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ORIGINAL ARTICLE CAD-based simulation of the hobbing process for the manufacturing of spur and helical gears V. Dimitriou & A. Antoniadis Received: 20 May 2007 /Accepted: 28 February 2008 / Published online: 29 April 2008 # Springer-Verlag London Limited 2008 Abstract Targeting an accurate and realistic simulation of the gear hobbing process, we present an effective and factual approximation based on three-dimensional computer-aided design. Hobbing kinematics is directly applied in one gear gap. Each generating position formulates a spatial surface path which bounds its penetrating volume into the workpiece. The three-dimensional surface paths generated from the combina- tionoftherelativerotationsanddisplacementsofhobandwork gear are used to split the subjected volume, creating concur- rently the chip and the remaining work gear solid geometries. Thedevelopedsoftwareprogram HOB3D simulatesaccurately the manufacturing of spur and helical gears, exploiting the modeling and graphics capabilities of a commercial CAD software package. The resulting three-dimensional solid geometrical data, chips and gears provide the whole geomet- ricalinformationneededforfurtherresearch,suchasprediction of the cutting forces, tool stresses and wear development as well as the optimization of the gear hobbing process. Keywords Gearhobbing . Manufacturingsimulation . CADmodeling 1 Introduction Gear hobbing is a widely applied manufacturing process for the construction of any external tooth form developed uniformly about a rotation center. Compared to conven- tional machining, such as turning and milling, the hobbing process is a sophisticated metal removal technology. Whileit is the most widely used process for the roughing of gears, its complexity and cost keep this technique poorly known. The kinematics principle of the process is based on three relative motions between the workpiece and the hob tool. For the production of spurs or helical gears, the workpiece rotates about its symmetry axis with a certain constant angular velocity, synchronized with the relative gear hob rotation. The worktable or the hob may travel along the work axis with the selected feed rate, depending on the hobbing machine that is used. For the simulation of the gear hobbing process variant approximations have been proposed for the development of numericalandanalytical models,aimingatthedetermination of the undeformed chip geometry, cutting force components and tool wear development 1, 2. The industrial weight of these three simulation results is associated with the optimization of the efficiency per unit cost of the gear hobbing process. The undeformed chip geometry is an essential 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 process 3. For the effective specification of the tolerances on hob design parameters and allowable alignment errors, Kim 4 proposes a method of representing the geometry of a hob tooth profile in parametric form and of determining the surface equation of a generated gear as a function of hob design parameters and generating motion specifications through a mathematical model of the generation process. The simulation of meshing of face hobbed spiral bevel and hypoid gears and mathematic models of tooth surface generations are presented by Fan, and a tooth contact analysis program was developed 5. Int J Adv Manuf Technol (2009) 41:347357 DOI 10.1007/s00170-008-1465-x V. Dimitriou Technological Educational Institute of Crete, Chania, Crete, Greece A. Antoniadis (*) Technical University of Crete, Chania, Crete, Greece e-mail: antoniadisdpem.tuc.gr The research work presented in 611 provided the basic knowledge for the numerical modeling of the gear hobbing process and later on newer approximations were proposed based on a similar modeling strategy 1217. These approximation methods are proposed for the deter- mination of the hobbing process results, but the main characteristic of these methods is the reduction of the actual three-dimensional process to planar models, primarily for simplification reasons. The application of these former approximations is leading to planar results, without to represent the exact solid geometry of the real chips and gears, with accuracy directly dependent from various input parameters such as the number of the calculation planes. Furthermore, any post-processing of the extracted chip and gear planar geometries, e.g., finite element analysis, requires additional data processing which leads to supple- mentary interpolations of the two-dimensional results. Focusingontherealisticandaccuratesimulationofthegear hobbing process, without inevitable modeling insufficiencies, we present an approach for the simulation of manufacturing spur and helical gears in this research work. A software program called HOB3D, originally presented in 18, is used for 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 on the same cutting principle. The resulting solid models output formats offer realistic parts, chips and work gears, easily managed for further individual research or as an input to any other CAD, CAM or FEA commercial software systems. 