Fast modeling of conjugate gear tooth profiles using discrete presentation by involute segments

Mechanism and Machine Theory 42 (2007) 751–762

Mechanism and Machine Theory www.elsevier.com/locate/mechmt

Fast modeling of conjugate gear tooth profiles using discrete presentation by involute segments Vasilios Spitas, Theodore Costopoulos *, Christos Spitas Laboratory of Machine Elements, National Technical University of Athens, Iroon Politechniou 9, 15780 Athens, Greece Received 31 March 2005; accepted 16 May 2006 Available online 2 August 2006

Abstract This paper introduces the method of the discretization of the gear tooth flank in involute segments for the determination of conjugate gear tooth profiles. Instead of following a point-to-point analytical approach to the problem of determining the path of contact and the geometry of the generating rack and the mating wheel, the actual tooth flank is considered to be composed of infinitesimal local involutes and therefore a closed solution can be achieved. Due to its simplicity, the method is faster than the standard theory of gearing and this is particularly useful in problems requiring iterative calculations of the tooth geometry such as gear optimization. The method is implemented on modified involute as well as loboid gears used in lobe pumps and it is verified against existing theories. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Involute; Local-involute; Gear modeling; Computational method

1. Introduction The calculation of the conjugate profiles of gear teeth has always been an important and challenging problem for the modeling of the kinematical and the statical or dynamical behavior of geared mechanisms. Many theories have been suggested to solve this problem [1–5] including the widely used ‘‘Theory of Gearing’’ by Litvin [6–10]. Despite the effort that has been made, issues related to the modeling of the generating surface of the gear tooth flank or of the families of conjugate gear tooth flanks are still open today and many works based on the theory of gearing are being published aiming to address specific tooth geometries [11–14]. The basic characteristic of these studies is that they usually offer implicit solutions requiring elaborate numerical methods to solve the complex differential equations and therefore they are characterized by increased computational time requirements. This paper introduces a new concept in gear flank modeling by using the principle of discretization of the flank in infinitesimal involute segments. According to this principle, the gear tooth flank is discretized not in

*

Corresponding author. E-mail address: [email protected] (T. Costopoulos).

0094-114X/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2006.05.007

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V. Spitas et al. / Mechanism and Machine Theory 42 (2007) 751–762

straight linear segments but in involute segments instead. Therefore, it is possible to obtain a closed nonimplicit solution for the calculation of the conjugate profile at only a fraction of the time needed by the standard conventional methods. Its concept is based on the fact that since the path of contact of an involute gear pair is a straight line then the path of contact of any gear pair can be discretized into linear segments and hence calculate the corresponding infinitesimal involutes on the meshing tooth flanks. For comparison, the standard theory of gearing is based on the fact that a given tooth flank is discretized into linear segments and hence the path of contact and the geometry of the mating gear are calculated. However, the involute has a compact and simple non-implicit mathematical representation which is independent of the coordinate system [1,2,4], used and the location of the rolling point, whereas the theory of gearing yields more complicated solutions which are coordinate system specific. One of the most important contributions of the proposed technique is that it offers a substantial reduction of the computational time needed for calculating conjugate tooth profiles [15] since it does not have to solve any equations as in the standard theory of gearing. This improvement is particularly useful in gear optimization algorithms where the computation of the conjugate profile must be repeated a number of times at each iterative step. This method can be easily expanded to cover not only geared mechanisms but also chain drives, cams, etc. and is very easy to integrate in a CAD software [16]. 2. Theory of infinitesimal involute segment modeling 2.1. The concept of the local involute Let us consider a two-dimensional gear tooth profile (not an involute in the general case), where the geometry of the working tooth profile yG = G(x) is given (Fig. 1). The problem of conjugate tooth geometry is to determine the path of contact yP = P(x), the tooth profile of the generating rack yR = R(x) and the tooth profile of the conjugate gear yW = W(x). Let us also consider a random point G(xG, yG) on the working gear tooth profile yG = G(x) and the Oxy Cartesian coordinate system, where O is the center of rotation of the gear and Oy coincides with the tooth centerline. At point G the tooth profile is approximated with an involute segment with corresponding pressure angle equal to aoG, such that the

Fig. 1. The concept of the local involute.


