Tooth Contact Analysis of Cylindrical Gears with an Unconventional Tooth Profile in and Technology

The paper presents mathematical models of cylindrical gear pairs with various types of tooth profiles, such as eccentric-cycloidal, concavo-convex Novikov type, and involute. Comparative analyses were provided for the aforementioned gear meshes, aimed at determining contact patterns, sliding velocity, and transmission errors. Tooth surface modifications were also considered. The results were compared with findings for a conventional involute gearing. It was found that the contact pattern in Novikov conformal gearing is 30% greater than in the involute gear pair, and 60% greater than in the eccentric-cycloidal gear pair. The smallest sliding velocity was obtained in Novikov gearing, which may be beneficial in terms of durability.


INTRODUCTION
Cylindrical gear pairs are commonly applied in nearly all branches of industry. Growing requirements placed on gear pairs, such as increased load-carrying capacity, improved noise behaviour, low weight or reliability, have prompted researchers and engineers to look for increasingly novel technological and design solutions. More and more attention is drawn to new manufacturing technologies like DMLS [1], as well as new tooth finishing techniques [2]. Novel types of calculation procedures aiming at gear mass reduction [3] as well as more accurate surface strength and wear prediction [4] are also within the interests of gear scientists and engineers. In addition, designers look for new tooth profiles which could provide an alternative to the commonly used involute profile. A recently published review paper [5] extensively discusses the literature concerning the non-involute external gears. The authors compared various types of gear meshes in terms of surface durability, root load capacity, gear efficiency, vibration, and sensitivity to assembly and manufacturing errors. Fourteen types of gear profiles were compared such as: cycloid gears [6], Novikov gears [7], triple circulararc gears [8], cosine gears [9], double circular arc involute gears [10], S-gears [11], and pure rolling helical gears [12]. From all referenced above types of tooth profiles, the greatest surface durability and the smallest heat load are assigned to pure rolling transmission, S-gears, and conformal gearings (Novikov gears or W-N gears). Pure rolling transmission operation is similar to Novikov gears. The continuity of meshing is realized by overlap. The tooth profiles may be chosen arbitrarily however the contact point should lay on point of tangency of pitch circles to avoid sliding velocity. In paper [13] authors proposed the tooth profiles of pure rolling transmission formed by circular arcs with the convexo-convex type of contact. They have investigated proposed gearing in terms of contact stresses and loaded transmission error Tooth Contact Analysis of Cylindrical Gears with an Unconventional Tooth Profile [13]. The pure rolling type of transmission was also investigated in gear-rack [14] and internal helical [15] gear mesh. S-gears are gears, which profi le is formed by a rack with an analytically defi ned profi le by power function [16]. The transverse line of action of this type of gearing is curvilinear and therefore may off er a greater contact ratio than conformal or pure rolling gearing. The advantage of this kind of gear mesh is reduced sliding velocity in comparison with the involute one which was reported in [17]. This type of gear profi le was proposed for polymer gears [18]. Novikov gears also offer reduced sliding in comparison with involute gears and due to conformal contact may be benefi cial in terms of surface strength. Successful application of Novikov gears [19] prompts the scientist to continue the research on this kind of gearing [7,20]. Another non-conventional tooth profi le is the so-called eccentric-cycloidal tooth profi le. This kind of gearing is a special case of helical cycloid gear drive where the pinion tooth profi le is formed by the circular arc and the conjugated gear wheel profi le is an internal equidistant of the epicycloid [21]. The-above referenced gear teeth profi le is not well researched and described in the literature. Therefore, the aim of this paper is to compare three types of cylindrical gear pairs: an involute gear pair, a Novikov gear pair [7,20], and an eccentric-cycloidal gear pair [21,22].

