Issue 
Mechanics & Industry
Volume 22, 2021



Article Number  47  
Number of page(s)  11  
DOI  https://doi.org/10.1051/meca/2021044  
Published online  17 November 2021 
Regular Article
High load capacity spur gears with conchoidal path of contact
^{1}
E.O. Paton Electric Welding Institute of the National Academy of Sciences of Ukraine, 11 K. Malevycha st., Kyiv 03150, Ukraine
^{2}
National Aviation University, Department of Engineering Science, 1 L. Huzara ave., Kyiv 03058, Ukraine
^{3}
Bialystok University of Technology, Faculty of Mechanical Engineering, 45C Wiejska st., Bialystok 15351, Poland
^{*} email: y.tsybrii@pb.edu.pl
Received:
22
September
2020
Accepted:
13
October
2021
The present study is devoted to investigation of spur gears with a conchoidal path of contact and a convexconvex contact between teeth. The load capacity and energy efficiency were evaluated using both theoretical and experimental approaches. The theoretical analysis showed that the conchoidal gear pairs are 5–21% stronger in terms of contact stress and have similar energy efficiency as compared to the involute gear pairs of the same configuration. Experiments were conducted on a gear test rig. Its energy efficiency was determined by measuring the active power of the motor driving the pinion shaft and controlling the torque at the gear shaft. The load capacity of the tested gear pair was estimated by analysing changes in the energy efficiency. It was found that the conchoidal gear pair has more than 20% higher load capacity and slightly higher energy efficiency, which agrees well with the mentioned theoretical results. Thereby, the study concludes a substantially higher load capacity of the conchoidal gears compared to the traditional involute ones.
Key words: Spur gears / involute gears / conchoidal path of contact / load capacity / contact strength / energy efficiency
© P. Tkach et al., Published by EDP Sciences 2021
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
Gears are widely used to transmit motion and energy in mechanical systems. It is difficult to imagine a modern transport machine that does not include gear trains. The most commonly used gears have teeth with an involute active profile generated by straight lines. The set of instantaneous contact points between such teeth forms a straight path of contact. The advantages of involute gears include high manufacturability, relatively small sensitivity to centre distance errors and simplicity of the reference profile. However, relatively large curvatures of the mating teeth result in large contact stresses, which strongly limits the load capacity of involute gears. Therefore, development of noninvolute gear drives has a significant scientific and practical interest.
Helical gears with a circulararc tooth profile were proposed by Wildhaber [1] and Novikov [2], subsequently called ‘WildhaberNovikov gears’. Novikov's idea is that the contact between gears should take place at a point moving along a tooth. The active profile of one tooth is generated by a convex arc, while that of the mating tooth is generated by a concave arc of a slightly larger radius. Accordingly, the convexconcave contact between teeth is characterised by a reduced curvature, which leads to smaller contact stresses. But a serious disadvantage of WildhaberNovikov gears can be found in increased bending stresses in teeth due to the point contact. A comprehensive analysis of this type of gears was performed in the studies [3,4].
WildhaberNovikov concept of gearing served as the basis for the development of various circulararc tooth profiles. One of the successful technical solutions is the application of double circulararc tooth profile helical gears [5] (p. 158). The reference profile for generating teeth active profiles is composed of two conjugated circular arcs: convex addendum arc and concave dedendum arc. Accordingly, the convex addendums interact with the concave dedendums of the respective pinion and gear teeth. The parameters of two circular arcs are sought to minimise both contact stresses and bending stresses in teeth [6]. The other known technical solutions include (but are not limited to) gears with basic rack of combined circular and involute profile [7], circulararc tooth pinion on involute tooth gear [8], pointline meshing gears [9], circulararc curvilinear tooth gears [10], stepped triple circulararc tooth gears [11], double circulararc tooth gears modified by tooth end relief with helix [12], double circulararc tooth gears optimised for harmonic drives [13].
