Mechanics & Industry 18, 405 (2017)© AFM, EDP Sciences 2017DOI: 10.1051/meca/2017028

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REGULAR ARTICLE

Determination of effective involute parameter limit in generationsimulation of gears manufactured by rack-type cuttersMahmut Cuneyt Fetvaci*

Istanbul University, Department of Mechanical Engineering, 34320 Istanbul, Turkey

� e-mail: f

Received: 4 October 2016 / Accepted: 30 May 2017

Abstract. This paper studies the computerized tooth profile generation of involute gears cut with rack-typecutters. Based on the theory of gearing the mathematical models of generating cutter with asymmetric involuteteeth and generated involute gears are given. Beveloid (conical involute) gears are considered for a generalizedtype of involute gearing for connecting parallel shafts. Effective limits of involute design parameters thatdetermine the actual tip circle radii of the generated gears are investigated. An approach based on contact pathof mating gears is proposed to eliminate further operations for standard value of tip circle radius. Computersimulation programs are developed to obtain graphs of generating tools and generated teeth surfaces.

Keywords: Beveloid gears / asymmetric involute profile / contact path / generation simulation / effectiveparameter limit

1 Introduction

Computer simulation of involute gear generating processhas been studied by various authors. A reliable numericalanalysis of gear tooth bending or contact stresses forsimulating the behavior of gears under working conditionsneeds precisely determined tooth profile of the machinedgear. Therefore, an accurate mathematical model of thegear tooth should be established.

In literature, mathematical modeling of involute gearsgenerated by rack- or pinion-type cutters have beenpresented [1–4]. Litvin proposed the Vector Method todevelop mathematical models for describing generatingand generated teeth surfaces of the various types ofgears [4]. This method can also consider functional orproduction-required tooth modifications. Based on Litvin’sVector Method, various authors studied the modeling,undercutting and stress analyzing of conventional andmodified gears [5–24].

Vector representation of the generating cutters includesthe topland, tip fillet and involute regions. According to thecutter geometry and an appropriate coordinate system, theposition vector of generating surface can be expressed. Thedesign parameter of the generating cutter surface whichdetermines the location of points on the related region hasupper and lower limits according to the geometry. Thenmatrix transformation, differential geometry and geartheory are applied to obtain gear tooth surface [4].

etvacic@istanbul.edu.tr

In the papers about mathematical modeling of thegenerating and generated teeth surfaces, the limits ofinvolute region design parameters are given for exactgeometry of the cutter but obtained gear teeth profile mayextend above the standard tip circle. This fact can beobserved in the papers about pinion-type genera-tion [6,15,22,23] and rack-type generation [5,7,12–14] forexample. There should be an effective upper limit of theinvolute parameter to calculate the exact tip radius of thegenerated gear. Path of contact for rack and gear may giveinformation about the effective parameter in case ofgenerating with a rack cutter. Thus, without a furtherCAD operation or an iterative method, the standard tipradius of the involute gear can be obtained.

This paper deals with the determination of the upperlimit of the rack cutter’s involute region that gives thestandard tip circle of the generated gear. The relationbetween length of path of recess and effective limit isinvestigated. A general model of involute gears (helicalbeveloid) is considered for verification and validation of theproposed approach. Beveloid gears can be used in trans-missions with intersecting or skewed axes or in trans-missions with parallel axes for backlash-free operation. Inprecision mechanisms gears should be designed to have nobacklash [25,26].

The rest of this paper is organized as follows. In the nextsections, the mathematical model of the generating cuttersis studied. The mathematical models: the locus of thegenerating-type cutter surfaces, the equation of meshingand the generated gear tooth surfaces, are given inSections 2 and 3. Converting to the two-dimensional

2 M.C. Fetvaci: Mechanics & Industry 18, 405 (2017)

modeling is presented in Section 4. The effective upperlimits of involute parameters for rack-type generation arepresented in Section 5. Computers graphs of generatingand generated surfaces are given in Section 6. Results arediscussed in Section 7. Finally, a conclusive summary ofthis study is given in Section 8.

Fig. 1. The normal section of the rack cutter.

Fig. 2. Formation of rack cutter surface for helical geargeneration.

Fig. 3. Formation of rack cutter surface for beveloid geargeneration.

2 Mathematical model of generating cutter

The normal section of a standard rack cutter with acomplete generating surface is shown in Figure 1. Regionsin the generating surface can be classified as the topland,tip fillet and involute regions. In the case of asymmetrictooth, these regions are designed according to correspond-ing sides pressure angles.

