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ORIGINAL ARTICLECAD-based simulation of the hobbing processfor the manufacturing of spur and helical gearsV. Dimitriou & A. AntoniadisReceived: 20 May 2007 /Accepted: 28 February 2008 / Published online: 29 April 2008# Springer-Verlag London Limited 2008Abstract Targeting an accurate and realistic simulation of thegear hobbing process, we present an effective and factualapproximation based on three-dimensional computer-aideddesign. Hobbing kinematics is directly applied in one geargap. Each generating position formulates a spatial surface pathwhich bounds its penetrating volume into the workpiece. Thethree-dimensional surface paths generated from the combina-tionoftherelativerotationsanddisplacementsofhobandworkgear are used to split the subjected volume, creating concur-rently the chip and the remaining work gear solid geometries.Thedevelopedsoftwareprogram HOB3D simulatesaccuratelythe manufacturing of spur and helical gears, exploiting themodeling and graphics capabilities of a commercial CADsoftware package. The resulting three-dimensional solidgeometrical data, chips and gears provide the whole geomet-ricalinformationneededforfurtherresearch,suchaspredictionof the cutting forces, tool stresses and wear development aswell as the optimization of the gear hobbing process.Keywords Gearhobbing.Manufacturingsimulation.CADmodeling1 IntroductionGear hobbing is a widely applied manufacturing process forthe construction of any external tooth form developeduniformly about a rotation center. Compared to conven-tional machining, such as turning and milling, the hobbingprocess is a sophisticated metal removal technology.Whileit is the most widely used process for the roughingof gears, its complexity and cost keep this technique poorlyknown. The kinematics principle of the process is based onthree relative motions between the workpiece and the hobtool. For the production of spurs or helical gears, theworkpiece rotates about its symmetry axis with a certainconstant angular velocity, synchronized with the relativegear hob rotation. The worktable or the hob may travelalong the work axis with the selected feed rate, dependingon the hobbing machine that is used.For the simulation of the gear hobbing process variantapproximations have been proposed for the development ofnumericalandanalytical models,aimingatthedeterminationof the undeformed chip geometry, cutting force componentsand tool wear development 1, 2. The industrial weight ofthese three simulation results is associated with theoptimization of the efficiency per unit cost of the gearhobbing process. The undeformed chip geometry is anessential parameter to determine the cutting force compo-nents, as well as to predefine the tool wear development,both of them important cost related data of hobbing process3. For the effective specification of the tolerances on hobdesign parameters and allowable alignment errors, Kim 4proposes a method of representing the geometry of a hobtooth profile in parametric form and of determining thesurface equation of a generated gear as a function of hobdesign parameters and generating motion specificationsthrough a mathematical model of the generation process.The simulation of meshing of face hobbed spiral bevel andhypoid gears and mathematic models of tooth surfacegenerations are presented by Fan, and a tooth contactanalysis program was developed 5.Int J Adv Manuf Technol (2009) 41:347357DOI 10.1007/s00170-008-1465-xV. DimitriouTechnological Educational Institute of Crete,Chania, Crete, GreeceA. Antoniadis (*)Technical University of Crete,Chania, Crete, Greecee-mail: antoniadisdpem.tuc.grThe research work presented in 611 provided thebasic knowledge for the numerical modeling of the gearhobbing process and later on newer approximations wereproposed based on a similar modeling strategy 1217.These approximation methods are proposed for the deter-mination of the hobbing process results, but the maincharacteristic of these methods is the reduction of the actualthree-dimensional process to planar models, primarily forsimplification reasons. The application of these formerapproximations is leading to planar results, without torepresent the exact solid geometry of the real chips andgears, with accuracy directly dependent from various inputparameters such as the number of the calculation planes.Furthermore, any post-processing of the extracted chip andgear planar geometries, e.g., finite element