearing Systems: Thebearingsystemhastowithstandhightemperatures,hotshutdown,soot loadingintheoil,contaminants,oiladditives,drystarts.Journalbearingsare manufacturedfromspeciallydevelopedbronzeorbrassbearingalloys.The manufacturingprocessisdesignedtocreategeometrictolerancesandsurface finishes to suit very high speed operation. Wastegate: A wastegate is a valve that diverts exhaust gases away from the turbine wheel in a turbocharged engine system. Diversion of exhaust gases causes the turbine to lose speed,whichinturnreducestherotatingspeedofthecompressor.Theprimary function of the wastegate is to stabilize boost pressure in turbocharger systems, to protecttheengineandtheturbocharger.Thewastegateiscontrolledbya 21 wastegate actuator in which the actuator is controlled by pressure coming from the intake manifold. Rotating assembly:Moststandardturbochargerdesignshavearotatingassemblythatconsistsofthe turbinewheel,thrustcollarassembly,compressorwheel,andshaftnut.Itis importantifanyone part of thisassemblyischanged, thatthe unitasawhole be rechecked for proper balance. This is the most important part of the turbocharger. Figure 8: Rotor Assembly of Turbocharger Other parts of the turbocharger are:1. Shaft nut 2. Back plate 3. Compressor piston ring 4. Star spring and Quad ring 5. Thrust bearing 6. Thrust collar 7. Bearing clip 8. Journal bearing 22 9. Heat shield 10. Turbine piston ring 1.5 BALANCING Balancing is the technique of correcting or eliminating unwanted inertia forces or momentsinrotatingorreciprocatingmassesandisachievedbychangingthe location of the mass centers. Unbalance in a rotor is the result of an uneven distribution of mass, which causes the rotor to vibrate. The vibration is produced by the interaction of an unbalanced masscomponentwiththeradialaccelerationduetorotation,whichtogether generateacentrifugalforce.Sincethemasscomponentrotates,theforcealso rotatesandtriestomovetherotoralongthelineofactionoftheforce.The vibrationwillbetransmittedtotherotor'sbearings,andanypointonthebearing willexperiencethisforceonceperrevolution.Theobjectivesofbalancingan engine are to ensure: 1.That the center of gravity of the system remains stationery during a complete revolution of the crank shaft. 2.Thecouplesinvolvedinaccelerationofthedifferentmovingpartsbalance each other. 1.6 TYPES OF UNBALANCE The location of the mass center and the principal inertia axes are determined by the distribution of mass within the part. Unbalance exists when the axis of rotation is notcoincidentwithaprincipalinertiaaxis.Itisimportanttodrawadistinction 23 betweenunbalanceandbalancecorrection.Unbalanceisamassproperty.It becomesacharacteristicofthepartwhenanaxisofrotationisdefined.Balance correction is a means to alter the mass properties to improve thealignment of the axis of rotation with the mass center and/or the central principal axis. Both can be expressed as weights and radii and have shared terminology. This section discusses unbalance as a mass property. 1.6.1 STATIC UNBALANCE A condition of static unbalance exists when the mass center does not lie on the axis of rotation. Static unbalance is also known asForce Unbalance. As defined, static unbalanceisanidealcondition,ithastheadditionalconditionthattheaxisof rotation be parallel to the central principal axis, i.e. no couple unbalance. Static unbalancehastheunitsofweightlengthormasslength.Commonunitsofstatic unbalanceareinozorg mm.Aworkpieceisinstaticbalancewhenthemass center lies on the axis of rotation. When this condition exists, the part can spin on the axis with no inertial forces; that is to say without generating centrifugal force. Even parts intended for static applications, such as speedometer pointers or analog metermovements, benefit from being in static balance in that the force of gravity will not create a moment greater at one angle than at another which causes them to benon-linear.Staticunbalancecanbecorrectedwithasingleweight.Ideallythe correction is made in the plane of the mass center and is sufficient to shift the mass centerontotheaxisofrotation.Itisimportanttoalignthecorrectionwiththe initial unbalance to move the mass center directly towards the axis of rotation. Static unbalance can be detected on rotating or non-rotating balancers. 