As shown in Figure 1, regions eg and fh of the rackcutter are straight lines that generate different workingsides of the involute gears. mn is the normal module of therack cutter with a asymmetric tooth profile. an1 and an2 arethe normal pressure angles. The position vector of regionseg and fh are represented in coordinate system Sn(Xn, Yn,Zn) as follows [13]:

Regn ¼

xegn

yegnzegn

8<:

9=; ¼

le cosan1

bc � le sinan1

0

8<:

9=; ð1Þ

Rfhn ¼

xfhn

yfhnzfhn

8<:

9=; ¼

lf cosan2

�bc þ lf sinan2

0

8<:

9=; ð2Þ

where le and lf are curvilinear parameters of rack cuttersurface which determine the location of points on workingsurface. le and lf are limited by�ha/cosan1� le� ht/cosan1and �ha/cosan2� lf� ht/cosan2 for left- and right-side ofthe rack cutter respectively.

To obtain a generalizedmathematical model of involutegears, additional coordinate systems should be used andappropriate coordinate transformations between thesesystems should be applied. In the first step, the rackcutter surface for the helical gear generation is simulated.The normal section of the rack cutter, attached to thecoordinate system Sn, is translated along the line OpOn asshown in Figure 2. Therefore, l ¼ jOpOn j is also one of thedesign parameters of the rack cutter surface, and b is thehelix angle (RH) of the generated helical gear. The planeYpZp can be regarded as the pitch plane of the rackcutter [7]. The coordinate transformation matrix fromcoordinate system Sn to Sp can be written as follows:

Mpn ¼1 0 0 00 cosb �sinb �l sinb0 sinb cos b l cos b0 0 0 1

2664

3775 ð3Þ

In the second step, the taper hobbing process issimulated. The generating cutter is inclined to the gear axisby the cone angle d as shown in Figure 3. As a result, therack cutter is represented in the plane axode coordinate

Fig. 4. Coordinate relationship between rack cutter andgenerated gear.

M.C. Fetvaci: Mechanics & Industry 18, 405 (2017) 3

system Sc(Xc, Yc, Zc) [7]. The coordinate transformationmatrix from coordinate system Sp to Sc can be written asfollows:

Mcp ¼cos d 0 sin d 00 1 0 0

�sin d 0 cos d 00 0 0 1

2664

3775 ð4Þ

The position vector of generating cutter represented incoordinate system Sc(Xc, Yc, Zc)can be written as follows:

Ric ¼ ½Mcn�Ri

n ¼ ½Mcp�½Mpn�Rin ð5Þ

where superscript i represents regions ac, bd, ce, df , eg andfh. We can write down the equations of generating surfacein open form as follows:

xic ¼ xi

n cos dþ yinsin d sinbþ l sin d cos byic ¼ yin cosb� l sinbzic ¼ �xi

n sin dþ yin cos d sinbþ l cos d cosbð6Þ

Tangent plane or tangent vectors determine theexistence of regular surfaces. The tangent plane of thegenerating cutter surface is composed of two vectors:∂Rc/∂lj and ∂Rc/∂l (partial derivatives). Surface normal isgiven by the cross product of the partial derivatives.According to differential geometry and gear theory, normalvectors of regions ac ∼ fh of the rack cutter surface can berepresented in coordinate system Sc by the followingequation [4].

Nic ¼

∂Ric

∂lj� ∂Ri

c

∂l¼

i j k∂xi

c

∂lj∂yic∂lj

∂zic∂lj

∂xic

∂l∂yic∂l

∂zic∂l

����������

����������ð7Þ

The unit normal vector of the rack cutter surface isrepresented by Equation (8) [4].

nic ¼

∂Ric

∂lj� ∂Ri

c∂l

∂Ric

∂lj� ∂Ri

c∂l

������

ði ¼ ac; . . . ; fhÞðj ¼ a; . . . ; fÞ ð8Þ

3 Mathematical model of generated gear

Figure 4 depicts the coordinate relationship between therack cutter and the generated gear. Coordinate systemSh(Xh, Yh, Zh) represents the fixed coordinate system.Sc(Xc, Yc, Zc)is the plane axode coordinate systemattached to the rack cutter and coordinate systemS1(X1,, Y1, Z1) is attached to the generated gear [7]. Whilethe gear blank rotates through an angle f1, the rack cuttertranslates the linear displacement S= rp1f1 along Yc axis.Therefore, the coordinate transformation matrix from Sc to