MATHEMATICAL MODEL OF CYLINDRICAL GEAR MESH
The analysis will concern cylindrical reduction gear pairs with parallel axes and external meshing. Moving coordinate systems 1 and 2 connected, respectively, with the drive gear (the pinion) and the driven gear (the gear wheel) were set up. The gears rotate around axes ! and ! by angles ! and ! in in the indicated directions of angular velocities ! and ! (Figure 1). Tooth surfaces are represented in appropriate coordinate systems ̅ ! (!) ( ! , ! ) and ̅ ! (!) ( ! , ! ) .
Moreover, fi xed coordinate system f connected with the transmission body was introduced, wherein axis z f corresponds to axis z 1 of the pinion's coordinate system. The gear wheel's rolling cylinders with diameters d t1 and d t2 are separated by centre distance a r . In order to take into account axis position deviations due to assembly or workmanship errors, or elastic deformations of shafts and bearings, the coordinate system connected with the gear wheel must be shifted along axis x f , y f , z f by Δa x , Δa y and Δa z , and then rotated relative to fi xed axes parallel to axes x f and y f , respectively, by angles κ x and κ y . For this purpose, additional coordinate system h was introduced ( Figure 2).
Allowing for errors, the gear will rotate around the new shifted and skewed axis z h =z 2 . In the context of Figures 1 and 2, transformation matrices from system f are given as (1) and (2): where: matrices M h2 and M �h are given as (3) and (4): !" = Rot& # , * · Rot( $ , ) · Tr( % + Δ $ , ) · Tr&Δ # , * · Tr(Δ & , ) * · Rot( $ , ) · Tr( % + Δ $ , ) · Tr&Δ # , * · Tr(Δ & , ) Then, tooth surfaces in the fi xed coordinate system can be expressed as (5) and (6): This study discusses three types of gear pairs with helical teeth, wherein the pinion has a righthand and the gear has a left-hand fl ank pitch line: (a) The involute gear pair where: r b1 , r b2 are base radii of, respectively, the pinion and the gear, t 1 , t 2 are gear and pinion tooth profi le parameters (rolling angles), ζ 1 , ζ 2 are the parameters of the helical line of the pinion and the gear, β is the inclination angle of the fl ank pitch line on the pitch cylinder, r 1 and r 2 are the pitch radii of, respectively, the pinion and the gear [23].
where: θ 1 , θ 2 are profi le parameters, ζ 1 , ζ 2 are helix parameters for the pinion and the gear, respectively, α t is the face pressure angle, ρ 1 is the convex tooth profi le radius, ρ 2 is the concave tooth profi le radius, while r 1 and r 2 are pitch circle radii [7].

TOOTH FLANK SURFACE MODIFICATION
Tooth fl ank surface modifi cation is a procedure wherein the tooth fl ank is deliberately tilted away from the nominal surface. Its purpose is to compensate for tooth workmanship errors, as well as gear position errors in the transmission box arising from elastic deformations of shafts, or bearing clearances (slackness). Standard [24] distinguishes between various types of modifi cation, the basic of which include: tooth profi le modification, tooth fl ank line modifi cation, and topology modifi cation. Tooth profi le modifi cation C α (Figure 3a)) is generally implemented in order to reduce uneven load along the pressure segment. Tooth fl ank line modifi cation C β (Figure 3b)) can compensate for shaft defl ections. The application of both modifi cations C α +C β (Figure 3c)) makes it possible to avoid edge contact, a phenomenon which may lead to stress concentration. In principle, any modifi cation may be considered as an appropriately defi ned topological modifi cation (Figure 3d)). Usually, however, topological modifi cation C Σ is understood as free-form modifi cation -i.e. freely defi ned on the tooth surface grid.
The modifi cation is conveniently defi ned and presented on coordinate grid ! = !
where: index i=1,2 refers to the tooth surface of, respectively, the pinion and the gear.
If surface parameters are assigned to the profi le and the tooth fl ank line, it may also be defi ned on their grid θ i , ζ i . If the modifi cation was not included when generating the surface, e.g. by shaping the honing tool or its additional motions, it may be implemented with the use of relationship (13)  In such a situation, instead of the non-modifi ed surface, the analyses make use of the surface with a modifi cation described by vector (13). Note that a positive modifi cation is the modification deep into the material (towards the tooth) whereas the negative one is a modifi cation to the outside of the material (towards the tooth space).

