BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI Publicat de Universitatea Tehnică „Gheorghe Asachi” din Iaşi Volumul 66 (70), Numărul 2, 2020 Secţia CONSTRUCŢII DE MAŞINI

AN APPROACH ON SIMULATION OF TOOTH PROFILE USED ON CYLINDRICAL GEARS

BY

MIHĂIȚĂ HORODINCĂ

“Gheorghe Asachi” Technical University of Iaşi, Faculty of Machine Manufacturing and Industrial Management

Received: April 6, 2020 Accepted for publication: June 10, 2020

Abstract. Some results on theoretical approach related by simulation of 2D involute tooth profiles on spur gears, based on rolling of a mobile generating rack around a fixed pitch are presented in this paper. A first main approach proves that the geometrical depiction of each generating rack position during rolling can be determined by means of a mathematical model which allows the calculus of Cartesian coordinates for some significant points placed on the rack. The 2D involute tooth profile appears to be the internal bordered by plenty of different equidistant positions of the generating rack during a complete rolling. A second main approach allows the detection of Cartesian coordinates of this internal envelope as the best approximation of 2D involute tooth profile. This paper intends to provide a way for a better understanding of involute tooth profile generation procedure.

Keywords: tooth profile; involute; generating rack; simulation.

1. Introduction

Gear manufacturing is a major topic in industry due to a large scale utilization of gears for motion transmissions and speed reducers. Commonly the flank gear surface on cylindrical toothed wheels (gears) is described by two

Corresponding author: e-mail: [email protected] 22 Mihăiţă Horodincă orthogonally : an involute as flank (in a transverse ) and a profile line. This profile line is a straight line (perpendicular on transverse plane on spur gears) or a cylindrical helix (on helical gears). There is a simple reason for the use of involute as flank line: it is generated by a rolling with linear cutting edges placed on a rack-type cutter (or an equivalent tool). A couple of linear cutting edges on this cutting tool generate several simultaneously. An important item in theoretical approach of gear manufacturing is the computer aided simulation of (involute) tooth profile generating process based on mathematical models, which is done in order to find the best ways to optimize the gear behaviour. For tooth profile generating process, the literature indicates that many specialized gear software are used, e.g. AutoCAD package (Kheifetc, 2016), ProEngineer Wildfire 5.0 (Rathod et al., 2011), GRAPHER2-D graphing system (Fetvaci and Imram, 2008), Working Model 2D (Simionescu, 2008), MAPLE (Cheng and Jian, 2014), Matlab and Solidworks (Anakhu et al., 2017), Scilab (Zhai et al., 2020). Regarding the mathematical models used for simulation of tooth profile three main approaches are available in literature. First one (Kheifetc, 2016; Rathod et al., 2011; Guo et al., 2014; Zhai et al., 2020) uses basically the mathematical equation of involute for each active tooth surface to define the tooth profile. Some other researchers (Simionescu, 2008; Cheng and Jian, 2014; Fetvaci and Imram, 2008; Zhai et al., 2020) use a second approach in tooth profile definition, as envelope of the generating rack positions during rolling, available also for non-circular (e.g. elliptic) spur gears with involute profile (Cheng and Jian, 2014) or for cylindrical spur gears with cosine (non-involute) profile (Luo et al., 2008; Statdfeld and Saewe, 2015). There is a third approach as mathematical model in simulation of tooth profile for symmetric and asymmetric involute teeth: the tooth profile occurs as the envelope of the trajectory paths of all the points placed on the generating rack as roulette-type curves, also called (Fetvaci, 2012; Fetvaci and Imram, 2008; Su and Houser, 2000). The second approach is privileged in this paper, which proposes some theoretical achievements in computer aided mathematical modelling of rolling (calculus and simulation) in order to produce in a simpler way the numerical description of 2D involute tooth profile of spur or helical gears, useful to verify and to validate the gear geometry before manufacturing. Based on the achievements proposed in this paper it is possible to generate by rolling any type of circular non-involute, symmetrical and non- symmetrical tooth profile, useful to define the flank gear surfaces. The 2D tooth profile thus generated can be used to build the 3D model of a spur or a helical gear by computer aided design. This 3D model can be used to produce the toothed wheel by cutting or by additive manufacturing processes (3D printing). Bul. Inst. Polit. Iaşi, Vol. 66 (70), Nr. 2, 2020 23