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local involute has the same tangent with the actual profile at that point. The normal (e1) to the profile at point G has an inclination equal to ðdG=dxÞ1 and its defining equation is ðxG ;y G Þ

1 y1 ¼ þB ðdG=dxÞðxG ;y G Þ

ð1Þ

where B is a constant. Point G(xG, yG) belongs to (e1) and therefore should verify Eq. (1) xG xG yG ¼ B and B ¼ GðxG Þ þ ðdG=dxÞðxG ;y G Þ ðdG=dxÞðxG ;y G Þ By substituting B, Eq. (1) becomes xG x y ¼ GðxG Þ þ ðdG=dxÞðxG ;y G Þ

ð2Þ

From the center O of the gear, line (e2) normal to (e1) is drawn, so that it intersects with it at point A(xA, yA). Since (e2) is parallel to the tangent to the profile at point G, passes through the center of the gear O(0, 0) and its inclination is ðdG=dxÞðxG ;y G Þ , its governing equation is dG y¼x ð3Þ dx ðxG ;y G Þ Point A(xA, yA) lies at the intersection of (e1) and (e2), therefore it should verify Eqs. (2) and (3) simultaneously dG xG xA xA ¼ GðxG Þ þ dx ðxG ;y G Þ ðdG=dxÞðxG ;y G Þ from where we finally obtain xG GðxG Þ þ ðdG=dxÞðxG ;y G Þ xA ¼ 1 ðdG=dxÞðxG ;y G Þ þ ðdG=dxÞðxG ;y G Þ

and

y A ¼ xA

dG dx ðxG ;y G Þ

ð4Þ

The radius of the local base circle rgG corresponding to the local involute at point G is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dG 2 2 rgG ¼ xA þ y A ¼ xA 1 þ dx ðxG ;y G Þ or equally rgG ¼

xG þ GðxG ÞðdG=dxÞðxG ;y G Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 dG 1þ dx ðxG ;y G Þ

ð5Þ

The local pressure angle corresponding to the local involute at G is 1 r gG aoG ¼ cos ð6Þ ro where ro is the rolling circle of the gear. At this point it should be stressed that in the general case ro can be variable i.e. gears with transmission errors or non-circular gears where the rolling point is not fixed or have a constant value, which is the usual case of most gears used in power transmissions. In the first case ro = ro(G) and in the latter and most common case it assumes the values N1 N2 and r02 ¼ r01 ð7Þ r01 ¼ a12 N1 þ N2 N1 where a12 is the center distance between gears 1 and 2 and N1, N2 are the number of teeth of gears 1 and 2 respectively. It is also worth noticing that the expressions (5) and (6) are invariant, meaning that they are

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independent of the coordinate system used. Therefore the above equations will remain the same if the coordinate system is such that axis Oy passes through the point where the rolling circle intersects with the tooth flank. 2.2. Determination of the conjugate tooth geometry and the path of contact Let us consider the tooth flank yG = G(xG) with respect to the coordinate system (O, xG, yG) illustrated in Fig. 2 and denote P as the point on the path of contact at which G comes in contact with the corresponding point of the mating rack/gear. C is the rolling point, which in the following analysis is assumed to be fixed and defined by Eq. (7), i.e. constant gear ratio is considered. Since the local involute segment corresponds to a straight segment on the path of contact, the corresponding point P of the path of contact lies on the intersection of circle (O, rG) and line CP with an inclination equal to aoG. The line normal to the tooth flank at the point of contact should pass through the rolling point C in order for the Law of Gearing [1–5] to be observed and constant transmission function is achieved. Point P lies on the intersection of circle (O, rG) and line CP, therefore it should satisfy both equations x2 þ y 2 ¼ r2G

and

y ¼ x tan aoG þ ro

The above equations yield 2sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 2 2 2 rG rgG rgG 5 4 1 ; xP ¼ rgG ro ro ro 3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi ro r r r G gG gG 5 1 y P ¼ rgG 14 rgG ro ro ro

ð8Þ

ð9Þ

The radii rgG and rG are given by Eqs. (5) and (8), respectively: The conjugate rack profile yR = R(x) and the tooth profile of the conjugate gear (wheel) yW = W(x) are determined from the calculated path of contact according to the well-known calculation methods [1–5].