S1 can be described as [4]:

½M1c� ¼cosf1 �sinf1 0 rp1ðcosf1 þ f1 sinf1Þsinf1 cosf1 0 rp1ðsinf1 � f1 cosf1Þ0 0 1 00 0 0 1

2664

3775 ð9Þ

Fundamental law of gearing states that the commonnormal to the transverse section of the cutter and the geartooth surface must pass through the instantaneous centerof rotation (pitch point). Therefore, the equation meshingcan be represented using coordinate system Sc(Xc, Yc, Zc)as follows [4]:

Xic � xi

c

nixc

¼ Y ic � yicniyc

¼ Zic � zicnizc

ð10Þ

where Xic, Y

ic and Zi

c are the coordinates of a point on theinstantaneous axis of gear rotation I–I represented incoordinate system Sc; xi

c, yic and zic are the surfacecoordinates of the rack cutter; symbols ni

xc, niyc and ni

zcsymbolize the components of the common unit normalrepresented in coordinate system Sc.

According to the theory of gearing [4], the mathemati-cal model of the generated gear tooth surface is acombination of the meshing equation and the locus(family) of the rack cutter surfaces. The locus of thecutter surface, expressed in coordinate system S1, can bedetermined as follows [4]:

Ri1 ¼ ½M1c�Ri

c; ði ¼ ac; . . . ; fhÞ ð11ÞAs a result, the mathematical model of the gear tooth

surfaces can be obtained by simultaneously consideringEquations (11) and (10).

Helical beveloid gear is the generalized model of theinvolute gears mounted on parallel axes. Straight conicalinvolute gears, helical gears and spur gears can berepresented by the equations by letting the helix angle,the cone angle, or both be equal to zero [10].

Fig. 5. Meshing diagram of a rack cutter and pinion.

Fig. 6. Helical gear tooth profile.

Fig. 7. Straight beveloid gear tooth profile.

4 M.C. Fetvaci: Mechanics & Industry 18, 405 (2017)

4 Tooth profile analysis on the plane ofrotation

In beveloid gears profile shift varies along the facewidth. Asa result the tooth geometry changes across the facewidth.Moreover helix angle also alters the geometry. Contactpath, zero topland and undercutting should be investigatedin several transverse planes. To express the profile of theimaginary rack cutter on the plane of rotation YcZc, thethird item of Equation (6) can be adopted as follows:

l ¼ ðzic þ xin sin d� yin cos d sinbÞ=ðcos d cosbÞ ð12Þ

Considering the first two items of Equation (6) togetherwith Equation (12), the profile of the two-dimensional rackcutter canbeobtainedand expressed on theplane of rotationby treating zc as a constant. An important macro-geometric property of helical beveloid gear is the differenttransverse pressure angles on left- and right sides that resultsasymmetric teeth profiles [7,10,25,27]. Transverse pressureangles are formulated in Equations (13) and (14).

at1 ¼ tan�1ð�tanb sin dþ tanan1 secb cos dÞ ð13Þ

at2 ¼ tan�1ðtanb sin dþ tanan2 secb cos dÞ ð14Þ

5 Contact path and determination ofeffective parameter limit

Figure 5 depicts the line of action in meshing of the rackcutter and the helical gear being generated at a transversesection. The upper end of path of contact is point T wherethe tip circle of the pinion intersects the line of action.PT isthe length of path of recess. From the gear geometry, PT

can be calculated as PT ¼ �rb tanat þffiffiffiffiffiffiffiffiffiffiffiffiffiffir2t � r2b

qwhere rb

is the base circle radius of the generated gear and at is thetransverse pressure angle. Therefore, the effective upperlimit of the involute parameter is calculated asheff ¼ PT sinat. As a result, when calculating the generat-ed profile we should use �ha/cosan1� le� heff/cosan1instead of �ha/cosan1� le� ht/cosan1.

The equation of effective limit can also be extended forbeveloid gears. We can write heff ¼ ðPT sinat � zc tan dÞcos d for a transverse section zc along the face width of abeveloid gear. The pressure angles on plane of rotation canbe calculated as tanat=tanan cos d for straight beveloidand tanat=∓ tanb sin d+tanan cos d secb for helical bev-eloid tooth where d is the cone angle and b is the helix angle.