2. A Mathematical Model of Generating Rack Position and Trajectory Used in Rolling Simulation

Some considerations related by rolling of circular involute tooth profile using symmetrical generating rack and gear nomenclature are presented (https://en.wikipedia.org/wiki/List_of_gear_nomenclature) in Fig. 1.

Fig. 1 – A graphical approach on rolling with a generating rack.

The main dimensional characteristics of generating rack (a conventional shape highlighted here in blue colour), related to datum, tip and root lines are: circular pitch p=πm (here m is module of gear), circular tooth thickness s, circular space thickness t (usually s=t=p/2), addendum hf (e.g. hf=1.25m), o dedendum ha (e.g. ha=1.1m) and pressure angle δ (usually δ=20 ). During rolling, the generating rack (placed as complex cutting edge on a rack-type cutter) move along the datum (pitch) line, with linear velocity v (here from right to left). The pitch line is to a pitch circle (having the radius Rp) in the pitch point Pp. The pitch circle rotates around its own centre (here in counterclockwise direction) with the angular velocity ω, strictly related by linear velocity v with the relationship:

v  Rp (1)

It is mandatory that, for a displacement l of generating rack, the pitch circle rotates with an angle β, so that the displacement l is disposed on the pitch circle as an arc with the length Rpβ=l. The derivative of this last relationship produces Eq. (1) because dl/dt=Rpdβ/dt or v=Rpω (if consider v=dl/dt and ω=dβ/dt). 24 Mihăiţă Horodincă

In these conditions, during rolling the tooth profile (as a conceptual shape, highlighted in red colour on Fig. 1) is generated in a plane attached to the pitch circle. In order to generate a tooth profile for a toothed wheel with Z teeth, the radius Rp of the pitch circle must be obligatory defined from the relationship Zp=Zπm=2πRp (with Z circular pitches p on the circumference of the pitch circle) as:

mZ R  (2) p 2

By mathematical point of view it is easiest to describe the rolling related to a fixed pitch circle. In this approach, the generating rack moves along the pitch line with the same velocity v from right to the left and the pitch line rotates with the angular velocity ω around the pitch point Pp, in clockwise direction (opposite to those depicted in Fig. 1).

Fig. 2 – Some geometrical considerations on the evolution of the coordinates of a point placed on the generating rack during rolling.

The key problem in tooth profile generation is to describe the evolution of the coordinates of an unspecified point placed on the generating rack (e.g. the point Pi1 on Fig. 1) during rolling, related to a fixed xOy Cartesian coordinate system. The geometrical considerations from Fig. 2 help to solve this problem. Suppose that the generating rack (Gr1 on Fig. 2 formally described with a single Bul. Inst. Polit. Iaşi, Vol. 66 (70), Nr. 2, 2020 25 tooth) is placed in the Cartesian coordinate system x1O1y1, with the pitch line placed onto x-axis x1O1 and the origin O1 placed in the pitch point Pp. Consider that the root line is tangent to the tip circle (Fig. 1) and the coordinates of significant points which define a first tooth on generating rack Gr1 (P11÷P51 on Fig. 1) in x1O1y1 system are given in Table 1.

