advanced computer aided design simulation of gear …. journals/12.20.pdfgear hobbing, as any...

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Dimitriou VasilisAdjunct Professor

Vidakis NectariosAssociate Professor

Antoniadis AristomenisProfessor

Technological Educational Institute of Crete,Romanou 3,

Chania 73133, Greece

Advanced Computer Aided DesignSimulation of Gear Hobbing byMeans of Three-DimensionalKinematics ModelingGear hobbing, as any cutting process based on the rolling principle, is a signally multi-parametric and complicated gear fabrication method. Although a variety of simulatingmethods has been proposed, each of them somehow reduces the actual three-dimensional(3D) process to planar models, primarily for simplification reasons. The paper describesan effective and factual simulation of gear hobbing, based on virtual kinematics of solidmodels representing the cutting tool and the work gear. The selected approach, in con-trast to former modeling efforts, is primitively realistic, since the produced gear andchips geometry are normal results of successive penetrations and material removal ofcutting teeth into a solid cutting piece. The algorithm has been developed and embeddedin a commercial CAD environment, by exploiting its modeling and graphics capabilities.To generate the produced chip and gear volumes, the hobbing kinematics is directlyapplied in one 3D gear gap. The cutting surface of each generating position (successivecutting teeth) formulates a 3D spatial surface, which bounds its penetrating volume intothe workpiece. This surface is produced combining the relative rotations and displace-ments of the two engaged parts (hob and work gear). Such 3D surface “paths” are usedto split the subjected volume, creating concurrently the chip and the remaining work gearsolid geometries. This algorithm is supported by a universal and modular code as well asby a user friendly graphical interface, for pre- and postprocessing user interactions. Theresulting 3D data allow the effective utilization for further research such as prediction ofthe cutting forces course, tool stresses, and wear development as well as the optimizationof the gear hobbing process. �DOI: 10.1115/1.2738947�

Keywords: gear hobbing, simulation, 3D-Modeling, CAD

IntroductionIn most of the demanding torque transmission systems, the key

omponents are premium well designed and properly fabricatedears. The gear hobbing process is widely applied for the con-truction of any external tooth form developed uniformly about aotation center. The kinematics principle of the process is based onhree relative motions between the workpiece and the hob tool. Toroduce spur or helical gears, the workpiece rotates about its sym-etry axis with certain constant angular velocity, synchronizedith the relative gear hob rotation. Depending on the hobbingachine used, the worktable or the hob may travel along the work

xis with the selected feed rate.Many advances have been made in the development of numeri-

al and analytical models for the simulation of the gear hobbingrocess, aiming to the determination of the undeformed chip ge-metry, cutting force components, and tool wear development1,2�. The industrial weight of these three simulation data is asso-iated with the optimization of the efficiency per unit cost of theear hobbing process. The undeformed chip geometry is an essen-ial parameter to determine the cutting force components, as wells to predefine the tool wear development, both of them importantost related data of the hobbing process �3�.

Gear hobbing is nowadays analytically understood, while datauch as chip geometry, cutting force components, mechanicaltresses, and wear performance may be determined with the aid of

Contributed by the Manufacturing Engineering Division. Manuscript receivedugust 18, 2006; final manuscript received November 3, 2006. Review conducted by
ong-Woo Cho.
ournal of Manufacturing Science and EngineeringCopyright © 20

aded 04 Jan 2008 to 147.27.11.190. Redistribution subject to ASME

consistent software codes �4–15�. Despite the research and indus-trial merit of such programs in the past, a variety of restrictionsand limitations arise, owing to the simplified modeling strategiesthat they imply. The basic principle, on which the FRS code wasbuilt in the 1970s, has been the determination of common areasbetween one cutting hob tooth profile and the work gear. Hereby,a two-dimensional �2D� space of a plane determined by the com-bination of work gear and hob’s kinematics was selected as thecalculation region. The accuracy of the chip dimensions approxi-mated by FRS code depends on various input parameters, such asnumber of calculation planes, the discrete interpolation and pro-jection done on the cutting section planes, and the calculated dis-continuity of chip thickness. Therefore it is clearly understood thatthe resulting plane chip geometry does not represent exactly thesolid geometry of a real chip. As a matter of these facts, transi-tional thickness variations may not be traced, if they belong to anintermediate calculation plane of the chosen ones. On the otherhand, any further postprocessing of the calculated chip and geargeometries requires additional data processing which leads tosupplementary interpolations of the 2D extracted results.