24 1.6.2 COUPLE UNBALANCE Couple unbalance is a specific condition that exists when the central principal axis ofinertiaisnotparallelwiththeaxisofrotation.Coupleunbalanceisoften presentedasdynamicunbalanceinengineeringclasses;howeverthistermis definedotherwisebyISO1925andisreservedforthemoregeneralcaseof combinedstaticandcoupleunbalance.Coupleunbalanceisanidealcondition.It carries the additional condition that the mass center lies on the axis of rotation - no staticunbalance.Coupleunbalancehastheunitsofweight length2or mass length2.Coupleunbalanceappearsastheoff-diagonaltermsintheinertia matrix for a rigid body.This is an indication that the inertial axes are not aligned with the principal axes. It can be expressed as a vector with direction perpendicular to the plane of the radius vector and thecouple arm vector. This is the axis about which the couple acts and is 900 or normal to the plane in which balance correction should be made. Couple correction requires that two equal weights be added to the work piece 180 apart in two correction planes. The distance between these planes is called the couple arm. The location of the correction planes is arbitrary provided the couple matches the unbalance. Whereas static unbalance can be measured with anon-rotatingbalancer,coupleunbalancecanonlybemeasuredonarotating balancer. 1.6.3 DYNAMIC UNBALANCE Dynamic unbalance is the case in which the central principal axis is not parallel to and does not intersect the axis of rotation. Dynamic unbalance is also referred to as two plane unbalance, indicating that correction is required in two planes to fully 25 eliminate dynamic unbalance. A two plane balance specification is normally expressed in terms of force per plane and must include the axial location of the correction planes to be complete. Dynamic unbalance captures all the unbalance which exists in a rotor. This type of unbalance can only be measured on a rotating balancer since it includes couple unbalance. Since dynamic unbalance is a combination of static and couple unbalance and since static and couple unbalance have different units, there are no unique units for dynamic unbalance. It can be expressed as static and couple or in terms of the balance corrections require. 1.6.4 QUASI-STATIC UNBALANCE Quasi- static unbalance is a special form of dynamic unbalance in which the static and couple unbalance vectors lie in the same plane. The central principal axis intersects the axis of rotation, but the mass center does not lie on the axis of rotation. This is the case where an otherwise balanced rotor is altered (weight added or removed) in a plane some distance from the mass center. The alteration creates a static unbalance as well as a couple unbalance. Conversely, a rotor with quasi-static unbalance can be balanced with a single correction of the right magnitude in the appropriate plane. 