Table 1 The Coordinates of Significant Points Involved in the Definition of a First Tooth on Generating Rack in x1O1y1 Cartesian Coordinate System Point Abscissa definition Ordinate definition i,Pi1 x1(Pi1) y1(Pi1)

i=1, P11 a (a conventional distance) ha

i=2, P21 a+[t-2hatan(δ)] ha i=3, P31 a+[t-2hatan(δ)]+(ha+hf)tan(δ) -hf

a+[t-2hatan(δ)]+(ha+hf)tan(δ)+ i=4, P41 -hf +[s-2hftan(δ)]

a+[t-2hatan(δ)]+(ha+hf)tan(δ)+ i=5, P51 ha +[s-2hftan(δ)]+(ha+hf)tan(δ)=a+t+s=a+p

Because between each two successive teeth is a distance equal with the circular pitch p, the coordinates of any other significant following point placed on the generating rack (for i>4) are simply defined as:

x1(Pi1)  x1(P(i4)1)  p y1(Pi1)  y1(P(i4)1) (3)

For example, according to last row in Table 1, the coordinates of point P51 are: x1(P51)=x1(P11)+p and y1(P51)=y1(P11). For a generating rack having N teeth, the number of significant points should be imax=5N-1. In order to generate completely a tooth profile with Z teeth, it is mandatory to have N>Z. Suppose that in Fig. 2 the pitch line (and x-axis x1O1 as well) rotates during rolling around the pitch circle, with an angle β. The coordinate system 훽 훽 훽 x1O1y1 has now a new position 푥1 푂1 푦1 . The generating rack has now a new β position: Gr1 . According with the rolling conditions previously discussed, the generating rack should move along the pitch line with a distance l=Rpβ. In the 훽 훽 훽 new coordinate system 푥1 푂1 푦1 the abscissa of significant points of generating β rack Gr1 decreases with this distance l, while the ordinates don’t change, as it follows:    (4) x1 (Pi1 )  x1(Pi1)  Rp y1 (Pi1)  y1(P(i1)

If the angle β progressively increases, any significant point Pi1 of the generating rack describes a trajectory path Ti in the fixed Cartesian coordinate   system xOy. The evolution of the coordinates x(Pi ) , y(Pi ) of any significant 26 Mihăiţă Horodincă point Pi1 in xOy coordinate system describes these trajectories. Also the   coordinates x(Pi ) , y(Pi ) are involved in the description of each position of generating rack.  In Fig. 2 the position of any point Pi is described in the xOy Cartesian  coordinate system by a vector rPi defined from vector equation:

   rP i  rO c  rO1  rP i1 (5)

The projections of each vector involved in Eq. (5) onto Ox axis and Oy axis are described in Table 2.

Table 2 The Projections of Vectors Involved in Eq. (4) onto Ox and Oy Axis

Vector Projection onto Ox axis Projection onto Oy axis    rPi x(Pi ) y(Pi )

rOc xc yc  rO1 Rp cos(  ) Rp sin(  )

[x(P )  Rp]cos( )  [x(P )  Rp]sin( )   i1 i1 rPi1 [y(Pi1)]sin( ) [y(Pi1)]cos( )

Here α is used to describe the first position of coordinate system x1O1y1, and  β β γ=α-β-π/2. The projection of vector rPi1 onto Ox axis is O1 B = O1 C − β β β 훽 β 훽 β BC = O1 C − ED = O1 Dcos 훾 − Pi1Dsin 훾 = 푥1 (Pi1)cos 훾 − 푦1 (Pi1)sin 훾 =[푥1 푃푖1 − 푅푝 훽]cos 훾 − 푦1 푃푖1 sin 훾 . This last result was already written in Table 2 (the 4th row, 2nd column). β β Similarly, the projection of vector onto Oy axis is Pi1B = Pi1E + β β β 훽 β 훽 β EB= Pi1E + DC = Pi1Dcos 훾 + O1 Dsin 훾 = 푦1 (Pi1)cos 훾 + 푥1 (Pi1)sin 훾 = 푦1 푃푖1 cos 훾 + 푥1 푃푖1 − 푅푝 훽 sin 훾 = 푥1 푃푖1 − 푅푝 훽 sin 훾 + th rd +푦1 푃푖1 cos 훾 . This last result was already written in Table 2 (the 4 row, 3 column). The projection of Eq. (5) onto x-axis of xOy coordinate system can be written using the results from 2nd column (Table 2) as:

x(P )  x  R cos(  ) [x(P )  R ]cos( ) [y(P )]sin( ) (6) i c p i1 p i1 Bul. Inst. Polit. Iaşi, Vol. 66 (70), Nr. 2, 2020 27