2 HOB3D modeling procedure The rolling principle between the hob and the workpiece makes the gear hobbing process different from conventional milling. As presentedin Fig. 1, the process problem is basically prescribed from the geometrical characteristics of the gear to be cut, the hob that will be used and the involved kinematics between them. The geometry of a resulting gear is basically described by six parameters: module (m), number of teeth (z 2 ), outside diameter (d g ), helix angle (h a ), gear width (W) and pressure angle (a n ). The correlation of these parameters automati- cally yields the module (m) of the hob tool, whereas other tool geometrical parameters as external diameter (d h ), number of columns (n i ), number of origins (z 1 ), axial pitch () and helix angle (), are options to be chosen. As soon as the geometrical parameters of the two combined parts are set, the kinematics chain has to be initialized. The helix angle of the hob and the work gear prescribe the setting angle ( s ) between the parts and the way that their relative motions will take place. Three distinct cutting motions are required: the tool rotation about its axis, the tool axial displacement and the workpiece revolution about its axis. By these means, the direction of the axial feed (f a ) prescribes two different hobbing strategies: the climb (CL) and the up-cut (UC). In case of helical gears, two additional variations exist, the tool helix angle () compared to the helix of the gear (h a ). If the direction of the gear helix angle is identical to the hob helix angle 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 input data the solid geometry of the work gear is created in the CAD environment and one hob tooth rake face profile is mathematically and visually formed. At the same moment the assembly of the effective cutting hob teeth (N) is determined and the kinematics of gear hobbing process is directly applied in one three-dimensional tooth gap of the Fig. 1 Essential parameters of gear hobbing 348 Int J Adv Manuf Technol (2009) 41:347357 gear due to the axisymetric configuration of the problem. Moreover, a three-dimensional surface is formed for every generating 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 volume and to create finally the chip and the remaining work gear solid geometries. 3 HOB3D simulation strategy In this approach every rotation and displacement that is taking place between the hob and the work gear during the simulationoftheprocessaredirectlytransferredtothehob.As presented in Fig. 3, the global coordinate system of the two parts is fixed to the center of the upper base of the workpiece, providing a steady reference system to the travelling hob. The presented scheme of kinematics has been adopted and in previous numerical approximation research works, mentioned at the introduction. Without any loss of generality, the hob is considered to have a tooth numbered-named: Tooth 0. The identification vector v 0 of Tooth 0 has its origin C H on the Y h axis of the hob and its end at the middle of the rake face, forming a module that equals to d h /2. This vector determines the direction of the Z h axis of the hob coordinate system X h Y h Z h and is initially placed as an offset of the global Z-axis, set at a vertical distance L 1 from the origin of the L 1 : Vertical distance hob - gear L 2 : Horizontal distance hob - gear wor kgear Z d E F G H O df h a Pitch circle d = helix arc of pitch circle d a /2 2 df sin(h ) a mz 2 Gear hobbing kinematics 1 : angleHob rotation 2 : Gear anglerotation Fig. 3 HOB3D simulation kinematics scheme CAD based Gear Hobbing Simulation Program HOB3D * d h : out diameter mmside Input Data Hob Geometry W orkgear Geometry Cutting Conditions m : module mm z 2 : number of teeth h a : helix angle deg m : module mm n i : number of columns z 1 : number of hob origins d g : outside diameter mm f a : axial feed mm/wrev t : depth of cut mm v : m/min Calculations Generation of Workgear solid geometry Determination of N effective cutting teeth Generation of 3D-spline path of hob tooth Mathematical description of the hob tooth rake face workgear path of a hob tooth Gear hobbing process - Results 3D surface geometrical descriptio of the hob tooth kinematic For active teeth Computation of the underformed chip geometry . tooth profile spline tooth revolving direction n solid . . cutting speed Fig. 2 Flowchart of the program HOB3D Int J Adv Manuf Technol (2009) 41:347357 349 fixed