analysis,requires additional data processing which leads to supple-mentary interpolations of the two-dimensional results.Focusingontherealisticandaccuratesimulationofthegearhobbing process, without inevitable modeling insufficiencies,we present an approach for the simulation of manufacturingspur and helical gears in this research work. A softwareprogram called HOB3D, originally presented in 18, is usedfor the guidance of an existent commercial CAD system,exploiting its powerful modeling and graphics capabilities.HOB3D is built in terms of a computer code in Visual Basic,extending this capability to other cutting processes based onthe same cutting principle. The resulting solid models outputformats offer realistic parts, chips and work gears, easilymanaged for further individual research or as an input to anyother CAD, CAM or FEA commercial software systems.2 HOB3D modeling procedureThe rolling principle between the hob and the workpiecemakes the gear hobbing process different from conventionalmilling. As presentedin Fig. 1, the process problem isbasically prescribed from the geometrical characteristics ofthe gear to be cut, the hob that will be used and theinvolved kinematics between them.The geometry of a resulting gear is basically described bysix parameters: module (m), number of teeth (z2), outsidediameter (dg), helix angle (ha), gear width (W) and pressureangle (an). The correlation of these parameters automati-cally yields the module (m) of the hob tool, whereas othertool geometrical parameters as external diameter (dh),number of columns (ni), number of origins (z1), axial pitch() and helix angle (), are options to be chosen.As soon as the geometrical parameters of the twocombined parts are set, the kinematics chain has to beinitialized. The helix angle of the hob and the work gearprescribe the setting angle (s) between the parts and theway that their relative motions will take place. Threedistinct cutting motions are required: the tool rotation aboutits axis, the tool axial displacement and the workpiecerevolution about its axis. By these means, the direction ofthe axial feed (fa) prescribes two different hobbingstrategies: the climb (CL) and the up-cut (UC). In case ofhelical gears, two additional variations exist, the tool helixangle () compared to the helix of the gear (ha). If thedirection of the gear helix angle is identical to the hob helixangle the type of the process is set to equi-directional (ED),if not to counter-directional (CD) one.As presented in Fig. 2, after the initialization of the inputdata the solid geometry of the work gear is created in theCAD environment and one hob tooth rake face profile ismathematically and visually formed. At the same momentthe assembly of the effective cutting hob teeth (N) isdetermined and the kinematics of gear hobbing process isdirectly applied in one three-dimensional tooth gap of theFig. 1 Essential parameters of gear hobbing348 Int J Adv Manuf Technol (2009) 41:347357gear due to the axisymetric configuration of the problem.Moreover, a three-dimensional surface is formed for everygenerating position (e.g., successive teeth penetrations),combining the allocation of the hob and the work gear,following a calculated spatial spline as a rail. These three-dimensional surface paths are used to identify the unde-formed chip solid geometry, to split the subjected volumeand to create finally the chip and the remaining work gearsolid geometries.3 HOB3D simulation strategyIn this approach every rotation and displacement that istaking place between the hob and the work gear during thesimulationoftheprocessaredirectlytransferredtothehob.Aspresented in Fig. 3, the global coordinate system of the twoparts is fixed to the center of the upper base of theworkpiece, providing a steady reference system to thetravelling hob. The presented scheme of kinematics hasbeen adopted and in previous numerical approximationresearch works, mentioned at the introduction.Without any loss of generality, the hob is considered tohave a tooth numbered-named: Tooth 0. The identificationvector v0of Tooth 0 has its origin CHon the Yhaxis of thehob and its end at the middle of the rake face, forming amodule that equals to dh/2. This vector determines thedirection of the Zhaxis of the hob coordinate systemXhYhZhand is initially placed as an offset of the globalZ-axis, set at a vertical distance L1from the origin of theL1: Vertical distance hob - gearL2: Horizontal distance hob - gearworkgearZd EFGHOdfhaPitch circled =helixarc ofpitch circleda/22 df sin(h )amz2Gear hobbing kinematics 1: angleHob rotation 2: Gear anglerotationFig. 