The projection of Eq. (5) onto y-axis of xOy coordinate system can be written using the results from 3rd column (Table 2) as:

 y(Pi )  yc  Rp sin(  ) [x(Pi1)  Rp]sin( ) [y(Pi1)]cos( ) (7)

The mathematical relationships (6) and (7) are the parametric equations of the trajectory Ti available for any (significant) point of the generating rack. The parameter is the angle β. During rolling, each position of generating rack (for each value β progressively increased between 0 and 2π) can be graphically depicted. First the   coordinates x(Pi ) , y(Pi ) of each significant point are calculated, after that a segment line between any two successive points is drawn. Finally the tooth profile occurs as an internal envelope of all positions of the generating rack. In this way the tooth profile is generated also during gear manufacturing process. The parametric equations (6) and (7) are also available to generate the trajectory of any point placed on generating rack (e.g. any point placed between any two successive significant points). The tooth profile occurs also as internal envelope of all these trajectories drawn for a progressive evolution of β between 0 and 2π. This is just a theoretical approach useful to study some properties of the tooth profile (e.g. the fillet between the gear flank and the bottom land). If a positive shift (or negative shift as well) of tooth profile is necessary (e.g. a positive shift used to eliminate the undercutting phenomenon) the generating rack is shifted, all the ordinates definition y1(Pi1) from Table 1 (involved in Eqs. (6) and 7)) should be increased (or decreased for negative shift if necessary) with the same quantity rsc·m, with rsc as rack shift coefficient.

3. Simulation Results

Based on these previous theoretical approaches, the tooth profile generating process can be graphically simulated on computer (e.g. using Matlab) for a toothed wheel having (as example) Z=30, m=25 mm (Rp=375 mm), t=s, ha=1.1·m, hf=1.25·m, rsc=0, with xc=0,yc=0, a=70 mm, and α=2π/3 radians. This simulation was done using Eq. (6) and Eq. (7) with the coordinates of significant points of generating rack (in x1O1y1 Cartesian coordinate system) given in Table 1. A first approach on rolling (partially) simulation is depicted in Fig. 3. Here a generating rack with only 3 teeth is used, for 31 angular equidistant positions (drawn in blue) which cover an angle of 0.31π (with an incremental variation Δβ=2π/200 of angle β between each two successive positions). The first position of generating rack is drawn in green; the last position is drawn in red, during rolling the pitch line rotates in clockwise sense. 28 Mihăiţă Horodincă

Fig. 3 – Some successive positions of generating rack during a partially simulation of rolling.

Fig. 4 presents the tooth profile as envelope of Npgr=300 angular equidistant positions of the generating rack (having N= Z+7 teeth) during a completely rolling (the range of β is 2π radians), in the same conditions as before.

Fig. 4 – Involute tooth profile. A result of a rolling with 200 equidistant generating rack positions.

Bul. Inst. Polit. Iaşi, Vol. 66 (70), Nr. 2, 2020 29

As Fig. 4 clearly indicates, the inner region bounded by all positions of generating rack is the 2D involute tooth profile (here depicted in red colour). A simple geometrical procedure is available to detect this tooth profile, based on the considerations from Fig. 5.

Fig. 5 – Some considerations on 2D involute tooth profile detection.

Let us assume that there is a rotary ray (here depicted with blue colour) having the endpoint Oc (Fig. 2) and a point placed on the pitch circle in an arbitrary position.

a) Npgr=300; Nprr=1,000 b) Npgr=300; Nprr=50,000

Fig. 6 – Two different scenarios on 2D tooth profile definition with low accuracy

related to Npgr and Nprr values (fillet radius on B area from Fig. 5).