1.7 SPECIFYING IMBALANCE: Unbalance can be specified in many forms. The most common is expressed as a weight of material to be added or removed at a specified correction radius. The weight units can be any convenient units; grams (g), ounces (oz), and kilograms (kg) are common units. Occasionally Newton's (N) are specified, but for practical 26 use must be converted to available weight scale units. Length units are often expressed in; inches (in), millimeters (mm), centimeters (cm), and meters (m). The most common combinations used to specify unbalance are ounce-inches (oz-in), gram-inches (g-in), gram-millimeters (g-mm), gram-centimeters (g-cm), and kilogrammeters (kg-m). 1.8 BALANCING LIMITS FOR TURBOCHARGERS Ideally a machine should be balanced until there is no unbalance at all. However for practical purposes, the time and cost required to balance the rotor should be taken into account. Therefore, it is appropriate to vary the permissibility of unbalance depending on the rotating machinery. The International Organization for Standardization (ISO) has published a standard ISO 1940/1 Balancing Quality Requirements of Rigid Rotors and has a guideline of the quality of balancing. This guideline shows what parameter is used to express the quality of balance and how to determine the balance acceptance limit for a particular rotor assembly. The balancing quality grade for turbochargers is G40. On the basis of this quality grade, the acceptance limits for balancing the compressor wheel, Turbine wheel and the turbocharger core assembly is decided. Compressor wheel balancing limits: 0.1 MMG Turbine wheel balancing limits: 0.1 MMG 27 Turbocharger core balancing limits:8mg at 140000 RPM15mg at 65000 RPM Table 1: Balancing quality grades BalanceProduct of the Relationship QualityRotor Types - General Examples Grade (1)(2) mm/s G 4 0004 000Crankshaft/drives(3) of rigidly mounted slow marine diesel engines with uneven number of cylinders (4) G 1 6001 600 Crankshaft/drivesof rigidly mounted large two-cycle engines G 630630 Crankshaft/drivesof rigidly mounted large four-cycle engines Crankshaft/drivesof elastically mounted marine diesel engines G 250250 Crankshaft/drivesof rigidly mounted fast four-cylinder diesel engines(4) G 100100 Crankshaft/drivesof fast diesel engines with six or more cylinders (4) Complete engines (gasoline or diesel) for cars, trucks and locomotives (5) G 4040 Car wheels, wheel rims, wheel sets, drive shafts Crankshaft/drivesof elastically mounted fast four-cycle engines with six or more cylinders (4) Crankshaft/drivesof engines of cars, trucks and locomotives G 1616 Drive shafts (propeller shafts, cardan shafts) with special requirements Parts of crushing machines Parts of agricultural machinery Individual components of engines (gasoline or diesel) for cars, trucks and locomotives G 6.36.3 Crankshaft/drivesof engines with six or more cylinders under special requirements Parts of process plant machines Marine main turbine gears (merchant service) Centrifuge drums

Paper machinery rolls; print rolls Fans Assembled aircraft gas turbine rotors Flywheels Pump impellers Machine-tool and general machinery parts 28 Medium and large electric armatures (of electric motors having at least 80 mm shaft height) without special requirements Small electric armatures, often mass produced, in vibration insensitive applications and/or with vibration-isolatingMountings Individual components of engines under special requirements G 2.52.5 Gas and steam turbines, including marine main turbines (merchant service) Rigid turbo-generator rotors Computer memory drums and discs Turbo-compressors Machine-tool drives Medium and large electric armatures with special requirements Small electric armatures not qualifying for one or both of the conditions specified for small electric armatures of balance quality grade G 6.3 Turbine-driven pumps G 11 Tape recorder and phonograph (gramophone) drives Grinding-machineDrives