6 Computer applications

Based on the mathematical models given in previoussections, computer programs can be developed to obtainthe coordinates of the generating cutters and generatedgears. In this study the program is written with GW-BASIC. Output of files is evaluated with GRAPHER toobtain computer graphs.

Figure 6 displays a transverse section of a helical gearwith symmetric involute teeth. In this example, designparameters are selected as: normal module mn=5mm,normal pressure angle an1= 20°, helix angle b=15° andnumber of teeth T=13. The proposed limit of the designparameter gives the exact length of involute profile that islimited by standard tip circle radius of the workpiece.

Figure 7 displays a transverse section of a straightbeveloid gear. In this example, design parameters areselected as: normal module mn=5mm, normal pressure

Fig. 8. Helical beveloid gear tooth profile.

Fig. 9. Helical beveloid gear generated by an asymmetric cutter.

M.C. Fetvaci: Mechanics & Industry 18, 405 (2017) 5

angle an1= 20°, cone angle d=15°, number of teeth T=25and with a cross-section constraint zc=� 10mm. Anegative profile shift coefficient appears near the heel soundercutting is observed. Transverse pressure angle iscalculated as at=19.37°.

Figure 8 displays a transverse section of a helicalbeveloid gear. In this example, design parameters areselected as: normal module mn=5mm, normal pressureangle an1= 20°, helix angle b=15°, cone angle d=15°,number of teeth T=25 and with a cross-section constraintzc=0. Asymmetric tooth profile on the plane of rotation isobserved. Transverse pressure angles on the left- and rightsides are at1= 16.41602° and at2= 23.42808°, respectively.Selected parameters in this example produces the left sideprofile be on the undercutting limit. As the conventionalinvolute parameter limit is used on the left side, it givesextended involute profile above the standard tip circle.

Figure 9 displays a transverse section of a helicalbeveloid gear cut by a rack cutter with asymmetric teeth.Also relative positions of the generating cutter with respectto gear blank during generation process are illustrated.Design parameters are selected as: normal module mn=5mm, left side normal pressure angle an1= 26.68735°, rightside normal pressure angle an2= 20°, helix angle b=15°,cone angle d=15°, number of teethT=25 and with a cross-section constraint zc=0. Transverse pressure angles arecalculated as at1= 23.42808° and at2= 23.42808° for left-and right sides, respectively. Therefore a symmetric profileon the plane of rotation can be obtained.

7 Results

Generating-type cutters are widely used in mass produc-tion of gears. In literature most of studies aboutmathematical modeling and analysis of gears are basedon Litvin’s vector approach [4]. This approach begins withthe parametric representation of tool surface in vectorform. As the tool surface is composed of different regions,the intervals of curvilinear parameters that determine theposition of a point in the region should maintain position,

tangent, or curvature continuity of adjacent regions. Thisstudy presents an effective upper limit of involuteparameter that gives the exact length of generated involuteprofile. As stated in Section 5, from the contact path ofrack and gear mechanism radial distance from pitch pointto the end point of contact can be defined as effectiveupper limit of involute parameter. This limit depends onthe transverse section through the face width. Proposedformula is derived for generalized model of involute gearsmounted on parallel axis.

In Section 6, illustrative examples are given. InFigures 6–8, the proposed upper limit of involuteparameter is used for generated profile on the right side.On the left side, generated involute profile ends beyond theactual tip circle radius of the gear blank. The extended partshould be trimmed.

Beveloid gears can be considered as involute gears witha linear variable profile shift along the face width. As theprofile shift changes from a negative value to a positive

Fig. 10. Solid model of a straight beveloid gear.

Fig. 11. 3-D printed straight beveloid gears.

6 M.C. Fetvaci: Mechanics & Industry 18, 405 (2017)

value, undercutting and zero topland can be observed inseveral transverse sections of generated gear. Moreover,helix angle causes asymmetric profile on the plane ofrotation. Figure 8 displays transverse section of a helicalbeveloid gear that has different pressure angle on left andright sides. As the transverse pressure angle decreases therisk of undercutting increases. To prevent undercutting inhelical beveloid gears rack cutter with asymmetric involuteteeth can be used. Figure 9 displays helical beveloid geargenerated by a rack cutter with an asymmetric tooth flank.

The author used the locus equation of the cutter andobtained illustrations displaying relative positions of thecutter during generation by manipulating rolling parame-ter as �p/4�f1�p/4 in the developed code. An exampleis illustrated in Figure 9. It has been reported in literaturethat the relative positions provide a basis to determine thechip geometry [28].