In this position the ray intersects some segments lines (between successive significant points) involved in the definition of generating racks. 30 Mihăiţă Horodincă

Between these intersection points (with calculated coordinates) there is a closest point to the endpoint Oc.

Fig. 7 – A high accuracy 2D tooth Fig. 8 – A 3D CAD model of a spur gear based profile definition with Npgr=10,000 on 2D involute tooth profile generated by and Nprr=50,000 (fillet radius on B computer aided simulation (Z=30, m=2 mm, area from Fig. 5) Npgr=1,000 and Nprr=10,000).

This point is certainly placed on the best approximation of 2D (BA2D) involute tooth profile. Then the ray is successively rotated with a small angular increment (Δθ) around the endpoint (in a range between 0 and 2π radians, with Nprr=2π/Δθ the integer number of positions for rotary ray). For each position of the ray, is similarly identified a point placed on BA2D tooth profile. Finally, the coordinates of several Nprr points placed on BA2D tooth profile are obtained. With an appropriate segment curve between each two successive points (usually a line segment) the BA2D tooth profile is completely defined. As simulation from Fig. 7 proves, a high accuracy of 2D involute tooth profile is obtained with high values for Npgr and Nprr (in opposition with the simulations with low accuracy, depicted in Fig. 6a and b). The 2D involute tooth profile can be transferred to an appropriate CAD software where is extruded (using a vertical straight line path) in order to build the 3D CAD model of a spur gear (as Fig. 8 indicates) available for additive manufacturing (as example). If the extrusion path is an appropriate helix, then the 3D model of a helical gear can be generated. In this way the 3D CAD model of both toothed wheels involved in any conjugate gear can be generated. The 2D involute tooth profile may also be used in spur gear fabrication by 2D contouring on a vertical CNC milling machine or by electrical discharge machining (Rivkin et al., 2020). Bul. Inst. Polit. Iaşi, Vol. 66 (70), Nr. 2, 2020 31

There are many supplementary resources in tooth profile simulation. Fig. 9 reveals a heavy undercutting undesirable phenomenon (Alipiev et al., 2013) which affects the fillet of gear due to a small number of teeth (Z=11). The length of involute on face flanks of gear is diminished due to the apparition of undercut root filet (as undercutting areas) caused by some typical positions of generating rack during rolling (e.g. 1 and 2).

Fig. 9 – An undercutting phenomenon on Fig. 10 – A possible error in tooth 2D tooth profile revealed by simulation on a profile detection located at the beginning toothed wheel having Z=11 teeth. of undercut area (Fig. 9).

Fig. 11 – Some relevant roulette-type Fig. 12 – A 2D tooth profile described as curves involved in the 2D profile an inner envelope bordered by definition from Fig. 9. roulette-type curves.

32 Mihăiţă Horodincă

Because of undercutting, there is a particular approach in the detection of 2D tooth profile: this is done successively for each tooth, the endpoint of rotating ray should mandatory be placed in a close proximity of the root cycle (e.g. Ep on Fig. 9), the angle of rotation of ray should have two appropriate limits in order to detect the profile of a single tooth. Otherwise (with the endpoint placed in Oc) the 2D tooth profile in undercut area cannot be properly described, because there are hidden parts of the envelope described by generating rack positions in relation with the rotary ray, as Fig. 10 indicates. A better approach on this undercutting phenomenon is related by the shape of symmetrical (left-right) curves TaL and TaR involved in description of the root fillets (Fig. 11) also called trochoidal fillets (Su and Houser, 2000). These curves (as roulette-type trajectory paths), are particular shapes of trajectory Ti described before with Eqs. (6) and (7), having the particularity that are described by significant points placed on generating rack: TaL is described by a point similar to P41 (Fig. 1), TaR is described by a point similar to P71.

Fig. 13 – Undercutting avoidance through Fig. 14 – Tooth pointing avoidance by tip positive shift of tooth profile (rsc=0.85). diameter diminished on gear blank.