As concerns spur gears, various alternative tooth shapes were proposed to improve the load capacity of gear drives. It is important to name, for instance, gears with zero relative curvature at many contact points [14], gears with linear profile modification [15], gears with quadratic parametric tooth profile [16], tooth profile relief design gears [17], asymmetric tooth gears [18], gears designed by usage of NURB deviation function technique [19], gears designed by Bspline curve fitting and sweep surface modelling [20], gears with multisegment path of contact [21], cosine tooth gears [22], sgears designed for wind power turbine operating conditions [23], gears with parabolic path of contact [24], gears with a constant relative curvature [25], gears with a circular arc contact path [26]. Although the mentioned technical solutions were shown to be effective for particular applications and operating conditions, their widespread implementation is limited by the difficulties related to manufacturing complexshaped teeth.
A relatively simple double circulararc reference profile for spur gears was proposed in the study [27]. The path of contact of these gears represents a conchoid of Nicomedes. Therefore, they are sometimes referred to as ‘conchoidal pathofcontact gears’ or ‘conchoidal gears’. The study [28] investigated the conchoidal gears with convexconcave contact between teeth and concluded their better meshing and performance characteristics compared to the involute gears. Their disadvantage, however, is an increased sensitivity to manufacturing and mounting errors [29] (p. 33). The study [30] showed that the tooth profiles of the conchoidal gears can be designed to allow a convexconvex contact, which can potentially eliminate the mentioned above disadvantage. To the best authors' knowledge, this potentially promising type of conchoidal gears has not been investigated in terms of load capacity and energy efficiency yet. With this in mind, the purpose of the present study was to theoretically and experimentally investigate the load capacity of the conchoidal pathofcontact spur gears with convexconvex contact between teeth as compared to the traditional involute gears.
2 Theory
2.1 Geometry of the conchoidal gears
Figure 1 presents a geometric description of the conchoidal gears under consideration. Figure 1a shows the generating surface defined in a coordinate system x_{g}O_{g}y_{g}. Figure 1b introduces the pitch circles of the pinion 1 and gear 2 and the path of contact. Note that the coordinate systems x_{1}O_{1}y_{1} and x_{2}O_{2}y_{2} are aligned with the respective pinion 1 and gear 2, whereas XOY is the fixed coordinate system with centre O at the pitch point.
The generating surface is based on the reference profile shown in Figure 1c. The curves O_{g}A and O_{g}B form the active profile of a tooth, while and form the active profile of the adjacent tooth. The curve AA^{′} is responsible for generation of the tooth root fillet.
The coordinates of the tooth active profile are expressed separately for the curve O_{g}A(1)curve O_{g}B(2)curve (3)and curve (4)
Here α is the reference profile angle varying from α_{w} to α_{max}; α_{w} is the reference profile angle at the pitch line (point O_{g}); α_{max} is the maximum reference profile angle (points A and B); ρ is the profile circle radius; x is the reference profile shift coefficient; parameters a and b are given by
It is important to note that equations (1)–(4) and the following equations incorporating geometric parameters are normalised for the module m equal to 1 mm.
Assume that the reference profile is symmetrical. Then it is sufficient to consider the active profile formed only by the curves O_{g}A and O_{g}B. The coordinates of the active profile, given by equations (1) and (2), are represented in a concise form:(5)where the upper and lower signs stand for the curves O_{g}A and O_{g}B, respectively.
In the meshing of the generating surface and a tooth being generated, the vector of the relative velocity must be perpendicular to the line normal to the generating surface at the contact point [6]. As applied to the conchoidal gears, this condition allows writing the equation of meshing as(6)which represents the relationship between α and the angle ϕ_{1,2} of rotation of the pinion/gear. Here r_{w1,2} is the pitch circle radius of the pinion/gear. As the pitch circles mesh without sliding, it is true that ϕ_{1}r_{w1} = ϕ_{2}r_{w2}.
The equation of the path of contact can be formulated in the coordinate system XOY based on equations (5) and (6) in the formwhich coincides with the expression for the conchoid of Nicomedes with angle (α + π ± π/2) and parameters (x ∓ a) and ρ.
From equation (6) one can also derive the equations for the pinion active profiles(7)
and gear active profiles(8)in the respective coordinate systems x_{1}O_{1}y_{1} and x_{2}O_{2}y_{2}.
Equations (6)–(8) allow thus finding the coordinates of the contact point and the points of the active profiles for specified values of ϕ_{1} and ϕ_{2}.