TRANSMISSION ERROR AND EASE-OFF TOPOGRAPHY
A method whereby surfaces can be given as discrete was used in order to determine transmission error and Ease-Off topography. Its concept was presented in [25]. The method was designed to analyse bevel and hypoid gear pairs, and implemented in KIMoS, a computer-aided bevel and hypoid gear pair design system by Klingelnberg. This section discusses the authors' implementation of the method.
It assumes a perfect mesh between pinion tooth Σ 1 and gear tooth surface Σ 2 , i.e. one in which the gears are rotated by angles resulting from theoretical gear ratio !" = " / ! = ! / " . Surfaces in the n th position are shown in Figure 4.
It is assumed that pinion rotation axis z 1 is aligned with axis z f of fixed coordinate system f, and the surfaces in that system are given by vectors (3.1) and (3.2). In each position n resulting from the discretization of gear rotation angles, between the surfaces there is a certain distance measured along the arc with radius The distance between fl ank surfaces is expressed as (15)  In each n th position there is a certain minimum distance !"# $%& with its corresponding angle !"# $%& . The relationship between minimum angle !"# $%& and gear rotation angle φ 2 determined discretely for all n positions constitutes the transmission error. In addition, in a specific n th position the distribution of distances k kor on surface Σ 1 is referred to as an instantaneous Ease-Off. The Ease-Off is generated in the form of the lowest-lying envelope of instantaneous Ease-Offs for all n posi- .

GEOMETRIC CONTACT PATTERN
In a gear pair, the contact pattern is the area where contact takes place between mating tooth surfaces. It is created by the elastic deformation of teeth due to forces exerted in the meshing. A geometric contact pattern is defined as an approximation of the actual contact area based exclusively on the geometric properties of surfaces in contact. In other words, it is determined without taking into account the relationship between the deformation and forces applied or the type of material used.
There are many methods of determining the geometric contact pattern. They include, among others, methods based on mathematical models [26][27][28] and methods employing CAD systems [29,30]. The use of CAD systems involves the mutual penetration of three-dimensional tooth models to the depth corresponding to the thickness of the gear marking compound applied in gear pair tests, or the expected deformations. The contact pattern has the form of a flaky construct generated through the logical multiplication of thus positioned solid shapes. To visualise the movement of the contact pattern, the above operation must be performed for consecutive discrete positions arising from the rotation angles of gear solids. This approach was detailed in [31]. Its equivalent is the method which involves finding the curve of the intersection of tooth surfaces supplied in an analytical way [27]. Another method is the one proposed in [26]. The contact pattern is defined as the area for which the distance between the points on the surfaces in contact, measured along the common normal, is lower than the set amount. In order to determine the contact pattern, the method uses an approach similar to determining Hertz contact stress for point contact [32,33]. It is based only on the main curvatures of the surfaces, and the contact pattern resulting from the local approximation of surfaces is always elliptical [34]. The main drawback of this approach is that it is constrained to only local analysis, without taking into account e.g. edge contact. Moreover, it is of limited use for non-elliptical contact patterns which occur e.g. in gears with an eccentric-cycloidal profile [22] or Novikov gears with a concavo-convex profile [35]. The last method, described in more detail in this paper, is one in which contact pattern is determined as a set of points for which the distance between tooth flank surfaces, measured along unit normal to the pinion tooth flank, is lower than the set value [28]. The method makes it possible to determine edge and non-elliptical patterns, as verified experimentally for cylindrical Novikov gears [7] and eccentric-cycloidal meshing [22].
Let us consider a gear for which mating tooth surfaces Σ 1 and Σ 2 are given in a fixed coordinate system f by vectors (5) and (6). The contact pattern is determined on the basis of vector equation where: k -the distance between tooth flanks measured along unit normal to the pinion tooth surface. The above equation can be solved on the basis of distance k and gear tooth surface parameters θ 2 , ζ 2 for all discrete pinion tooth surface parameters θ 1 , ζ 1 . The points for which distance k is smaller than the assumed value form the contact pattern. Assuming that gears revolve around axis z, the contact pattern is conveniently presented as a projection on plane r y1 , z f , where !" = #( " ($) ) & + ( " ($) ) & is an arbitrary radius of the pinion. Solving vector equation (16) representing a system of three algebraic, usually non-linear equations, is problematic and time-consuming even if numerical methods are employed [36,37]. For this reason, it is recommended that the solution should be approximated by means of a geometric method. The author proposes a method ( Figure 6) wherein the gear tooth surface is presented as a grid of triangles, which can be accomplished by Delaunay triangulation [38].
Next, for each point on pinion tooth surface M 1 , we shall determine point M 2 , where the gear tooth surface triangle intersects with a straight line normal to the pinion tooth surface, which can be accomplished by the algorithm described in [39]. The points for which distance |M 1 M 2 | is lower than the set distance are plotted on the chart, forming the gear pair's contact pattern.