All the points placed on generating rack generate a roulette-type trajectory Ti. Except the trochoidal fillets, each other trajectory is tangent to the 2D tooth profile in a single point (for a certain value of parameter β involved in Eqs. (6) and (7). A point placed properly on the pitch line and generating rack describes the trajectory TbL, the next point describes the trajectory TbR. Both these trajectories are involutes having the pitch circle as base circle (in Eqs. (6) and (7) the ordinate y(Pi1)=0). A point placed on the root line and generating rack describes the trajectory Tc. Bul. Inst. Polit. Iaşi, Vol. 66 (70), Nr. 2, 2020 33

It is easy to imagine many different other types of trajectories (e.g. TbcL and TbcR). The 2D involute tooth profile can be described as the envelope externally bordered by the trajectories of all the points of the generating rack during rolling, as described in Fig. 12 (a single tooth of a toothed wheel with Z=40, m=25). The trochoidal fillets here describe the transition curve from the involute face flank to root (dedendum) circle (fillet radius). Because of this fillet radius, a gear and its mate need mandatory a clearance between (https://en.wikipedia.org/wiki/List_of_gear_nomenclature). It is interesting to remark that the transition between involute face flanks and tip (dedendum) circle is done always with an edge, without any transition curve. The undesirable undercutting phenomenon (revealed previously in Figs. 9 and 11) can be avoided with a positive shift rsc·m of generating rack. In simulation of 2D shifted tooth profile (Fig. 13) the rack shift coefficient rsc is increased until undercutting disappears (here rsc=0.85). Unfortunately, for a small number of teeth, the top land of the gear is reduced at a line segment and on 2D shifted profile the top of each tooth (normally an arc of circle) is reduced at a single point (as tooth pointing phenomenon). This inconvenient is simply eliminated if the diameter of tip circle (cylinder) is diminished (as Fig. 14 indicates). We should mention that usually the top lane of the gear is a cylindrical surface as a part of the gear blank, during rolling (and cutting) process the root line of rack-type cutter is not tangent at tip circle (as Fig. 1 indicates). Many other items in 2D tooth profile generation are approachable in this way, as a challenge for a future research (e.g. gears with non-involute and non-symmetrical flank faces, gears with harmonic shape of generating rack, etc.). Many considerations from this paper are useful in generating by rolling of 2D profile of any other object with rotational symmetry in cross section.

4. Conclusions

The computer aided simulation of 2D involute tooth profile due to rolling with a generating rack, provides mainly two theoretical achievements useful in gear manufacturing and optimisation. First relevant achievement proposed in this paper is a mathematical model of the trajectory described by a point placed on generating rack during rolling. This model allows two kinds of geometrical description of a 2D involute tooth profile: as the envelope of all positions of the generating rack (a privileged approach in this paper) or as the envelope of the trajectory paths (roulette-type curves) described by all the points placed on the generating rack (this approach was mainly used in this paper to describe the gear fillet). The second important achievement consists of a method to detect the best approximation of the coordinates of points involved in 2D tooth profile description (the inner envelope bordered by generating rack positions). The 34 Mihăiţă Horodincă intersection points of a rotary ray (with the endpoint in the origin of pitch circle) with the line segments which depicts the positions of generating rack (or cutting edges on rack-type cutter as well) are found by calculation. The closest point to the endpoint of the ray is with a good approximation a point placed on 2D tooth profile. Each position of the rotary ray in a range between 0 and 2π provides the coordinates of a point. The utility of these theoretical approaches was investigated and proved by some simulations of 2D tooth profiles and roulette-type curves, including the removing of undercutting phenomenon effects during rolling. These researches are useful to optimize many known spur (or helical) gear manufacturing processes using rack-type cutters, the gear manufacturing by 3D printing being a direct beneficiary of this work.