Depending on the geometric parameters, the contact between teeth in a conchoidal gear pair can be of convexconcave or convexconvex type. The present study focusses on investigation of the convexconvex contact which takes place under the condition [30] that
The following analysis requires specification of the parameters of the simulated reference profiles and gear pairs. The angle α_{w} of 20° is the most common, while it can be 25° or even 28° in special gear transmissions [31] (p. 75, 143). Table 1 describes three reference profiles, codenamed as ‘α20’, ‘α25’ and ‘α28’, with α_{w} close to the mentioned values.
Figure 2 shows the teeth profiles of the pinion 1 and gear 2 generated by the reference profiles of Table 1. The profiles of the pinion 1 are determined due to equation (7) in the coordinate system x_{1}O_{1}y_{1} (solid lines), while the profiles of the gear 2 are determined due to equation (8) in the coordinate system x_{2}O_{2}y_{2} (dashed lines). In both cases, the values of ϕ_{1,2} are calculated by equation (6). For the sake of clarity, the profiles of the gear 2 are shown not in its own coordinate system but are meshed with the respective profiles of the pinion at ϕ_{1} = 0. The geometry of the teeth corresponds to u = z_{2}/z_{1} = 72/18 and is normalised for m = 1 mm.
The shift of the reference profile has a significant influence on the performance of the gear pair. Table 2 presents three simulated cases: zero shift (‘x0’), 0.3 shift (‘x0.3’) and 0.5 shift (‘x0.5’).
Finally, Table 3 presents the parameters of the simulated gear pairs and operating conditions. Two gear pairs, codenamed as ‘u2.4’ and ‘u4’, have the same parameters except for the number z_{1} of the pinion teeth and, accordingly, gear ratio u = z_{2}/z_{1}.
Fig. 1
Geometry of conchoidal gears: (a) generating surface; (b) pinion 1 and gear 2; (c) reference profile. 
Parameters of the simulated reference profiles normalised for m = 1 mm.
Fig. 2
Geometry of the teeth active profiles of the pinion 1 and gear 2. 
Shift coefficient x of the simulated reference profile normalised for m = 1 mm.
Parameters of the simulated gear pairs and operating conditions.
2.2 Contact strength
The instantaneous mechanical contact of two gear teeth is often treated as the contact between two cylinders with parallel axes. The relationship between the stress σ_{H} and normal force F_{n} in the contact zone is then defined by the Hertz equation. If the pinion and gear are made of the same steel with elastic modulus E, this equation takes the form [5](9)where R_{r} is the reduced curvature radius of the active profiles of the mating teeth at the contact point; b_{w} is the effective face width.
For the conchoidal gear pair, it yields from equations (5)–(8) that
On the other hand, following the study [28], one can derive a similar expression for the involute gear pair:
The normal force F_{n} in the contact zone is expressed via the pinion torque T_{1} as(10)where k = 1 for onepair contact and k = 0.5 for doubletooth contact.
Substitution of equation (10) into equation (9) leads to the equation(11)
where the coefficient θ given bydepends solely on the geometry of the active profiles. According to equation (11), θ is inverse proportional to σ_{H} and, thereby, can serve as the measure of contact strength.
Based on the expressions of R_{r con} and R_{r inv} above, one easily finds θ for the conchoidal gear pair: (12) and for the involute gear pair: (13)
Note that θ_{con} of equation (12) is a function of α, while θ_{inv} of equation (13) is a function of x_{g}.
Figure 3 illustrates the variation ranges of the ratio θ_{con}/θ_{inv} calculated by equations (12) and (13) for the reference profiles and gear pairs specified in Tables 1–3. The limit values of each variation range correspond to the limit points of the path of contact. It is seen that for all combinations under consideration, θ_{con}/θ_{inv} is noticeably larger than 100%. The lower limit of θ_{con}/θ_{inv} varies between 105% and 121%, implying that the conchoidal gear pairs have 5–21% higher contact strength compared to the involute gear pairs of the same configuration. An exceptional case is ‘α20 u4 x0’, implying α_{w} = 20°, u = 4 and x = 0, at which the upper limit of θ_{con}/θ_{inv} is about 200%. This case is characterised by a close to zero curvature radius at the tooth dedendum of the pinion and, therefore, is eliminated from consideration. It is notable that θ_{con}/θ_{inv} decreases substantially as α_{w} increases.