3 HOB3D simulation kinematics schemeCAD based Gear Hobbing Simulation Program HOB3D*dh: out diameter mmside Input DataHobGeometryWorkgearGeometryCuttingConditionsm : module mmz2: number of teethha: helix angle degm : module mmni: number of columnsz1: number of hob originsdg: outside diameter mmfa: axial feed mm/wrevt : depth of cut mmv : m/minCalculationsGeneration of Workgear solid geometryDetermination of N effective cutting teethGeneration of 3D-splinepath of hob toothMathematical description of the hob tooth rake faceworkgearpath of ahob toothGearhobbingprocess-Results3D surface geometrical descriptioof the hob tooth kinematicFor active teethComputation of theunderformed chipgeometry.toothprofilesplinetooth revolvingdirectionnsolid.cutting speedFig. 2 Flowchart of theprogram HOB3DInt J Adv Manuf Technol (2009) 41:347357 349fixed XYZ global system, determining the region where thecutting starts. Once the simulation parameters are settled,the work gear solid model is generated and the assemblyof the effective cutting hob teeth is enabled. The momentthat the gear hobbing simulation starts is considered as timezero. At this time (t=0) the planes YZ and YhZhare paralleland their horizontal distance, steady for the whole simula-tion period, is set to: L2 dh=2dg=2C0t. Thedistance L2practically determines the cutting depth, userdefined as an input parameter. To determine the settingangle s, the XhYhZhhob coordinate system is rotated aboutthe Xhaxis, so that the simulation process becomescompletely prescribed.Using the spatial vector v0, it is easy to compute theidentification vectors viof each of the N effective teethrelatively to v0, taking into account the hob geometricalCAD based Gear hobbing kinematicsYZXChip solidgeometryidentificationChip solid geometryWorkgearchip crosssectiondetail Ahmaxpathofahob toothchiptoothcuttingplanecutting planeAtoothrevolvingpositionrake facen1iCHin2idhhob revolvingdirection3d splinepathworkgearworkgearviFig. 4 Three-dimensional kinematics scheme in the CAD environmentHOB3D algorithmic diagramInitializationof input dataInitialization of programparametersMathematical determinationof Vector v0Generationof Work Gear cylinderMathematical determinationof spatial vectorsv , (C n ) and (C n )iH1i H2iApplication of forwardkinematics to vectorsv , (C n ) and (C n )iH1i H2iMathematical description of3D spline points,3D planesand corresponding profilesCreation of 3D spline points,3D planes and profiles,3D spline trajectoryGeneration of i-cutting tooth3D surface pathSubtraction of i-chipssolid geometryi=li=fNumerical implementation CAD system implementationEndNYi = i+1Determination of N effectivecutting teethFig. 5 Algorithmic diagram ofthe program HOB3D350 Int J Adv Manuf Technol (2009) 41:347357input parameters. The independent parameter 1counts therotational angle of hob tool about its axis Yhduring thecutting simulation. The parameter 2declares the rotationalangle of the hob tool about the work gear and fathe axial feedof the hob. 2and faare dependent from 1and their valuesare determined according to the values of 1angle (see alsothe upper part of Fig. 3). The forward kinematics of each ofthe N effective cutting hob teeth occur in one gear tooth space(gap). Hereby, after the determination of v0and consideringthat the identification vector of the first of the N cutting hobtooth vfis determined relatively to v0,the forward kinematicsare firstly applied to vfand sequentially to the followingvectors vi, until the last cutting hob tooth of the work cycle vl,simulating precisely the real manufacturing process.In case of simulating helical gears fabrication, adifferential angular amount is added on the rotating systemof the gear, in order to increase or decrease the angularvelocity of the rotating workpiece, ensuring the propermeshing of the hob-cutting and gear-cut angles. Dependingon the type of the hobbing process a new angular parameterd has to be inserted to the whole kinematical chain for theacceleration or deceleration of 2. The calculation of theparametrical value of d is taking place on the pitch circleas it is detailed schematically presented in the lower part ofFig. 