Fig. 2. Path of contact and local pressure angle.


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Fig. 3. Proof of the method.

A strict proof of the proposed method is cited below: Let G be a random point on the gear tooth flank illustrated in Fig. 3, C be the rolling point of the gear pair defined by Eq. (7) and O the center of the gear. Consider also the line normal to the flank at point G that intersects with the rolling circle ro at point a. According to the local involute concept this line is tangent to the local base circle rgG. When the gear is revolved about its center so that point a coincides with the rolling point C, then point G will coincide with P on the path of contact according to Relaux [1]. b (angles with perpendicular sides) meaning that the normal to the flank at point P of b ¼ aoG ¼ P Cx It is A0 OC the path of contact passes through the rolling point thus satisfying the Law of Gearing. It is also true that when the gear is revolved so that point a coincides with C, the pressure angle that corresponds to the local r involute at G having rgG as its base circle, will be given by the equation aoG ¼ cos1 rgGo . As expected, the proposed theory should be reduced to an identity if the gear tooth flank is an involute. Indeed, the defining equation of the involute curve has the following parametric form [1–5]: x ¼ rg ðsin x x cos xÞ;

y ¼ rg ðcos x þ x sin xÞ and

dy 1 ¼ dx tan x

Substituting the above expressions in Eq. (6) yields for the local base circle 2 3 cos x þ x sin x 6sin x x cos x þ 7 7 ¼ rg cos x ¼ rg rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffitan x rgG ¼ rg 6 4 5 cos x 1 1þ 2 tan x which is constant and independent on the location of the point on the tooth flank. The theory of infinitesimal involute flank modelling can solve the inverse problem i.e. the determination of the conjugate gear tooth profiles and the corresponding generating rack profile when the path of contact P(x) and the rolling circle ro of the gear are given. For this polar coordinates must be used as described below. From an arbitrary point (xp, yp) on the given path of contact the local pressure angle is calculated y ro y P ¼ xP tan aoG þ ro ) tan aoG ¼ P xP

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hence qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rG ¼ x2P þ y 2P ¼ x2G þ G2 ðxG Þ and aoG ¼ tan1

y P ro xP

ð10Þ

rgG ¼ ro cos aoG

ð11Þ ð12Þ

The local involute angle and involute function are calculated rgG rG uG ¼ tan aG aG

aG ¼ cos1

ð13Þ ð14Þ

and therefore point G(rG, uG) in polar coordinates is the corresponding point of the gear. If the rack profile yR = R(x) is given, the path of contact is first calculated [1,4] and then the gear flank is calculated using the above methodology. It should be noted that although the tooth flank is approximated by a local involute at each point, the method itself is precise and not approximate. The theory does not demand that the tooth flank is divided into finite involute segments, in which case the density of the nodes and the lengths of the local involutes would affect the precision of the method, but the coordinates and the tangent at each flank point (x, y, dx/dy) are sufficient to define the corresponding point of the path of contact in a unique way. The well-established theory of gearing which is again precise and not approximate uses the same data (i.e. coordinates and normal vectors to the flank points) but proceeds with classical solution of the kinematical equations in cartesian or polar coordinates. It should also be noticed that Eq. (9) for the determination of the path of contact are analytical and non-implicit. 3. Case studies of non-involute gear geometries 3.1. Parallel axes cycloidal teeth for power transmission Consider the cycloidal spur gear tooth illustrated in Fig. 4. The working flanks of such teeth are composed of an epicycloidal addendum and a hypocycloidal dedendum. Point O is the center of the gear, point A is the intersection of the epicycloidal curve BA and the outside circle (O, rk1) and point B lies at the intersection of the epicycloidal and the hypocycloidal curve. Point C marks the beginning of the circular fillet and point D is the point of tangency of the circular fillet with the root circle (O, rf). In such gears the rolling point should be point B, otherwise interference would occur [1]. Starting from the top, the coordinates of point A should satisfy both equations of the epicycloid (15) and the outside circle (O, rk1) (16) uðrg1 þ r1 Þ uðrg1 þ r1 Þ xepi ¼ r1 sin ð15Þ þ ðr1 þ rg1 Þ sin u; y epi ¼ r1 cos þ ðr1 þ rg1 Þ cos u r1 r1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y k ¼ r2k1 x2k ð16Þ By solving the above system of equations, the value of u(A) = uk and the coordinates of point A are determined ! 2 2 2 r1 ðr þ r Þ þ r r 1 g1 1 k1 uk ¼ cos1 ð17Þ 2 rg1 2r1 r1 þ rg1 uk ðrg1 þ r1 Þ uk ðrg1 þ r1 Þ ð18Þ xA ¼ r1 sin þ ðr1 þ rg1 Þ sin uk ; y A ¼ r1 cos þ ðr1 þ rg1 Þ cos uk r1 r1 The epicycloidal curve is calculated from the standard equation (15), where angle u = 0 ! uk.


