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An Approach on Simulation of Involute Tooth Profile Used on Cylindrical Gears

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An Approach on Simulation of Involute Tooth Profile Used on Cylindrical Gears 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 INVOLUTE 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 circle 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 envelope 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 curves: an involute as flank line (in a transverse plane) 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 curve 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 involutes 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 trochoids (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 tangent 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(i4)1) p y1(Pi1) y1(P(i4)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.
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