Fig. 3
Ratio between contact strength coefficients θ_{con} and θ_{inv}. 
2.3 Energy efficiency
The active profiles of the mating teeth slide relative each other at the contact point with velocity υ_{g}. This is inevitably accompanied by the friction process which is characterised by friction coefficient μ. Introduce the energy efficiency of a gear pair as follows:(14)where ω_{1} is the angular velocity of the pinion; ϕ_{11} and ϕ_{12} are the angles of the pinion rotation which correspond to the start and end of the meshing phase. The start contact point and end contact point in the meshing of the pinion and gear are defined in the generating process by the points of the reference profile with angle α_{max1} at ϕ_{1} = ϕ_{11} and angle α_{max2} at ϕ_{1} = ϕ_{12}, respectively. The values of α_{max1} and α_{max2} are found by the equations below:
The integral term in equation (14) indicates the fraction of the input power that transforms into friction heat over one meshing phase from ϕ_{11} to ϕ_{12}.
For the conchoidal gear pair, the sliding velocity is derived from equations (5) and (6) aswhile for the involute gear pair, it can be expressed in our notation as
For steelonsteel gear teeth contacts, the friction coefficient μ is well approximated by the empirical function [32,33](15)where BHN is the Brinell hardness number; R_{a} is the roughness parameter; ν is the oil kinematic viscosity; υ_{Σ} is the sum of the velocity components of the active profiles at the contact point perpendicular to the line of contact [31] (p. 182). The quantities in equation (15) should be specified in the following units of measurements: BHN and E in the same units; F_{n} in N; b_{w} in mm; R_{r} and R_{a} in cm; ν in mm^{2}/s; υ_{g} and υ_{Σ} in cm/s.
The formula to calculate υ_{Σ} for the conchoidal gear pair is obtained from equations (5) and (6) in the formwhile for the involute gear pair, it reads
Figure 4 shows the values of the ratio between the energy efficiencies η_{con} and η_{inv} calculated by equation (14) for the respective conchoidal and involute gear pairs. It is seen that η_{con} is up to 0.1% larger than η_{inv} for all considered combinations, suggesting that the conchoidal gear pairs have similar energy efficiency as the involute gear pairs of the same configuration. This result is explained by the fact that μ of equation (15) takes a smaller value in the case of conchoidal gear pair, which is due to the inequalities υ_{Σ con} > υ_{Σ inv} and R_{r con} > R_{r inv}.
Fig. 4
Ratio between energy efficiencies η_{con} and η_{inv}. 
3 Experimental study
The foregoing theoretical analysis showed that the conchoidal gear pairs have a substantially higher contact strength and similar energy efficiency if compared to the involute gear pairs with the same parameters. These findings should undergo experimental validation to be accepted. This section describes an experimental study of the conchoidal and involute gear pairs in terms of the mentioned characteristics.
3.1 Toothcutting tool
A hob cutter was designed and manufactured for cutting conchoidal gear pairs. It was made of P18 steel due to GOST 1926573 (EN X75WCrV1841) and heat treated to Rockwell hardness HRC 65. The parameters of the hob cutter are presented in Table 4, while its tooth profile is schematically shown in Figure 5. The parameters of the reference profile were calculated using the formulae given in Table 5.
Hob cutter parameters.
Fig. 5
Hob cutter tooth profile. 
Reference profile parameters.
3.2 Tested gear pairs
Using the hob cutter, 3 conchoidal gear pairs were manufactured. Further 3 involute gear pairs were manufactured using a standard hob cutter of the same module. All gears were cut out of 40X steel (EN 41Cr4) due to GOST 454371 364. Table 6 presents its mechanical properties. The conchoidal and involute gear pairs had the same configuration presented in Table 7. Note that they were manufactured with the reference profile shift corresponding to ‘x0.5’ (see Tab. 2).
The pinion and gear of the tested gear pair (conchoidal or involute) were installed on the shafts supported by bearings inside a gear box. An oil bath lubrication method with natural convection was used. The gears were lubricated by an AK15 motor oil. The level of oil in the gear box was regularly monitored.
Gear steel mechanical properties.
Gear pair parameters.