Y

757

y A

rk

Outside circle

Addendum (Epicycloid)

. .. .

B

ro

ε1

ε2 E

D

θf

Dedendum (Hypocycloid)

rs

C

ϕk

..

ϕs

Pitch circle

rf

Fillet (Circular)

Root circle

u x O Fig. 4. Geometry of a cycloidal gear tooth.

The coordinates of point B are xB ¼ 0;

y B ¼ ro1 ¼ rg1

ð19Þ

Point C is the point where the circular fillet starts and it is the point of intersection of the hypocycloidal curve and the lower working circle (O, rs1). The radius of the lower working circle is calculated from the following equation: rs1 ¼ ro1 cf to

ð20Þ

Hence the coordinates of point C must satisfy both the equations of the hypocycloidal curve (24) and the lower working circle (22). uðr1 rg1 Þ uðr1 rg1 Þ ð21Þ xhyp ¼ r1 sin ðr1 rg1 Þ sin u; y hyp ¼ r1 cos ðr1 rg1 Þ cos u r1 r1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y s ¼ r2s1 x2s ð22Þ By solving the above system of equations, the value of u(C) = us and the coordinates of point C are determined ! r1 ðr1 rg1 Þ2 þ r21 r2s1 1 /s ¼ cos ð23Þ 2 rg1 2 r1 ðr1 rg1 Þ us ðr1 rg1 Þ us ðr1 rg1 Þ ð24Þ ðr1 rg1 Þ sin us ; y C ¼ r1 cos ðr1 rg1 Þ cos us xC ¼ r1 sin r1 r1 The hypocycloidal curve is calculated from the standard equation (21), where angle u = us ! 0. Point D is the point of tangency of the circular fillet with the root circle (O, rf). Its coordinates are p p þ hf ; y D ¼ rf 1 sin þ hf xD ¼ rf 1 cos 2 2

ð25Þ

Line e1 is perpendicular to the hypocycloidal curve at point C (Eq. 12), and line e2 s perpendicular to the tangent of the root circle at point D. The governing equations of these lines are

758


dxðxC ;y C Þ ðx xC Þ þ y C dy p ðe2 Þ : y ¼ tan þ hf x 2 ðe1 Þ : y ¼

The intersection of these two lines is point E, which is the center of the circular fillet with coordinates dxðxC ;y C Þ þ yC xC p dy þ h ¼ tan ð26Þ ; y xE ¼ f xE p dx E 2 ðxC ;y C Þ þ hf þ tan 2 dy As point E is the center of the circular fillet, the distances (CE) and (ED) (the radius of the fillet) must be equal hence 2

2

2

CE2 ¼ ðxE xC Þ þ ðy E y C Þ

ED2 ¼ ðxE xD Þ þ ðy E y D Þ ;

2

ð27Þ

Angle hf is such that EC becomes equal to ED. The points of the circular fillet are calculated by the equation: x ¼ ðECÞ cos u þ xE ; y ¼ ðECÞ sin u þ y E