RESULTS AND DISCUSSION
The proposed computation methods were visualised based on cylindrical gear pairs with data listed in Table 1.
Parameters of individual profi les together with their transverse sections were shown in Table 2.
The simulations enabled us to obtain Ease-Off charts, transmission charts, and contact patters for the involute gear pair (Figure 7), Novikov gear pair (Figure 8), as well as eccentric-cycloidal ( Figure 9) gear pair. In all cases, the modifi cation which involves tooth fl ank line and profi le crowning allowed us to avoid interference, as displayed by the consistent distribution of Ease-Off topography with minimum values present in the expected tooth contact areas. Moreover, transmission error did not occur, and the shape of the transmission curve was parabolic. In all cases, contact patters were determined for gear marking compound thickness k=5μm. Their areas as a function of pinion rotation angle were juxtaposed in Figure 10.
Used modifi cation for involute gears results in a parabolic-like transmission chart (Figure 7b)), which is desirable and consistent with results obtained by other researchers [40,41]. Contact patterns are elliptical (Figure 7 c)) and also correspond to localized bearing contact for involute helical gearing [42]. In the case of the Novikov type of gearing, the transmission chart is fl attened (Figure 8b)) with parabolic segments at the beginning and at the end of meshing. This shape of the transmission chart results from applied lead modifi cation, which must be chosen not to reduce the overlap ratio (maintaining the constant transmission). Due to conformal contact Ease-Off plot (Figure 8a)) takes lower values than in the case of involute gearing (Figure 7a)). Contact patterns in Novikov gear are not elliptical and take the shape which is typical for high conformal gear pairs as reported in [20]. The transmission chart obtained for eccentric cycloidal gear mesh (Figure 9b)) is similar to that obtained for involute gearing (Figure 7b)). This similarity results from the fact that both of these gears are characterized by transverse and overlap ratios. Contact patterns are not elliptical (Figure 9c)). Its shape and location correspond to contact lines for this kind of gear mesh [21].
The largest contact area was found for the Novikov gear pair (A=8.5 mm 2 ), due to the convexo-concave tooth contact. In other gear pairs, maximum areas of contact were A=6.57 mm 2 for    involute and A=5.32 mm 2 for eccentric-cycloidal gear. In addition, a reduced contact ratio in the Novikov gear relative to the other two gear pairs can be observed, caused by the character of tooth contact in the pair. Linear contact in the involute gear and the eccentric-cycloidal gear pair implies the presence of both face and overlap contact ratio, while in the Novikov type (with point contact), meshing takes place only as a result of the overlap.

CONCLUSIONS
The simulations allowed us to draw the following conclusions. Tooth modification involving crowning the addendum and the flank line enables avoiding edge contact. A suitably chosen modification ensures consistent transmission (no transmission error). The Novikov gear pair was characterised by the largest contact area (approx. 30% greater than for the involute gear pair, and 60% greater than for the eccentric-cycloidal gear pair).
Furthermore, assuming that the pinion's rotation speed is n 1 =100 rpm, the maximum sliding velocities are as follows: 102 mm/s for the involute gear pair, 77 mm/s for the Novikov gear pair and 99 mm/s for the eccentric-cycloidal gear pair. The sliding velocity is lower nearly about 30% for Novikov gears. As it is known sliding velocity has a direct impact on the contact temperature which in turn is related to the wear of teeth.
Taking the above into account and assuming, for the sake of simplicity, that maximum contact stress is inversely proportional to the area of the contact ellipsis, the Novikov gear enables us to obtain a nearly 1.30-fold stress reduction. Therefore, we may posit a thesis that the Novikov gear pair will offer a greater surface load-carrying capacity (or durability arising from this capacity) than other types of meshing analysed in this paper, which may be tested in further research, both in the FEA environment and on test benches.