3.3 Gear test rig
The tests were carried out using a gear test rig shown in Figure 6. The principal units of the test rig were an electric motor, the gear box with the tested gear pair inside and an electromagnetic powder brake. The motor of rated power 15 kW applied torque T_{1} to the pinion shaft and drove it with angular velocity ω_{1} of 153.9 rad/s. The brake allowed controlling the torque T_{2} at the gear shaft by measuring displacements of an elastic element. The motor, gear box and brake were dead mounted to a massive basement with bolts. The shafts of the mentioned units were connected by flexible couplings. An accurate alignment of the shafts was provided. The measurement system of the test rig allowed measuring the active power P = T_{1}ω_{1} of the motor, the gear torque T_{2} and the temperature ϑ of the oil in the gear box.
Fig. 6
Gear test rig. 
3.4 Research methodology
Each gear pair was tested according to a multistep procedure developed in the Central Scientific Research Institute of Machine Building and Metalworking (former Soviet Union) intended for conventional gear transmissions, as described in Table 8.
Zero step was a 2hour runningin of the gear pair with the brake off (idle mode). At each next loading step, the gear torque T_{2} was increased so that the corresponding pinion torque T_{1} increased by 0.2[T_{1}], where [T_{1}] is the allowable pinion torque. The value of [T_{1}] was determined based on equation (9) with account of different factors:where [σ_{H}] is the allowable contact stress (see Tab. 6); K_{Hα} is the factor of nonuniform load distribution between the teeth; K_{Hβ} is the factor of nonuniform load distribution along the contact lines; K_{HV} is the factor of the internal dynamic load in the meshing; Z_{E} is the factor in view of the mechanical properties of the gears, ; Z_{H} is the factor in view of the shape of the mating teeth surfaces at the pitch point; Z_{ϵ} is the factor in view of the total length of the contact lines. In the equation above, [T_{1}] is in Nm, [σ_{H}] is in MPa, b_{w} and r_{w1} are in mm.
At the loading steps 1 to 4, the gear pair was tested for 8 hours per day, with total duration of 24 hours. Accordingly, the state of the active surfaces of the gears was examined after each 8 test hours. After the loading step 4, the used oil was drained from the gear box, the gear box and the components inside it were cleaned, and fresh oil was added. At the loading steps 5 and 6 with respective loads [T_{1}] and 1.2 [T_{1}], the gear pair was tested for 8 hours per day, with total number of pinion cycles of N_{H lim} and 1.5N_{H lim}, respectively. Here N_{H lim} is the base number of cycles (see Tab. 6).
The energy efficiency η_{e} of the test rig was determined from the known values of the motor active power P, pinion angular velocity ω_{1} and gear torque T_{2} as follows:
It is noteworthy that η_{e} represents the product of the energy efficiencies of all units of the test rig, including the gear box. Since the test rig operated under the same kinematic and loading conditions regardless of gear pair type, one can compare the energy efficiencies of the conchoidal and involute gear pairs based on the ratio between the values of η_{e con} and η_{e inv} obtained by testing the respective conchoidal and involute gear pairs.
Experimental multistep procedure.
4 Results and discussion
The experimental data are shown in Figures 7 and 8. Each point indicates the average stationary value obtained from testing 3 gear pairs at a specific loading step.
Figure 7 shows the dependence of the energy efficiencies η_{e con} and η_{e inv} on the pinion torque T_{1}. It is seen that η_{e con} is systematically 0.1–0.2% larger than η_{e inv}, implying that the conchoidal gear pair is slightly more energy efficient than the involute one. The value of η_{e inv} at T_{1} =37.2 Nm, however, is not in line with the general trend. An aftertest visual examination showed that the active surfaces of the involute gears were subjected to severe pitting, whereas the conchoidal gears were damaged insignificantly. This allows to explain the unexpected decrease in η_{e inv} at T_{1} = 37.2 Nm.
The data presented in Figure 7 suggest that T_{1} of about 31 Nm corresponds to the load capacity of the involute gear pair. On the other hand, the stable growth of η_{e con} allows stating that the load capacity of the conchoidal gear pair is as minimum as 37.2 Nm, which is 20% larger. This correlates well with the theoretical finding that the conchoidal gear pairs have 5–21% higher contact strength compared to the involute gear pairs of the same configuration.