1 y C y E 1 xC xD where / ¼ tan ; 3p=2 þ tan . x x y y C

E

C

ð28Þ

D

With respect to the coordinate system (Oxy) as illustrated in Fig. 2, the previous coordinate system must be rotated at an angle u, about the origin O, so the new axis Oy would be now the axis of symmetry of the tooth. The angle u is calculated by the following equations: u¼

So ro1

ð29Þ

where So is the thickness of the tooth on the pitch circle. Then all the previous coordinates of the points and the curves must be recalculated using the well-known formula of the transformation of coordinates x ¼ x0 cos u y 0 sin u;

y ¼ x0 sin u þ y 0 cos u

3.2. Circular addendum two-lobed root blowers Let us consider now the root blower illustrated in Fig. 5. The geometrical modeling of this blower is suggested by Litvin and Feng [13]. Each lobe is composed by an addendum and a dedendum. The addendum is a circular arc of radius q and center (xade, yade). The dedendum is generated by the addendum of the mating rotor, during meshing. y

A

Pitch circle

ro θ1

ρ

.

B

O

a

Fig. 5. Geometry of a cycloidal two-lobed rotor.

x


759

To model that blower, point A must be calculated first. As illustrated in Fig. 5 point A coordinates are ro ro xA ¼ pffiffiffi ; y A ¼ pffiffiffi ð30Þ 2 2 Angle h1 is calculated by y ro h1 ¼ cos1 A ¼ cos1 pffiffiffi q 2q and

ð31Þ

ro ro 1 ro1 a ¼ q sin h1 þ pffiffiffi ¼ q sin cos pffiffiffi þ pffiffiffi 2 2p 2

After calculating distance a the coordinates of the center of the circle (addendum), point B are ro 1 ro xB ¼ a ¼ q sin cos pffiffiffi þ pffiffiffi ; y B ¼ 0 2q 2

ð32Þ

ð33Þ

and the addendum is calculated by the following equations:

p 3p h1 ; þ h1 xade ¼ q cos h þ xB ; y ade ¼ q sin h; h ¼ ð34Þ 2 2 To calculate the dedendum, we must first calculate the path of contact, by the addendum, and then the flank of the conjugate gear, using the equations of the proposed theory. Then the flank must be rotated about point A for p rad using the following equations: xded ¼ ðx xA Þ cos p ðy y A Þ sin p;

y ded ¼ ðx xA Þ sin p þ ðy y A Þ cos p

ð35Þ

After calculating the addendum and the dedendum of the first lobe the other lobes are calculated using simple geometrical transformations. 4. Results and discussion The above presented non-involute spur gear tooth geometries were generated using specially developed computer programmes. The flanks of all generating gear teeth were discretized in 1001 points and the normal vectors were calculated using B-spline approximation.

Fig. 6. Cycloidal rack and pinion in mesh.

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Fig. 7. Cycloidal pinion and wheel in mesh.

The cycloidal pinion and its rack are presented in Fig. 6 and a pair of two cycloidal spur gears are illustrated in Fig. 7. In both figures it is worth noticing that there is no backlash or radial clearance between the meshing teeth since they are generated by each other, therefore forming a whole ‘‘family’’ of gears (i.e. rack, pinion and wheel) sharing the same path of contact. The common path of contact and its generation using the proposed method is illustrated in Fig. 8. The theory of infinitesimal involute flank modelling can readily compute the mating gear and the geometry of the generating rack for a two or three lobed root blower of circular arc addendum. In this case only the geometry of the addendum is known (circular arc) and the dedenda of both gears are calculated using the novel theory. In Fig. 9 a rack–pinion pair with two lobes is shown. The corresponding path of contact for a two-lobed circular arc addendum root blowers in mesh is illustrated in Fig. 10. The path of contact which was computed as an intermediate step of the algorithm has a distinct ‘‘8’’ shape.

y

C

O

Fig. 8. Path of contact of the cycloidal spur gear family.

x


761

y

C

O

x

Fig. 9. A two-lobed blower with circular arc addendum in mesh with its generating rack.

O2

O2

.... .

C O1

O1

Fig. 10. A two-lobed circular arc addendum blower meshing pattern and its relevant path of contact.