REFERENCES

Alipiev O., Antonov S., Grozeva T., Generalized Model of Undercutting of Involute Spur Gears Generated by Rack-Cutters, Mechanism and Machine Theory, 64, 39-52 (2013). Anakhu P.I., Abioye A.A., Bolu C.A., Azeta J., Modeling of the Kinematic Geometry of Spur Gears Using Matlab, The 4th International Conference on Mechatronics and Mechanical Engineering (ICMME 2017), MATEC Web of Conferences 153, 03004 (2018). Cheng W.H., Jian L., Tooth Profile Design and Kinematic Machanism Simulation of Higher-order Elliptic Gears Based on MAPLE, Journal of Applied Sciences, 14, 4, 352-367 (2014). Fetvaci C., Computer Simulation of Involute Tooth Generation, Mechanical Engineering, Dr. Murat Gokcek (Ed.), IntechOpen, 503-526 (2012). Fetvaci C., Imrak E., Mathematical Model of a Spur Gear with Asymmetric Involute Teeth and Its Cutting Simulation, Mechanics Based Design of Structures and Machines, 36, 1, 34-46 (2008). Guo X.L., Zhao H.S., Lu J.B., Parametric Modeling of Involute Spur Gear Based on AutoCAD, Advanced Materials Research, 945–949, 845-848 (2014). Kheifetc A.L., Geometrically Accurate Computer 3D Models of Gear Drives and Hob Cutters, Procedia Engineering, 150, 1098-1106 (2016). Luo S., Wub Y., Wang J., The Generation Principle and Mathematical Models of a Novel Cosine Gear Drive, Mechanism and Machine Theory, 43, 1543-1556, (2008). Rathod A., Patel V., Agrawal P.M., A Parametric Modeling of Spur Gear Using ProEngineer, National Conference on Recent Trends in Engineering & Technology, B.V.M. Engineering College, V.V. Nagar, Gujarat, India (2011). Rivkin A., Nekrasov A., Sobolev A., Arbusov M., The Features of the Gears Profile Forming by the Electrical Discharge Machining, Materials Today: Proceedings (in press) (2020). Bul. Inst. Polit. Iaşi, Vol. 66 (70), Nr. 2, 2020 35

Simionescu P.A., Interactive Involute Gear Analysis and Tooth Profile Generation Using Working Model 2D, 115th American Society for Engineering Education (ASEE) Annual Conference and Exposition, Pittsburgh PA, 13.781.1- 13.781.13 (2008). Stadtfeld H.J., Saewe J.K., Non-Involute Gearing, Function and Manufacturing Compared to Established Gear Design, Gear Technology, January-February 2015, 42-51 (2015). Su X., Housser D.R., Characteristics of Trochoids and their Application to Determining Gear Teeth Fillet Shapes, Mechanisms and Machine Theory, 35, 2, 291-304 (2000). Zhai G., Liang Z., Fu Z., A Mathematical Model for Parametric Tooth Profile of Spur Gears, Mathematical Problems in Engineering, Article ID 7869315, 2020 (2020). https://en.wikipedia.org/wiki/List_of_gear_nomenclature

UN STUDIU AL PROCESULUI DE GENERARE A PROFILELOR EVOLVENTICE UTILIZATE PE ROȚILE DINȚATE CILINDRICE

(Rezumat)

În lucrare se prezintă o serie de abordări teoretice legate de simularea profilelor 2D ale roților dințate cilindrice cu profil evolventic, folosind o sculă de tip cremalieră care rulează pe cercul de rostogolire fix. Prima abordare demonstrează că descrierea geometrică a fiecărei poziții a cremalierei generatoare este posibilă pe baza unui model matematic care permite calculul coordonatelor punctelor semnificative de pe sculă. Profilul 2D al roții dințate apare ca anvelopa internă mărginită de pozițiile cremalierei în timpul unei rulări complete. A doua abordare permite detecția coordonatelor punctelor de pe această anvelopă, ca fiind cea mai bună aproximare a profilului 2D al danturii evolventice. Lucrarea propune un studiu care dorește să asigure o mai bună înțelegere a procedurii de generare a profilului evolventic pe roți dințate cilindrice.