G 0.40.4 Small electric armatures with special requirements Spindles, discs and armatures of precision grinders Gyroscopes 2.MAIN PROJECT 2.1 OBJECTIVE OF THE PROJECT Themainobjectiveoftheprojectistoreducetheinitialimbalanceintheturbo chargerbyreorientingtherotorcouples.Thepartsoftheturbochargerwillbe assembled in a predetermined orientation. In this orientation the imbalance present in the turbocharger will be the least. The positive effects of the project are:1.The initial imbalance in the turbocharger can be reduced 2.Itwillreducetheamountofmaterialtoberemovedfromtheturbocharger core. 3.The number of cuts and correction runs will decrease. 4.The cycle time of balancing the turbocharger can be reduced.5.The number of rejection will decrease. 29 For the purpose of the project we are concerned onlywith rotor assembly, i.e. the compressorwheel,turbinewheelandshaft.Themainaimistofindoutthe orientationofthecompressorwheelwithrespecttotheturbinewheelsuchthe maximum amount of forces and couples are nullified. Figure 9: Rotary components taken into consideration 2.2 ASSEMBLY PROCEDURE AT TEL The turbocharger assembly mainly has the following stages: 1.Core assembly2.Balancing3.Assembly of Housings4.Waste Gate Actuator setting5.Final Inspection30 TheindividualcomponentsarestoredinHeavydutystorageRacksystems& automated Vertical storage systems. These parts are moved to the assembly using conveyors systems and other lean management material handling systems.Core Assembly: CoreassemblycanbecalledasaheartoftheTurbocharger;TheCoreassembly providessupportsforthejournalbearingsandprovideslubricationtotherotary parts.Therearesemi-automaticcoreassemblystationswithspecialcontrolsfor errorproofing.TELisalsoequippedwithautomaticcoreassemblylinefor passenger cars which has higher operational speed. There are controls at each stage for ensuring Zero defects. Balancing: Inadditiontoindividualcomponentlevelbalancing,basedontheapplication balancingisalsodoneasacoreassembly.Thebalancinghelpsinreducingthe vibrationsandnoiselevelsduringoperation.TELhasfacilityforfullyautomatic highspeedcorebalancingandsemi-automaticbalancing.Thesemachinesarea masterpiecedemonstrationofthecollaborationstechnologicalcompetence.The corescanbebalancedtohigherspeedsupto200,000RPMbasedonthe application requirement. Assembly of Housings: TheCoreAssembliesaremountedandfastenedatrequiredorientationonthe Compressorhousing&turbinehousing.Theseoperationsaredoneinindividual work stations with error proofing control. TEL also has a semi-automatic conveyor assemblylineforassemblingthehousingstothecoreassembly.Thishaspallet mechanismforfeedingthematerialsandassemblyisdoneonthepallet.The 31 traceabilityisestablishedbyinterfacing2Dreaderswhichreadstheunique turbocharger serial no. at each station and records the parameters and values at that stage. The tightening process is controlled with a DC tightening tool, the feedback ismonitoredandthetorquevaluesarestoredinthedatabase.Theassemblyline also has sensors and vision systems to detect the presence of various components. Waste Gate Actuator setting: TheWasteGateActuatorsettingisdonetoenablethewastegateflaptoopenat the required operational boost pressure. TEL uses a custom built Setting Rig made bythetechnologydevelopmentwingforperformingthisoperation.Thereare variouscontrolslikedatatraceability,characteristicmonitoring,errorproofing control etc. in these rigs. TEL also has the state of art Flow setting rig for Variable geometry Turbochargers Final Inspection: Final Inspection of turbochargers is carried out at the End of each line to ensure the complianceoftheTurbochargertothespecifiedrequirements.FinalInspection actsasafirewalltoimprovetherapiddetectionandcontrolmechanism.Final Inspectionisdonemanuallybytrainedpersonalsandwiththeaidofautomated vision systemThe projects attempts reduce the initial imbalance during the core assembly stage. This is going to be done by placing the compressor wheel at a particular angle with respect to the turbine wheel. 