5. Conclusion In this paper a new approach for calculating conjugate tooth profiles of gears in two dimensions is presented. The theory is based on the concept of discretizing the tooth flank in ‘‘local involutes’’ and the derived equations provide a quick, simple and non-implicit way to calculate from a given tooth geometry the path of contact, the tooth of the generating rack and the tooth form of the mating wheel. The method can be applied on both involute and non-involute gears and can be easily programmed on a computer. The proposed method is generally faster (up to six times depending on the problem) than the other existing generic gear theories including the widely used theory of gearing. These advantages are derived from the fact that the standard discretization of the tooth flank in linear segments (i.e. a point and an inclination) suggested by the theory of gearing is substituted by a discretization into infinitesimal local involutes. In this way the corresponding point on the mating wheel/rack and the relevant point on the path of contact are calculated using simple and closed analytical formulae instead of having to solve complex equations which often exhibit branching points. In the examples presented in the previous paragraph it was demonstrated that the theory works satisfactorily without any special adaptation in the case of non-involute gear tooth forms with either open or closed paths of

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contact such as cycloidal power transmission gearing and two-lobed root blowers with circular addenda. Even the conditions for avoiding undercutting can be reduced to the well-known limits of undercutting used for the involute. The method can be particularly useful when the geometry of the mating gears is to be calculated in an iterative procedure such as in gear optimization. The new method can be easily expanded to cover not only geared mechanisms but also chain drives, cams, etc. and is easy to integrate in a CAD software as indicated in the examples above. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

E. Buckingham, Analytical Mechanics of Gears, Dover Publications Inc., New York, 1988. J.R. Colbourne, The Geometry of Involute Gears, Springer-Verlag, New York, 1987. E. Wildhaber, Conjugate pitch surfaces, American Machinist 90 (13) (1946) 150–152. R. Errichello, D.P. Townsend, Gear tooth calculations, in: D.P. Townsend (Ed.), Dudley’s Gear Handbook, Mc-Graw Hill, New York, 1992. T. Yeh, D.C.H. Yang, S. Tong, Design of new tooth profiles for high-load capacity gears, Journal of Mechanism and Machine Theory 36 (2001) 1105–1120. F.L. Litvin, Theory of Gearing, NASA Reference Publication 1212, AVSCOM Technical Report, 88-C-035, 1989. F.L. Litvin, Gear Geometry and Applied Theory, Prentice-Hall, Englewood Cliffs, NJ, 1994. F.L. Litvin, A. Fuentes, Gear Geometry and Applied Theory, Cambridge University Press, 2004. F.L. Litvin, P. Feng, Computerized design and generation of cycloidal gearings, Journal of Mechanism and Machine Theory 31 (7) (1996) 891–911. F.L. Litvin, Development of Gear Technology and Theory of Gearing, NASA Reference Publication 1406, ARL-TR-1500, 1998. D. Vecchiato, A. Demenego, J. Argyris, F.L. Litvin, Geometry of a cycloidal pump, Journal of Computer Methods in Applied Mechanics and Engineering 190 (2001) 2309–2330. A. Demenego, D. Vecchiato, F.L. Litvin, N. Nervegna, S. Manco´, Design and simulation of meshing of a cycloidal pump, Journal of Mechanism and Machine Theory 37 (2002) 311–332. F.L. Litvin, P.-H. Feng, Computerized design and generation of cycloidal gearings, Journal of Mechanism and Machine Theory 31 (7) (1996) 891–911. S.-H. Tong, D.C.H. Yank, On the generation of new lobe pumps for higher pumping flowrate, Journal of Mechanism and Machine Theory 35 (2000) 977–1012. V. Spitas, T. Costopoulos, C. Spitas, A quick and efficient algorithm for the calculation of gear profiles based on flank involutization, in: Proceedings of the 4th GRACM Congress on Computational Mechanics, Patra, Greece, 2002. G.K. Sfantos, V.A. Spitas, T.N. Costopoulos, Application of the theory of gear tooth flank Involutization on external modified involute gear teeth, NTUA, Dep. Mech. Engineering, Laboratory of Machine Elements, Technical Report No. TR-SM-0203, 2002.

Fast modeling of conjugate gear tooth profiles using discrete presentation by involute segments

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