Figure 8 shows the influences of T_{1} on the gear box oil temperatures ϑ_{con} and ϑ_{inv} measured for the respective conchoidal and involute gear pairs. According to the presented values, ϑ_{inv} is substantially larger than ϑ_{con}, i.e., the involute gear pair generates more friction heat than the conchoidal one. This confirms qualitatively the results of Figure 7. It is also seen that the increase in ϑ_{inv} is large at T_{1 }= 37.2 Nm, which is in agreement with the behaviour of η_{e inv}.
For the nominal mode of T_{1} = 31 Nm, ϑ_{con} is 17% smaller than ϑ_{inv} (see Fig. 8). For the sake of comparison, flash temperatures arising in the contact points of the tested gear pairs operating in the nominal mode were simulated based on Blok's formula [5]. The simulations show that as compared to the involute gear pair, the flash temperature in the conchoidal gear pair is 27% lower for the pinion tooth addendum on gear tooth dedendum contact, 56% lower for the pinion tooth dedendum on gear tooth addendum contact, and on average 20% higher for the teeth contact in the vicinity of the pitch point which is about 0.4m in height.
Fig. 7
Gear test rig energy efficiency η_{e} vs pinion torque T_{1}. 
Fig. 8
Gear box oil temperature ϑ vs pinion torque T_{1}. 
5 Conclusion
The conchoidal pathofcontact spur gears of convexconvex contact type were systematically investigated regarding their load capacity and energy efficiency. The theoretical analysis conducted for different values of the reference profile parameters and gear ratio showed that the conchoidal gear pairs are 5–21% stronger in terms of contact stress and have similar energy efficiency if compared to the traditional involute gear pairs of the same configuration. These findings were experimentally validated using a gear test rig in which the energy efficiency was determined by measuring the active power of the motor driving the pinion shaft for a controlled value of the torque at the gear shaft. The load capacity of the tested gear pair was estimated by analysing the behaviour of the energy efficiency under intensifying loading conditions. It was found that the load capacity of the conchoidal gear pair is higher by more than 20%, while its energy efficiency is slightly higher. A good agreement between the obtained theoretical and experimental results confirms a substantially higher load capacity of the conchoidal gears.
This research did not receive any specific grant from funding agencies in the public, commercial, or notforprofit sectors.
Notation
x: Reference profile shift coefficient
x_{g}O_{g}y_{g}: Coordinate system aligned with reference profile
N_{H lim}: Base number of cycles
R_{r}: Reduced curvature radius
[T_{1}]: Allowable pinion torque
α_{w}: Reference profile angle at pitch line
α_{max}: Maximum reference profile angle
η: Simulated energy efficiency of gear pair
η_{e}: Measured energy efficiency of gear test rig
θ: Contact strength coefficient
ρ_{a}: Tooth tip fillet radius
ρ_{f}: Tooth root fillet radius
[σ_{H}]: Allowable contact stress
ω_{1}: Pinion angular velocity
_{p}: Related to reference profile
_{0}: Related to toothcutting tool
_{1,2}: Related to pinion and gear
_{con}: Related to conchoidal gear pair
_{inv}: Related to involute gear pair
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Cite this article as: P. Tkach, P. Nosko, O. Bashta, Y. Tsybrii, O. Nosko, High load capacity spur gears with conchoidal path of contact, Mechanics & Industry 22, 47 (2021)
All Tables
All Figures
Fig. 1
Geometry of conchoidal gears: (a) generating surface; (b) pinion 1 and gear 2; (c) reference profile. 

In the text 
Fig. 2
Geometry of the teeth active profiles of the pinion 1 and gear 2. 

In the text 
Fig. 3
Ratio between contact strength coefficients θ_{con} and θ_{inv}. 

In the text 
Fig. 4
Ratio between energy efficiencies η_{con} and η_{inv}. 

In the text 
Fig. 5
Hob cutter tooth profile. 

In the text 
Fig. 6
Gear test rig. 

In the text 
Fig. 7
Gear test rig energy efficiency η_{e} vs pinion torque T_{1}. 

In the text 
Fig. 8
Gear box oil temperature ϑ vs pinion torque T_{1}. 

In the text 
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