32 2.3 BALANCING PROCEDURE FOLLOWED AT TEL Foraturbochargertobecompletelybalanced,theindividualcomponentsmaking theturbochargershouldbebalancedwithinthelimits.Mainlythecompressor wheel and the turbine wheel need to be balanced very carefully. When all the parts oftheturbochargerarematedacertainamountofstackupunbalanceis introduced into the completed turbo. The turbocharger requires balancing to much finerlimitsandthiscannotbeachievedbybalancingofindividualcomponents alone.Thesolutiontothisproblemistoachieveamoreprecisestateofbalance which can only be attained by balancing the complete core assembly over a speed rangeclosetothemaximumoperatingspeedoftheunit.Balancingofthe componentsandthefinalcoreassemblyaredonebyremovingmaterialintwo selected planes until it is within the balancing limits. 2.3.1 COMPRESSOR WHEEL BALANCING The compressor wheel is made of an aluminum alloy. Material is removed at two planes,thenutplaneandthehubplane.Paintismarkedonthebladesofthe compressorwheelandthispaintactsasthereference.Thecompressorwheelis rotated at high speeds and the imbalance value is checked. The balancing machine decides the amount ofmaterial to be removed and the angle at which it has to be removed. The final imbalance value and the angle with respect to the reference are noted.Thecompressorwheelisbalancedtothelimitsof0.1MMG.Thefinal imbalance values got from the compressor wheel is F1 and 1 in plane 1, F2 and 2 in plane 2.Plane 1: Nut plane of compressor wheel 33 Plane 2: Hub plane of compressor wheel Figure 10: Compressor wheel balancing 2.3.2 TURBINE WHEEL BALANCING Theturbinewheelismadeofanickelalloy.Forbalancingtheturbinewheel, materialisremovedattwoplanes,thenutplaneand the hubplane oftheturbine wheel. Paint is marked on the blades of the turbine wheel and this paint acts as the reference.Theturbinewheelisrotatedathighspeedsandtheimbalancevalueis checked. The balancing machine calculatesthe amount of material to be removed and the angle at which it has to be removed with respect to the paint reference. The finalimbalancevalueandtheanglewithrespecttothereferencearenoted.The Turbinewheelisbalancedtothelimitsof0.1MMG.Thefinalimbalancevalues got from the compressor wheel is F3 and 3 in plane 3, F4 and 4 in plane 4. Plane 3: Hub plane of Turbine wheel Plane 4: Nut plane of Turbine wheel PLANE 1 PLANE 2 REFERENCE PAINT 34 Figure 11: Turbine wheel balancing 2.3.3 TURBOCHARGER CORE BALANCING: Since balancing the individual components is not enough for producing a completely balanced turbocharger, the turbocharger core needs to be balanced as a whole in the High Speed Core Balancing Machine. Material is removed only from the compressor wheel side in planes 1 and 2. Material is removed until the turbocharger imbalance is within the limits. Turbocharger core balancing limits:8mg at 140000 RPM15mg at 65000 RPM PLANE 3 PLANE 4 35 Figure 12: Turbocharger Core balancing 2.4 BENCHMARK DATA: 1.The turbocharger being studied is KP35 MSIL (Maruti Swift) 2.The compressor wheel and the turbine wheel are balanced individually. 