To overcome such modeling insufficiencies, the present re-search work introduces a new approach, which exploits powerfulmodeling capacities of up-to-date CAD software environments.Hereby, a software module called HOB3D has been developedfrom scratch and embedded to an existent commercial CAD sys-tem. The developed algorithm is built in terms of a computercode, which is supported by a user friendly graphical interface.HOB3D provides the extend ability to other cutting processesbased on the same cutting principle. The models output formats
provided from the “parent” CAD system �.ipt, .sat, .iges, .dxf etc�
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license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

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ffer realistic solid parts, chips, and work gear, which can beasily managed for further investigations individually, or as annput to other commercial CAD, CAM, or FEA environments.

Gear Hobbing Features and Simulation StrategyGear hobbing differs significantly from conventional milling,

ince its kinematics is governed by the rolling principle betweenhe hob and the workpiece. To optimize computationally this cut-ing scheme, the implementation of each geometric feature, kine-

atics and cutting conditions to the developed algorithm, is criti-al.

The process problem is basically prescribed from the geometri-al characteristics of the gear to be cut, the hob that will be used,nd the involved kinematics between them. As presented at Fig. 1,he final geometry of a resulting gear is basically described by sixarameters: module �m�, number of teeth �z2�, outside diameterdg�, helix angle �ha�, gear width �W�, and pressure angle �an�.he correlation of these parameters automatically yields the mod-le �m� of the hob tool, whereas other tool geometrical parametersexternal diameter �dh�, number of columns �ni�, number of ori-ins �z1�, axial pitch �� and helix angle ��, are options to behosen. As soon as the geometrical parameters of the two com-ined parts are set, the kinematics chain has to be initialized. Theelix angle of the hob and the work gear prescribe the settingngle ��s� between the parts and the way that their relative mo-ions shall take place. Three distinct cutting motions are required:he tool rotation about its axis, the tool axial displacement, and theorkpiece revolution about its axis. By these means, the directionf the axial feed �fa� prescribes two different hobbing strategies:he climb �CL� and the up-cut �UC�. In case of helical gears, twodditional variations exist, the tool helix angle �� compared tohe helix of the gear �ha�. If the direction of the gear helix angle isdentical to the hob helix angle the type of the process is set toquidirectional �ED�, if not to counterdirectional �CD� one.

Focusing on the generation of highly precise geometric modelsf undeformed solid chips, as well as on the elimination of anyoss of data occurring from the processing of the geometricalquations involved with the gear hobbing process, the CAD basedrogram HOB3D has been introduced. The essential input dataoncern the determination of the hob and work gear geometriesnd the cutting parameters that take place for the completion ofhe simulation process. When the values of the input parametersre set, the work gear solid geometry is created in the CAD envi-onment and one hob tooth rake face profile is mathematically and

Fig. 1 Intrinsic principles and

isually formed. At the same moment the assembly of the effec-

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tive cutting hob teeth �N� is determined, as presented in Fig. 2.The kinematics of gear hobbing process is directly applied in onethree-dimensional �3D� tooth gap of the gear, considering the axi-symetric configuration. Moreover, a 3D surface is formed for ev-ery generating position �e.g., successive teeth penetrations�, com-bining the allocation of the two involved parts �hob and workgear� and following a calculated spatial spline as a rail. These 3Dsurface “paths” are used to identify the undeformed chip solidgeometry, to split the subjected volume, and to create finally thechip and the remaining work gear continuous solid geometries.

3 HOB3D Simulation ProcessIn the developed simulation every rotation and displacement

between the hob and the work gear, for modeling enhancementreasons, are directly transferred to the hob. By this implementa-tion a global coordinate system that is placed on the center of theupper gear base is always fixed and provides a steady referencepoint to monitor the traveling hob.

3.1 Mathematical Formulation. Without any loss of gener-ality, the hob is considered to have one and only tooth �numbered-named “tooth 0”�. The identification vector �v0� of “tooth 0” hasits origin �CH� on the Yh axis of the hob and its end at the middleof the rake face, forming a module that equals to �v0 � =dh /2. Theidentification vector determines the direction of the Zh axis of thehob coordination system �XhYhZh� and is initially placed as anoffset of the global Z axis, set at a vertical distance L1 from theorigin of the XYZ global system. The right part of Fig. 3 illustrateshow the parameter L1 determines the region where the cuttingstarts. Once the simulation parameters are settled, the work gearsolid model is generated and the assembly of the effective cuttinghob teeth is enabled. The moment that the gear hobbing simula-tion starts is considered as time zero. At this time �t=0� the planesYZ and YhZh are parallel and their horizontal distance, steady forthe whole simulation period, is set to value L2=dh /2+dg /2− t.Distance L2 practically determines the cutting depth, user definedas an input parameter. To determine the setting angle ��s�, theXhYhZh hob coordinate system is rotated about the Xh axis, so thatthe simulation process becomes completely prescribed. Using thespatial vector v0, it is easy to compute the identification vectors viof effective teeth N, with −k� i�n and N=n+k+1, k ,n�Z+ tak-ing into account the hob geometrical input parameters, relative tov0.