3, constituting a section of significant importance.As illustrated at Fig. 4, the prescribed kinematics chainis used for the construction of a three-dimensional splinepath in the CAD environment. This spline path occurs fromthe interpolation of points generated from the vivector ofthe cutting hob teeth, that is properly transformed androtated by the help of the simulation parameters 1, 2and fa. Following the same tactic, the unit vectors (CHn1)iand (CHn2)i,described in the same Figure, are shifted androtated for the generation of a plane properly positionedinto the three-dimensional space, for every revolvingposition of the i-th cutting tooth.The profile of the cutting hob tooth is formed on thetwo-dimensional space that is created by each one of theseFig. 6 Maximum chip thickness on the revolving positions of every generating position of a full work cycleInt J Adv Manuf Technol (2009) 41:347357 351spatial planes, as presented in the middle of Fig. 4. By theproper lofting of the constructed open profiles following therail of the constructed spatial spline, a three-dimensionalopen surface is created in one gear-gap tooth space. Thissurface path represents the generating position of the i-thcutting tooth and bounds its penetrating volume into theworkpiece. With the aid of this surface path, the solidgeometry of a chip is identified for every generatingposition, using the Boolean operations and the graphicscapabilities of the CAD environment. The chip geometry isrestricted by the external volume of the instantly formedworking gear gap, bounded outside the created surface. Theidentified solid geometry is then subtracted from theworkpiece leading to the generation of the continuousthree-dimensional solid geometries of the chip and theremaining work gear. Because of the output form of thesolid resulting parts of the simulation process, any kind ofpost-processing is primitively enabled. As can be seen atthe lower right section of Fig. 4 in detail A, the maximumthickness of an extracted solid geometry of a chip at acertain revolving position is detected. All of the imagespresented in Fig. 4 were captured from the CAD environ-ment during a simulation performed from HOB3D. Thewords, phases and arrows were added afterwards with thehelp of commercial image processing software for the betterexplanation of the images.All of the previously mentioned simulation tasks arecontrolled by the developed program HOB3D that manipu-lates the modeling capabilities of the CAD system, forcingthe generation of the geometrical entities in it, collecting theoccurring geometrical data from it, and making all of theprogrammed numerical calculations. The continuous interac-tion of the CAD system with the numerical calculationsenvironment of the program is algorithmically described atFig. 5. After the insertion of the input data from the user theprogram parameters are initialized. The work gear cylindergeometry is generated and the number N of the effectivecutting teeth is computed. The spatial vector v0is mathe-matically determined. The cutting tooth number parameter iis set to f (first cutting tooth number) and the vector viandthe correspondent unit vectors (CHn1)iand (CHn2)i,essentialfor the construction of the spatial planes, are relative to v0mathematically determined. The forward kinematics of thehobbing procedure is applied to these vectors. The resultingnumerical data, prescribing the 3D spline points and the 3Dplanes, are used for the generation of the correspondingentities in theCAD environment.Thegenerated3Dpointsareinterpolated for the construction of the 3D spline in the CADenvironment. For each created spatial plane, the correspon-dent tooth profile is mathematically formed and sketched inthe CAD system. The profiles are lofted properly, followingthe 3D spline trajectory forming the 3D surface path of thei-cutting tooth. The solid geometry included in the spatialsurface is subtracted from the work gear and saved as thei-chip geometry (see the enlargement detail of Fig. 4). Thei-counter takes the value i+1 and the same procedure isrepeated until i became equal to l (last cutting tooth number).For the reduction of the computational effort and time,the overall rotation of the hob about its axis Yhis restrictedfrom 0 to 180 degrees (01180) for every generatingposition of each effective cutting hob tooth i. This way onlythe motions that affect the resulting solid geometries aretaking place, without any influence to the process suffi-ciency, during the entire simulation process. After thecompletion of the sequential construction of every spatialsurface and the subtraction of the chip solid geometries, thegear gap is generated