Like as in manufacturing cylindrical involute gears,hobbing tools are used for beveloid gearing. Conventionalgear hobbing machines are arranged with the methodsproposed byMitome [27]. They are the inclined-work-arbortaper hobbing method and the table-sliding taper hobbingmethod. Also generating grinding wheels can be used.Hobbing or grinding is simulated by an imaginary rackcutter in computer applications.

The coordinates of the points on the gear tooth can alsobe imported into a CAD software to obtain 3-D models ofgears for 3D Printing. Then it is exported to 3D PrintingMachines where physical model of gears can be obtained.

Figure 10 displays a solid model of a straight beveloid gear.In this study, prototypes of straight beveloid gears are alsomanufactured by a 3-D printer that runs on fuseddeposition modeling (FDM) technology. Figure 11 displaysstraight beveloid gears made from polylactide (PLA).

8 Conclusions

Generating-type of cutters is widely used in massproduction of involute gears. In this study, mathematicalmodels of rack cutters and beveloid gears are given, takinginto account the asymmetric involute tooth profile. In theopen literature obtained gear teeth involute profile mayextend above the standard tip circle when using upper limitof involute parameter for the profile of cutter.Major findingin this paper is the concept of analytically derived effectiveupper limit of involute parameter. This approach elimi-nates CAD operations or iterative methods needed tocorrect the involute profile exceeding the limit that occurswhen using design parameters in the conventional range.Examples for different cases are presented to validate theproposed approach.

Undercutting and zero topland should be checked inbeveloid gears due to varying profile shift from toe to heel.For helical beveloid gears cutters with asymmetric involuteteeth can be used to prevent the undercutting on therelated side.

The relative positions of the generating cutter can alsobe illustrated. Chip geometry can be described from therelative positions. The description of chip geometry can beused predicting the tool life and cutting forces formanufacturing in further studies.

In addition to hobbing and grinding, the additivemanufacturing methods (3D printing) can also be used forproducing beveloid gears. The mathematical model givenin this study can also be used to obtain 3-D models in aCAD software. 3D CAD models can be obtained fromseveral transverse sections by using extrude/sweepoperations.

Beveloid gears have found wide application in marinetransmission and automotive industry. Beveloid gearing isa cheaper means of limiting backlash in precisionmechanisms such as industrial robots. In further studiesbased on beveloid gears, mechanical aspects such as frictionand wear can be considered. A wear-modified modelproposed by Brauer and Andersson can be extended tohelical beveloid gears [26]. Modifications such as tipchamfering, parabolic tooth and root free-cut can beapplied to the mathematical model of generating cutter.These aspects are beyond to the scope of this study.

Nomenclature

bc

tool setting of the rack cutter ha tool setting of the rack cutter ht tool setting of the rack cutter mn normal module mt transverse module lj curvilinear coordinates of rack cutter where sub-

script j= a, b, c, d, e, f

M.C. Fetvaci: Mechanics & Industry 18, 405 (2017) 7

[M1c]

coordinate transformation matrix from coordinatesystem Sc to S1

nc

unit normal vector of rack cutter surface Ri

c

position vector rack cutter surface for geargeneration

Rin

position vector of rack cutter normal section where

superscript i represents regions ac, bd, ce, df , egand fh of the rack cutter surface

Ri1

the family of the rack cutter surfaces

rb

radius of base circle of generated gear rp1 radius of pitch circle of generated gear rt radius of tip circle of generated gear Sc rack cutter coordinate system Sf fixed coordinate system S1 gear blank coordinate system T number of teeth of generated gear an1 normal pressure angle of the rack cutter (left side) an2 normal pressure angle of the rack cutter (right side) at transverse pressure angle b helix angle d cone angle f1 motion parameter r1 left tip fillet radius of the cutter r2 right tip fillet radius of the cutter

References

[1] E. Buckingham, Analytical mechanics of gears, McGraw-Hill, New York, 1988

[2] C. Salamoun, M. Suchy, Computation of helical or spur gearfillets, Mech. Mach. Theory 8 (1973) 305–323

[3] J.R. Colbourne, The geometry of involute gears, Springer-Verlag, New York, 1987

[4] F.L. Litvin, Gear geometry and applied theory, PrenticeHall, New Jersey, 1994