In addition, the independent parameter �1 counts the rotational

tures of gear hobbing process
fea
angle of hob tool about its axis Yh during the cutting simulation.

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ince �2 declares the rotational angle of hob tool about the workear and fa the axial feed of hob, their values are dependent on �1nd can be determined according to this angle values �see also theeft part of Fig. 3�.

As mentioned before, the forward kinematics of each of the Nffective cutting hob teeth occur in one gear tooth space �gap�.ereby, after the determination of v0 �tooth 0� and considering

hat v−k is relative to v0 the forward kinematics are first applied to−k and afterwards to the following vectors vi, simulating pre-isely the real manufacturing process. For the determination of thenvolved hob-gear kinematics, the following transformational ma-rices Rx,a �rotation about X axis with � angle�, Ry,� �rotationbout Y axis with � angle�, Rz,� �rotation about Z axis with �ngle�, are used:

Fig. 2 The flow chart of the develope

Fig. 3 Gear hobbing simula

ournal of Manufacturing Science and Engineering


Rx,� = �1 0 0

0 cos�� − sin��0 sin�� cos��

� ,

Ry,� = � cos�� 0 sin��0 1 0

− sin�� 0 cos�� ,

Rz,� = �cos�� − sin�� 0

sin�� cos�� 0

0 0 1� �1�

OB3D gear hobbing simulation code

tion kinematics scheme

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In case of simulating helical gears fabrication, a differentialngular amount is put on the rotating system of the gear, in ordero increase or decrease the angular velocity of the rotating work-iece, ensuring the proper mechanical functionality �meshing� ofhe hob-cutting and gear-cut angles. Depending on the hobbingype process a new angular parameter �d�� has to be inserted tohe whole kinematical chain for the acceleration or deceleration of2. The calculation of the parametrical value of d� is taking placen the pitch circle as presented in Fig. 4.

3.2 Exploitation of Modeling and Graphical Features ofhe CAD Environment. Using the kinematics scheme describedn the previous paragraph, a spatial spline path is created insidehe CAD system, by interpolating the points generated from theransformation of each vi vector of each cutting hob tooth relativeo the hob axis Yh �using �1� and the global Z axis �using �2�, asell as the displacement fa �see Fig. 5�a��. Thereby, the unit vec-

ors �CHn1�i and �CHn2�i, described in Fig. 5�b�, are rotated andhifted for the creation of a plane properly positioned into the 3Dpace, for every revolving position of a certain cutting tooth i. Therofile of the cutting hob tooth is formed on the 2D space that isreated on its rake plane. This process takes place for every re-olving position, as graphically illustrated in Fig. 5�c�.

By shifting these open cutting hob tooth profiles and using thepatial spline path as a rail, a 3D surface path is constructed in one

Fig. 4 Determination of d� angle in

Fig. 5 Generation principle of hob too

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gear tooth space. This path represents the generating position of“tooth i”. With the aid of this surface path, the solid geometry ofa chip is identified for every generating position, using the bool-ean operations and the graphics capabilities of the CAD environ-ment, as presented in Fig. 6�a�. The chip geometry is restricted bythe external volume of the instantly formed working gear gap,bounded outside the created surface. The identified solid geometryis then subtracted from the workpiece, as it is presented in Fig.6�b�, generating the continuous 3D geometries of the chip and theremaining work gear.

Bearing in mind the solid outputs form of the simulation pro-cess, their direct postprocessing is primitively enabled. Fig. 6�b�illustrates the chip solid geometry that is produced during a cer-tain revolving position, whereas the detected maximum chipthickness is presented in detail A.

To save computational time and resources, in every generatingposition of each effective cutting tooth i, the overall rotation of thehob about its symmetry axis Yh is restricted �0° ��1�180° �. Inthis way, during the entire simulation process, only motions thataffect the resulting solid geometries are taking place, without anyinfluence to the process sufficiency. After the completion of onework cycle, i.e., the termination of every spatial surface path andthe subtraction of the chip solid geometries, the produced gear gapis finally generated. Fig. 7 illustrates a cross section �detail A� of

se of helical gear cutting simulation
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he generated gear gap, formed by the collective work of everyenerating position. As can be seen in Fig. 7�c�, the remainingear solid geometry holds complete geometrical information, bothor the removed and the remaining “material.”