XYZ global system, determining the region where the cutting starts. Once the simulation parameters are settled, the work gear solid model is generated and the assembly of the effective cutting hob teeth is enabled. The moment that the gear hobbing simulation starts is considered as time zero. At this time (t=0) the planes YZ and Y h Z h are parallel and their horizontal distance, steady for the whole simula- tion period, is set to: L2 dh=2dg=2C0t. The distance L 2 practically determines the cutting depth, user defined as an input parameter. To determine the setting angle s , the X h Y h Z h hob coordinate system is rotated about the X h axis, so that the simulation process becomes completely prescribed. Using the spatial vector v 0 , it is easy to compute the identification vectors v i of each of the N effective teeth relatively to v 0 , taking into account the hob geometrical CAD based Gear hobbing kinematics Y Z X Chip solid geometry identification Chip solid geometry Work gear chip cross section detail A h max pathofa hob tooth chip tooth cutting p lane cutting plane A tooth revolving position rake face n 1i C Hi n 2i d h hob revolving direction 3d spline path work gear work gear v i Fig. 4 Three-dimensional kinematics scheme in the CAD environment HOB3D algorithmic diagram Initialization of input data Initialization of program parameters Mathematical determination of Vector v 0 Generation of Work Gear cylinder Mathematical determination of spatial vectors v , (C n ) and (C n ) iH1i H2i Application of forward kinematics to vectors v , (C n ) and (C n ) iH1i H2i Mathematical description of 3D spline points,3D planes and corresponding profiles Creation of 3D spline points, 3D planes and profiles, 3D spline trajectory Generation of i-cutting tooth 3D surface path Subtraction of i-chips solid geometry i=l i=f Numerical implementation CAD system implementation End NY i = i+1 Determination of N effective cutting teeth Fig. 5 Algorithmic diagram of the program HOB3D 350 Int J Adv Manuf Technol (2009) 41:347357 input parameters. The independent parameter 1 counts the rotational angle of hob tool about its axis Y h during the cutting simulation. The parameter 2 declares the rotational angle of the hob tool about the work gear and f a the axial feed of the hob. 2 and f a are dependent from 1 and their values are determined according to the values of 1 angle (see also the upper part of Fig. 3). The forward kinematics of each of the N effective cutting hob teeth occur in one gear tooth space (gap). Hereby, after the determination of v 0 and considering that the identification vector of the first of the N cutting hob tooth v f is determined relatively to v 0, the forward kinematics are firstly applied to v f and sequentially to the following vectors v i , until the last cutting hob tooth of the work cycle v l, simulating precisely the real manufacturing process. In case of simulating helical gears fabrication, a differential angular amount is added on the rotating system of the gear, in order to increase or decrease the angular velocity of the rotating workpiece, ensuring the proper meshing of the hob-cutting and gear-cut angles. Depending on the type of the hobbing process a new angular parameter d has to be inserted to the whole kinematical chain for the acceleration or deceleration of 2 . The calculation of the parametrical value of d is taking place on the pitch circle as it is detailed schematically presented in the lower part of Fig. 3, constituting a section of significant importance. As illustrated at Fig. 4, the prescribed kinematics chain is used for the construction of a three-dimensional spline path in the CAD environment. This spline path occurs from the interpolation of points generated from the v i vector of the cutting hob teeth, that is properly transformed and rotated by the help of the simulation parameters 1 , 2 and f a . Following the same tactic, the unit vectors (C H n 1 ) i and (C H n 2 ) i, described in the same Figure, are shifted and rotated for the generation of a plane properly positioned into the three-dimensional space, for every revolving position of the i-th cutting tooth. The profile of the cutting hob tooth is formed on the two-dimensional space that is created by each one of these Fig. 6 Maximum chip thickness on the revolving positions of every generating position of a full work cycle Int J Adv Manuf Technol (2009) 41:347357 351 spatial planes, as presented in the middle of Fig. 4. By the proper lofting of the constructed open profiles following the rail of the constructed spatial spline, a three-dimensional open surface is created in one gear-gap tooth space. This surface path represents the generating position of the i-th cutting tooth and bounds its penetrating volume into the workpiece. With the aid of this surface