[5] C.-B. Tsay, Helical gears with involute shaped teeth:geometry, computer simulation, tooth contact analysisand stress analysis, J. Mech. Des. 110 (1988) 482–491

[6] S.-L. Chang, C.-B. Tsay, Computerized tooth profile gener-ation and undercut analysis of noncircular gears manufac-tured with shaper cutters, J. Mech. Des. 120 (1998) 92–99

[7] C.-C. Liu, C.-B. Tsay, Tooth undercutting of beveloid gears,J. Mech. Des. 123 (2001) 569–576

[8] C.-C. Liu, C.-B. Tsay, Mathematical models and contactsimulations of concave beveloid gears, J. Mech. Des. 124(2002) 753–760

[9] G. Figliolini, J. Angeles, The synthesis of elliptical gears gen-erated by shaper-cutters, J. Mech. Des. 125 (2003) 793–801

[10] J. Brauer, A general finite element model of involute gears,Finite Elem. Anal. Des. 40 (2004) 1857–1872

[11] K.J. Huang, H.W. Su, Approaches to parametric elementconstructions and dynamic analyses of spur/helical gearsincluding modifications and undercutting, Finite Elem.Anal. Des. 46 (2010) 1106–1113

[12] C.-F. Chen, C.-B. Tsay, Tooth profile design for themanufacture of helical gear sets with small numbers ofteeth, Int. J. Mach. Tool Manuf. 45 (2005) 1531–1541

[13] S.-C. Yang, Mathematical model of a helical gear withasymmetric involute teeth and its analysis, Int. J. Adv.Manuf. Technol. 26 (2005) 448–456

[14] S.-C. Yang, Study on an internal gear with asymmetricinvolute teeth, Mech. Mach. Theory 42 (2006) 977–994

[15] M.-F. Tsay, Z.-H. Fong, Novel profile modification method-ology for moulded face-gear drives, J. Mech. Eng. Sci. 221(2007) 715–725

[16] S.-Z. Wu, S.-J. Tsai, Contact stress analysis of skew conicalinvolute gear drives in approximate line contact, Mech.Mach. Theory 44 (2009) 1658–1676

[17] C. Fetvaci, Computer simulation of helical gears withasymmetric involute teeth, J. Fac. Eng. Archit. Gazi Univ.25 (2010) 441–447

[18] C. Fetvaci, Generation simulation of involute spur gearsmachined by pinion-type shaper cutters, Stroj. Vestn. – J.Mech. Eng. 56 (2010) 644–652

[19] W.-L. Chen, C.-B. Tsay, Mathematical model and toothsurfaces of recess action worm gears with double-depth teeth,Mech. Mach. Theory 46 (2011) 1840–1853

[20] Y.-C. Chen, C.-B. Tsay, Stress analysis of a helical gear setwith localized bearing contact, Finite Elem. Anal. Des. 38(2002) 707–723

[21] J.-H. Kuang, W.-L. Chen, Determination of tip parametersfor the protuberance preshaving cutters, Mech. Mach.Theory 31 (1996) 839–849

[22] C.-B. Tsay, W.-Y. Liu, Y.-C. Chen, Spur gear generationby shaper cutters, J. Mater. Process. Technol. 104 (2000)271–279

[23] C. Fetvaci, Computer simulation of involute tooth genera-tion, in: M. Gokcek (Ed.), Mechanical engineering, InTech,Rijeka, Croatia, 2012, pp. 503–526

[24] C. Fetvaci, Computer simulation of helical gears generatedby rack-type cutters, Arab. J. Sci. Eng. 36 (2011) 1321–1332

[25] J. Börner, K. Humm, F. Joachim, Development of conicalinvolute gears (beveloids) for vehicle transmission, GearTechnol. 22(6) (2005) 28–35

[26] J. Brauer, S. Andersson, Simulation of wear in gears withflank interference � a mixed FE and analytical approach,Wear 254 (2003) 1216–1232

[27] K. Mitome, Conical involute gear, part 1: design andproduction system, Bull. JSME 26(212) (1983) 299–305

[28] K.-D. Bouzakis, S. Kombogiannis, A. Antoniadis, N. Vidakis,Gear hobbing cutting process simulation and tool wearprediction models, J. Manuf. Sci. Eng. 124 (2002) 42–51

Cite this article as: M.C. Fetvaci, Determination of effective involute parameter limit in generation simulation of gearsmanufactured by rack-type cutters, Mechanics & Industry 18, 405 (2017)