3.The parts of the turbocharger are assembled at arbitrary positions to form the turbocharger core. 4.TheentireturbochargercoreisbalancedinaHighSpeedCoreBalancing machine until the balancing values are within the limits. Usingtheabovementionedprocedureforbalancing,thebenchmarkreadingsfor 150 turbochargers were noted and the observations were: Parameters noted:1.Incoming imbalance and angle 2.Outgoing imbalance and angle 3.Cycle time 4.Number of correction runs 5.Number of cuts on the compressor wheel PLANE 2 PLANE 1 36 1.Rejectionrate:Thisisthescenarioinwhichtheturbochargercannotbe balancedforvariousreasonsandissentbacktotheinventoryforthe compressorwheeltobechanged.Anattemptismadetobalancethe turbocharger with the new compressor wheel. Rejection rate = 24/150 = 16% 2.Averagecycletime:Thisistheaveragetimetakentobalancethe turbocharger.Thetimeismeasuredfromtheinstanttheturbochargeris loaded onto the High Speed Core Balancing Machine till the point it is taken out of the machine. Average Cycle Time = 109 seconds per turbocharger Figure 13: Benchmark readings 37 3.Averagecorrectionruns:ThisisthenumberoftimestheHighSpeedCore BalancingMachineattemptstobalanceaturbocharger.Themachine removessomeamountofmaterialinthefirstcorrectionrunandifthe turbochargerisstillnotbalanced,themachinestartsthesecondrunand attemptstobalancetheturbocharger.Themaximumnumberofcorrection runs is set at 6 by TEL. Average Correction Runs = 1.8 runs per turbocharger 4.Average number of cuts: This is the number of cuts made on the compressor wheel after the balancing is finished.Average Number of cuts = 3.2 cuts per turbocharger 2.5 REASONS FOR REJECTION: INITIALLIMIT:Thisisthescenarioinwhichtheincomingimbalancein theturbochargeristoohigh(M>150mg).Itisnotpossibletoremoveso much material from the turbocharger core. If the imbalance level is too high, the depth of cut will exceed the recommended value. The turbocharger core is rejected immediately. MANUFACTURING LIMITS: When the depth of cut of the material to be removed is too high, the machine rejects the turbocharger. NUMBER OF CORRECTION RUNS: This is the number of times the High SpeedCoreBalancingMachineattemptstobalanceaturbocharger.The 38 machine removes some amount of material in the first correction run and if the turbocharger is still not balanced, the machine starts the second run and attemptstobalancetheturbocharger.Thecompanyhassetthemaximum number of correction runs to 6. If the turbocharger is not balanced within 6 correctionrunsitisrejected.Theoldcompressorwheelisreplacedwitha new one. 2.6 SOLUTION: DYNAMIC BALANCING OF ROTATING MASSES 2.6.1 BALANCING OF SEVERAL MASSES ROTATING IN DIFFERENT PLANES Whenseveralmassesrevolveindifferentplanes,theymaybetransferredtoa referenceplaneand thisreference planeisaplane passing througha pointon the axisofrotationandperpendiculartoit.Whenarevolvingmassinoneplaneis transferred to a reference plane, its effect is to cause a force of same magnitude to the centrifugal force of the revolving mass to act in the reference plane along with acouple ofmagnitudeequalto theproductoftheforceand thedistance between the two planes. Inordertohaveacompletebalanceoftheseveralrevolvingmassesindifferent planes: 1.Theforcesinthereferenceplanemustbalance,i.e.,theresultantforce must be zero and 2.Thecouplesaboutthereferenceplanemustbalancei.e.,theresultant couple must be zero. 39 A mass placed in the reference plane may satisfy the first condition but the couple balanceissatisfiedonlybytwoforcesofequalmagnitudeindifferentplanes. Thus, in general two planes are needed to balance a system of rotating masses. Position of planes of masses: Choose a reference plane at O so that the distance of the planes 1, 2, 3 and 4 from OareL1,L2,L3andL4respectively.ThereferenceplanechosenisplaneL. Choose another plane M between plane 3 and 4 as shown. Plane M is at a distance of Lm from the reference plane L. The distances of all theotherplanestotheleftofLmaybetakenasnegative(-ve)andtotheright may be taken as positive (+ve). The magnitude of the balancing masses mL and mM in planes L and M may be obtained by following the steps given below. Figure 14: Dynamic Balancing of rotating masses 40 Tabulate the given data as shown after drawing the sketches of position of planes ofmassesandangularpositionofmasses.Theplanesaretabulatedinthesame order in which they occur from left to right. Step 1: Table 2: Tabulation for Dynamic balancing method CentrifugalDistance Couple/ 2 PlaneMass (m)Radius (r)force/2from Ref. (m r) plane L (L) (m r L) 1m1r1m1 r1- L1- m1 r1 L1 LmLRL mL rL00 2m2r2m2 r2L2m2 r2 L2