path, the solid geometry of a chip is identified for every generating position, using the Boolean operations and the graphics capabilities of the CAD environment. The chip geometry is restricted by the external volume of the instantly formed working gear gap, bounded outside the created surface. The identified solid geometry is then subtracted from the workpiece leading to the generation of the continuous three-dimensional solid geometries of the chip and the remaining work gear. Because of the output form of the solid resulting parts of the simulation process, any kind of post-processing is primitively enabled. As can be seen at the lower right section of Fig. 4 in detail A, the maximum thickness of an extracted solid geometry of a chip at a certain revolving position is detected. All of the images presented in Fig. 4 were captured from the CAD environ- ment during a simulation performed from HOB3D. The words, phases and arrows were added afterwards with the help of commercial image processing software for the better explanation of the images. All of the previously mentioned simulation tasks are controlled by the developed program HOB3D that manipu- lates the modeling capabilities of the CAD system, forcing the generation of the geometrical entities in it, collecting the occurring geometrical data from it, and making all of the programmed numerical calculations. The continuous interac- tion of the CAD system with the numerical calculations environment of the program is algorithmically described at Fig. 5. After the insertion of the input data from the user the program parameters are initialized. The work gear cylinder geometry is generated and the number N of the effective cutting teeth is computed. The spatial vector v 0 is mathe- matically determined. The cutting tooth number parameter i is set to f (first cutting tooth number) and the vector v i and the correspondent unit vectors (C H n 1 ) i and (C H n 2 ) i, essential for the construction of the spatial planes, are relative to v 0 mathematically determined. The forward kinematics of the hobbing procedure is applied to these vectors. The resulting numerical data, prescribing the 3D spline points and the 3D planes, are used for the generation of the corresponding entities in theCAD environment.Thegenerated3Dpointsare interpolated for the construction of the 3D spline in the CAD environment. For each created spatial plane, the correspon- dent tooth profile is mathematically formed and sketched in the CAD system. The profiles are lofted properly, following the 3D spline trajectory forming the 3D surface path of the i-cutting tooth. The solid geometry included in the spatial surface is subtracted from the work gear and saved as the i-chip geometry (see the enlargement detail of Fig. 4). The i-counter takes the value i+1 and the same procedure is repeated 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 Y h is restricted from 0 to 180 degrees (0 1 180) for every generating position of each effective cutting hob tooth i. This way only the motions that affect the resulting solid geometries are taking place, without any influence to the process suffi- ciency, during the entire simulation process. After the completion of the sequential construction of every spatial surface and the subtraction of the chip solid geometries, the gear gap is generated, formed by the collective work of every generating position. md znhzt f=5mm, =125mm, =1, =12, =0, =30, =11mm, =4mm/wrev h1ia2 a Chip Thickness Development for Up-Cut and Climb Hobbing Generating Positions Chip Thickness 0.6 mm 0.4 0.3 0.2 0.1 0 -18 14-2-10 6 -18 14 -2-10 6 Up-Cut hobbing Climb hobbing Generating Positions Fig. 7 Maximum chip thickness of the generating positions of a full work cycle for UC and CL gear hobbing cases 352 Int J Adv Manuf Technol (2009) 41:347357 4 HOB3D simulation results The developed program HOB3D is used for the simulation of manufacturing of spur and helical gears, and its post- processing code is used for the determination of the solid chip thickness development. The proposed program was verified and validated in a previous research work 18, by the usage of the resulting three-dimensional solid geome- tries of the gear gaps generated by HOB3D. These gap profiles were compared to the standard ones introduced by Petri 19 and DIN 3972 20 and the calculated mean error was less than 10 m for the working depth of produced gears. Such negligible deviations satisfy the computational accuracy expectations, and verify the sufficiency of the developed code. 4.1 Simulation of the manufacturing process of spur gears HOB3D is used for the simulation of manufacturing of a spur gear. The input data of the UC case processed and the output chip solid geometries of every generating position are presented in Fig. 6. The cutting direction