Generation and Storage of Solid Chip EntitiesThe main data-entry window of the HOB3D graphical interface

s presented in Fig. 8. The specific data entry includes the sectionsf hob and work gear data, as well as selected cutting conditionsnd the user defined postdata. An up-cut �UC� case for the simu-ation of a spur gear fabrication is used, in order to demonstratehe program functionality.

The input data described at the previous sections are manuallynserted in their corresponding input boxes. The highlighted inputoxes, also recognized by an asterisk, refer to modeling param-ters, which are automatically calculated and proposed byOB3D algorithm, offering simultaneously the possibility to beser defined. The picture at the middle bottom part of the inputata window visualizes the strategy selected for each simulatingcenario. As can be observed at the output section, the programses a working folder, set by the user, to save in proper formatshe chosen outputs. The simulation parameters can also be savednd reopened by selecting appropriate fields, while a simulationeport comprising the user selected output results may be option-lly created after the completion of the simulation process.

Figure 9 illustrates the resulted solid geometry of the chip pro-uced from the generating position gp=−4 of the above-presentedest case. The normally continuous chip solid geometry, for visu-lization purposes, has been sliced in each of the 91 assignedevolving positions belonging to the kinematics transformations of

Fig. 6 Solid chip geom

Fig. 7 Visual gear gap g



hob tooth “−4.” The intersecting planes of the revolving positionsalong with the chip geometry are illustrated in the left part of Fig.9, while the sliced solid geometry chip cross sections, on the samerevolving positions, are presented in the middle part. It is evidentthat every revolving position generates the same profile on twosuccessive sliced parts of the chip. This is the reason why eachsliced part has the name of two successive generating positions.The maximum thicknesses of each cross section �hmax� are iden-tified, as shown at the three inserted details their development isdescribed in the right diagram of Fig. 9.

The output chip solid geometries at 15 characteristic generatingpositions of two different test cases, UC and CL, produced by theactivation of the HOB3D code, are presented in the left and rightparts of Fig. 10, respectively. Except for the direction of the axialfeed fa all other input data are identical, for both examined cases.Examining each of the resulting chips, it is obvious that so manyof the extreme geometrical changes of the chip shapes are suffi-ciently determined, even if the generated chip solid is parted frommore than one domain.

The resulting 3D solid geometries for both gear gaps, generatedfor the same presented cases, are ideal to verify and validate theproposed algorithm. Since the gear gaps are generated by subtract-ing the solid volume of each chip from the initially cylindricalwork gear solid geometry, every cross section of them, passingparallel to the XY plane, offers a polyline which corresponds tothe calculated profile �see Fig. 7 right part�. These two producedgap profiles are compared to the standard ones introduced by Petri�16� and DIN 3972 �17�, as presented in the left part of Fig. 11. Todetermine the calculated profile deviation, the dichotomy betweenthem and the nominal profile is calculated and the occurring error

y identification process
etr
eneration by HOB3D

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s measured at its direction, as presented in detail A of Fig. 11. Thealculated error of each test case is plotted to the graphs presentedt the right part of Fig. 11, illustrating a mean error less than0 �m for the working depth of produced gears, for the UC andL cases, respectively. Such negligible deviations satisfy the com-utational accuracy expectations, and verify the sufficiency of theeveloped code.

ConclusionsThe paper illustrated an advanced and validated simulationethod for gear hobbing process, based on a commercial CAD

nvironment. In contrast to former research attempts, in theresent investigations, the kinematics of gear hobbing is directly

Fig. 8 HOB3D

Fig. 9 Solid chip formation at specific

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applied in one tooth 3D space by the construction of spatial sur-face paths, for every generating position. The kinematics involvesthe rotations and displacements of the two rolling parts �hob andwork gear� for every possible manufacturing case of gear hobbingprocess. These 3D surface “paths” are used to divide the subjectedvolume and directly create the chip and the remaining work gearcontinuous solid geometries. The algorithm is supported by acomputer code with the aid of a user friendly graphical interface.Considering the quality and the format of the resulting solid ge-ometries, it is implicit that further manipulation and postprocess-ing of them is effortlessly achievable, without further postprocess-ing. The confirmation of the validity and accuracy of the proposedmethod has been accomplished by comparing the produced gear

a-entry window

generating position in gear hobbing



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ap profile with theoretical ones, expressing a faultless agreementetween them. The results of the present work hold significantndustrial and research interest, including the accurate predictionf dynamic behavior and tool wear development in gear hobbingrocedure.