3m3r3m3 r3L3m3 r3 L3

MmMrMmM rMLMmM rM LM 4m4r4m4 r4L4m4 r4 L4

Step 2:Drawthespacediagramorangularpositionofthemasses.Sincealltheangular position of the masses are given with respect to mass 1, take the angular position of mass 1 as 1 = 00. In order to balance the system the summation of the couple and the summation of the centrifugal forces should individual be zero. For dynamic balancing the conditions required are, mr + mM rM + mL rL = 0 ---------- (I) for force balance 41 mrl + mN rN lN = 0 -------------- (II) for couple balance Balancing the couple: Resolvethecouplesintotheirhorizontalandverticalcomponentsandfindtheir sums.Sum of the horizontal components gives, mrl cos + mM rM lM cosM = 0 (1) Sum of the vertical components gives, mrl sin + mM rM lM sinM = 0 (2) Square and add (1) and (2) to find the value of mass mM Divide (2) by (1) to get the value of M. Balancing the forces:Resolvetheforcesintotheirhorizontalandverticalcomponentsandfindtheir sums. Sum of the horizontal components gives, mr cos + mL rL cosL + mM rM cosM = 0 (3) Sum of the vertical components gives, 42 mr sin + mL rL sin L + mN rN sinN = 0 (4) Squaring and adding (3) and (4), we get the value of mL Dividing (4) by (3), we get the value of L TheendresultisthatML,L shouldbeaddedinplaneLandMM,M, shouldbe added in plane M for the entire system to be balanced. 2.6.2 APPLYING THE ABOVE METHOD TO TURBOCHARGER KP35 Plane 1: Compressor wheel nut plane Plane 2: Compressor wheel hub plane Plane 3: Turbine wheel hub plane Plane 4: Turbine wheel nut plane M1, R1 & 1: Imbalance mass, radius and its angle at plane 1 M2, R2 & 2: Imbalance mass, radius and its angle at plane 2 M3, R3 & 3: Imbalance mass, radius and its angle at plane 3 M4, R4 & 4: Imbalance mass, radius and its angle at plane 4 MA, RA & A: Balancing mass, radius and its angle to be placed at plane A MB, RB & B: Balancing mass, radius and its angle to be placed at plane B L2, L3, L4: Dimensions of the turbocharger Let plane A be the reference plane. 43 Dimensions of the turbocharger: L2 = 24.1mm L3 = 55.1 mm L4 = 24.17 mm

Let the force F = M*R Since mass can be removed only the compressor wheel side (plane 1 and plane 2), the balancing planes A and B are picked as follows. Plane A: Balancing plane A on the compressor wheel nut plane Plane B: Balancing plane B on the compressor wheel hub plane M11 M1 M1 M1 M4 M3M2 R3 L4L3L2 R11 M1 M1 M1 R21 M1 M1 M1 R41 M1 M1 M1 COMPRESSOR WHEELTURBINE WHEEL A B MB1 M1 M1 M1 M41 M1 M1 M1 M31 M1 M1 M1 MA1 M1 M1 M1 M21 M1 M1 M1 M11 M1 M1 M1 Figure 15: Dynamic Balancing of rotating masses method applied to KP35 Turbocharger 44 Table 3: Dynamic Balancing of Masses applied to Turbocharger KP35 PlaneMass (M) Radius (R) Centrifugal force (M*R) Length from reference plane (L) Couple (M*R*L) 1M1R1M1R100 AMARAMARA00 2M2R2M2R2L2M2R2L2 BMBRBMBRBL2 MBRBLB 3M3R3M3R3L2 + L3 M3R3L3 4M4R4M4R4L2 + L3 + L4M4R4L4 BALANCING THE COUPLE: Summation of Horizontal components: FL Cos()+FBL2 Cos(B) = 0 FBL2 Cos(B) = ConstantFB Cos(B) = C1Equation1 Summation of Vertical components: FL Sin()+ FBL2 Sin(B)= 0 45 FBL2 Sin(B)= Constant FB Sin(B)= C2Equation2 (Equation 2) divided by (Equation 1) tan(B) = C2/C1 B = tan-1(C2/C1) Substitute B value in Equation 1FB = C1/ Cos(B) (OR) Perform (Equation 2)2 + (Equation 1) 2 (FB )2= C12 +C22 FB = (C12 +C22)0.5 Substitute FB in Equation 1B = cos-1(C1)/ FB BALANCING THE FORCE: Summation of Horizontal components: F Cos() + FA Cos(A) = 0 FA Cos(A)= C3Equation 3 Summation of Vertical components: 46 F Sin() + FA Sin(A) = 0 FA Sin(A) = C4Equation 4 Equation 4 divided by Equation 3 tan(A) = C4/C3 A = tan-1(C4/C3) Substitute A value in Equation 3 FA= C3/ Cos(A) (OR) Perform (Equation 3)2 + (Equation 4) 2 (FA )2= C32 +C42 FA = (C32 +C42)0.5 Substitute FA in Equation 3A = cos-1(C3)/ FA NowthatthevaluesofFAandFBareknownforaparticularorientationofthe compressorwheelandtheturbinewheel,themainobjectiveistoreducethe balancingmass(i.e.FAandFB)byfindingtheoptimumorientationofthe compressor wheel with respect to the turbine wheel. The value of FA and FB needs to be calculated for every orientation or position of thecompressorwheelwithrespecttotheturbinewheel.Thenthe 47 orientation/positionatwhichtheresultantofFAandFBistheleastneedstobe picked.Rotatethecompressorwheelby1owithrespecttotheturbinewheel.Thiswill resultinthevaluesof1 and2 changeby1o each.CalculatethenewFAandFB values for the new 1 and 2 and check if the sum of the new FAand FB is lower than the previous sum of FA and FB. If the new sum is lesser than the old sum, then theoldorientationofthecompressorwheelisreplacedwiththeneworientation. Now repeat the above method by varying 1 and 2 from 0-360o and choose the 1 and 2 2.7 PROGRAM ON PYTHON Variables used in the python code: C1 = Constant from Equation 1 C2 = Constant from Equation 2 C3 = Constant from Equation 3 C4 = Constant from Equation 4 F1 & t1: Imbalance and angle at plane 1 (nut plane of compressor wheel) F2 & t2: Imbalance and angle at plane 2 (hub plane of compressor wheel) F3 & t3: Imbalance and angle at plane 3 (hub plane of turbine wheel) F4 & t4: Imbalance and angle at plane 4 (nut plane of turbine wheel) Fa & ta: Balancing value and angle to be placed at plane A Fb & tb: Balancing value and angle to be placed at plane B 48 Fres: resultant of Fa and Fb Fminres: Least value of Fres T1min & t2min: optimum value of t1 and t2 for which Fres is the least. 2.7.1 FORMULAE USED IN PYTHON

C1=-1*(F2*L2*cos(t2)+F3*(L2+L3)*cos(t3)+F4*(L2+L3+L4)*cos(t4))/L2 C2=-1*(F2*L2*sin(t2)+F3*(L2+L3)*sin(t3)+F4*(L2+L3+L4)* sin(t4))/L2 tb = tan-1(C2/C1) Fb = C1/cos(tb) C3 = -1 * (F1 * cos(t1) + F2 * cos(t2) + F3 * cos(t3) + F4 * cos(t4) + Fb * cos(tb)) C4 = -1 * (F1 * sin(t1) + F2 * sin(t2) + F3 * sin(t3) + F4 * sin(t4) + Fb * sin (tb)) ta = tan-1(C4/C3) Fa = C3/cos(ta) Fres = ((Fa * Fa) + (Fb * Fb) + abs((2 * Fa * Fb * cos(ta-tb))))0.5 49 2.7.2 PROGRAM

from math import * import cmath F1 = input(" Enter F1 : ") t1= input(" Enter t1 : ") F2 = input(" Enter F2 : ") t2= input(" Enter t2 : ") F4 = input(" Enter F4 : ") t4= input(" Enter t4 : ") F3 = input(" Enter F3 : ") t3= input(" Enter t3 : ") t3 = 360-t3 t4 = 360-t4 Fa = 0 Fb = 0 50 L2 = 24.1 L3 = 55.1 L4 = 24.17 ta = 0 tb = 0 C1 = 0 C2 = 0 C3 = 0 C4 = 0 Fminres = 150 Fres = 0 i = 0 Famin = 0 Fbmin = 0 while(i < 360): 51 C1 = -1 * (F2 * L2 * cos(radians(t2)) + F3 * (L2 + L3) * cos(radians(t3)) + F4 * (L2+L3+L4) * cos(radians(t4)))/L2 C2 = -1 * (F2 * L2 * sin(radians(t2)) + F3 * (L2 + L3) * sin(radians(t3)) + F4 * (L2+L3+L4) * sin(radians(t4)))/L2

tb = atan(C2/C1) Fb = C1/cos(tb) C3 = -1 * (F1 * cos(radians(t1)) + F2 * cos(radians(t2)) + F3 * cos(radians(t3)) + F4 * cos(radians(t4)) + Fb * cos(radians(tb))) C4 = -1 * (F1 * sin(radians(t1)) + F2 * sin(radians(t2)) + F3 * sin(radians(t3)) + F4 * sin(radians(t4)) + Fb * sin (radians(tb))) ta = atan(C4/C3) Fa = C3/cos(ta) F = ((Fa * Fa) + (Fb * Fb) + abs((2 * Fa * Fb * cos(ta-tb)))) Fres = pow(f , 0.5) 52

if(Fres