omenclatureCL � climb hobbingUC � up-cut hobbingED � equidirectional hobbing

Fig. 11 Algorithm verification as with the aidnal theoretical one

Fig. 10 Typical solid chips

CD � counterdirectional hobbing



GP � generating positionRP � revolving positionfa � axial feed �mm/wrev�t � cutting depth �mm�v � cutting speed �m/min�m � work gear and hob tool module �mm�ni � number of hob columnsz1 � number of hob originsz2 � number of work gear teethdg � external work gear diameter �mm�ha � gear helix angle �°�

the calculated gear gap profile and the nomi-

up-cut and climb hobbing

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eferences�1� Klocke, F., and Klein, A., 2006, “Tool Life and Productivity Improvement

Through Cutting Parameter Setting and Tool Design in Dry High-Speed BevelGear Tooth Cutting,” Gear Technol., May/June, pp. 40–48.

�2� Rech, J., 2006, “Influence of Cutting Edge Preparation on the Wear Resistancein High Speed Dry Gear Hobbing,” Wear, 261�5–6�, pp. 505–512.

�3� Rech, J., Djouadi, M. A., and Picot, J., 2001, “Wear Resistance of Coatings inHigh Speed Gear Hobbing,” Wear, 250, pp. 45–53.

�4� Sulzer, G., 1974, “Leistungssteigerung bei der Zylinderradherstellung durchgenaue Erfassung der Zerspankinematik,” dissertation, TH Aachen, Aachen,Germany.

�5� Gutman, P., 1988, “Zerspankraftberechnung beim Waelzfraesen,” dissertation,TH Aachen, Aachen, Germany.

�6� Venohr, G., 1985, “Beitrag zum Einsatz von hartmetall Werkzeugen beimWaelzfraesen,” dissertation, TH Aachen, Aachen, Germany.

�7� Joppa, K., 1977, “Leistungssteigerung beim Waelzfraesen mit Schnellarbeitss-tahl durch Analyse, Beurteilung und Beinflussung des Zerspanprozesses,” dis-sertation, TH Aachen, Aachen, Germany.

�8� Tondorf, J., 1978, “Erhoehung der Fertigungsgenauigkeit beim Waelzfraesendurch Systematische Vermeidung von Aufbauschneiden,” dissertation, THAachen, Aachen, Germany.

�9� Antoniadis, A., 1988, “Determination of the Impact Tool Stresses During Gear
Hobbing and Determination of Cutting Forces During Hobbing of Hardened
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Gears,” dissertation, Aristoteles University of Thessaloniki, Thessaloniki,Greece.

�10� Antoniadis, A., Vidakis, N., and Bilalis, N., 2002, “Failure Fracture Investiga-tion of Cemented Carbide Tools Used in Gear Hobbing—Part I: FEM Model-ing of Fly Hobbing and Computational Interpretation of Experimental Re-sults,” ASME J. Manuf. Sci. Eng., 124�4�, pp. 784–791.

�11� Antoniadis, A., Vidakis, N., and Bilalis, N., 2002, “Failure Fracture Investiga-tion of Cemented Carbide Tools Used in Gear Hobbing—Part II: The Effect ofCutting Parameters on the Level of Tool Stresses—A Quantitive ParametricAnalysis,” ASME J. Manuf. Sci. Eng., 124�4�, pp. 792–798.

�12� Sinkevicius, V., 1999, “Simulation of Gear Hobbing Geometrical Size,” Kau-nas University of Technology Journal “Mechanika,” 5�20�, pp. 34–39.

�13� Sinkevicius, V., 2001, “Simulation of Gear Hobbing Forces,” Kaunas Univer-sity of Technology Journal “Mechanika,” 2�28�, pp. 58–63.

�14� Komori, M., Sumi, M., and Kubo, A., 2004, “Method of Preventing CuttingEdge Failure of Hob due to Chip Crush,” JSME Int. J., Ser. C, 47�4�, pp.1140–1148.

�15� Komori, M., Sumi, M., and Kubo, A., 2004, “Simulation of Hobbing forAnalysis of Cutting Edge Failure due to Chip Crush,” Gear Technol., Sept./Oct., pp. 64–69.

�16� Petri, H., 1975, “Zahnhuß–Analyse bei außenverzahnten Evolventenstirnraed-ern : Teil III Berechnung,” Antriebstechnik, 14�5�, pp. 289–297.

�17� DIN 3972, 1992, Bezugsprofile von Verzahnwerkzeugen fuer Evolventen-
Verzahnungen nach DIN 867, Taschenbuch 106, Beuth Verlag.


advanced computer aided design simulation of gear …. journals/12.20.pdfgear hobbing, as any...

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