Multi-objective optimization of hypoid gears to improve operating characteristics

Mechanism and Machine Theory 146 (2019) 103727

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Research paper

Multi-objective optimization of hypoid gears to improve operating characteristics Vilmos V. Simon Faculty of Mechanical Engineering, Department for Machine and Product Design, Budapest University of Technology and Economics, ˝ Muegyetem rkp. 3, H-1111 Budapest, Hungary

a r t i c l e

i n f o

Article history: Received 7 September 2019 Revised 29 November 2019 Accepted 29 November 2019

Keywords: Multi-objective optimization Hypoid gear Operating characteristics Genetic algorithm

a b s t r a c t In this paper a multi-objective optimization method of hypoid gears correlating to the operating characteristics is presented. Optimal design of hypoid gears demands that multiple objectives be simultaneously achieved. Four objectives considered in this study are the minimization of the maximum tooth contact pressure, transmission error and the average temperature in the gear mesh, and the maximization of the mechanical efficiency of the gear pair. The goals of the optimization are achieved by the optimal modification of meshing teeth surfaces. In practice, these modifications are introduced by applying the appropriate machine tool setting for the manufacture of the pinion and the gear, and/or by using a tool with an optimized profile. The proposed optimization procedure relies heavily on the loaded tooth contact analysis for the prediction of tooth contact pressure distribution and transmission errors, and on the mixed elastohydrodynamic analysis of lubrication to determine temperature and efficiency. A fast elitist nondominated sorting genetic algorithm (NSGA-II) is applied to solve the model. The effectiveness of the method is demonstrated by using hypoid gear examples. The obtained results have shown that by the optimization considerable improvements in the operating characteristics of the gear pair are achieved. © 2019 The Author. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license. (http://creativecommons.org/licenses/by-nc-nd/4.0/)

1. Introduction More strength, less transmission error, higher efficiency and lower temperature in the gear mesh are major demands in the design of hypoid gear transmissions. These goals can be achieved by introduction of optimal tooth surface modifications. In practice, these modifications are introduced by applying the appropriate machine tool setting for the manufacture of the pinion and the gear, and/or by using a tool with an optimized profile. Therefore, the main goal of this study is to systematically define optimal tool geometry and machine tool settings to simultaneously minimize tooth contact pressure, angular displacement error of the driven gear and average temperature in the gear mesh, and to maximize the efficiency of the gear pair. During the last decades many research works have been directed towards the design and manufacture of spiral bevel and hypoid gears with optimal tooth surface modifications to reduce the maximum tooth contact pressure and transmission error [1–43]. The most relevant papers to the present work are as follows: Ding et al. [34] propose a novel multi-objective correction of machine settings correlating to the loaded tooth contact performance using nonlinear interval optimization algorithm for spiral bevel gears. The research [35] deals with the multi-objective optimization of gear tooth E-mail address: [email protected] https://doi.org/10.1016/j.mechmachtheory.2019.103727 0094-114X/© 2019 The Author. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license. (http://creativecommons.org/licenses/by-nc-nd/4.0/)

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V.V. Simon / Mechanism and Machine Theory 146 (2019) 103727

Nomenclature a c e f g ig1 p r pro f 1 , r pro f 2 rt1 , rt2 T Taverage

β δ φ2 ηe f f φ1 , φ2 ψ

pinion offset, mm sliding base setting, mm basic radial, mm basic machine center to back increment, mm basic offset, mm velocity ratio in the kinematic scheme of the machine tool for the generation of the pinion tooth surface tooth contact pressure, oil film pressure, Pa radii of the circular arc head-cutter profile, mm radii of the head-cutter for pinion and gear finishing, mm oil film temperature, K average temperature, K tilt angle, deg swivel angle, deg angular displacement error of the driven gear, arcsec efficiency of the gear pair rotation angles of the pinion and the gear rolling through mesh, deg cradle angle, deg

surface to help design tooth corrections in order to simultaneously optimize several objective physical quantities. Wang et al. [36] propose a methodology for optimizing the loaded contact pattern of spiral bevel and hypoid gears by a surrogate Kriging-based model. The paper [37] presents a six sigma (6σ ) robust multi-objective optimization of machine-tool settings for hypoid gears having higher quality requirements. In the work [38] Artoni details an algorithmic framework inspired by deterministic multi-objective optimization methods, specially combined with a direct-search global optimization algorithm to obtain globally Pareto-optimal solutions. In the paper published by Mogal and Wakchaure [39] attempt has been made to optimize worm and worm wheel with multiple objectives, which takes gear ratio, face width of worm and worm wheel and pitch circle diameters of worm and worm wheel as design variables by using the Genetic Algorithm (GA). A novel fitness predicted genetic algorithm is developed by Qui et al. [40] to improve the herringbone gear performance over a wide range of operating conditions. Regular mechanical and critical tribological constraints (scuffing and wear) are optimized by Patil et al. [41] to obtain a Pareto front for the two-stage gearbox using a specially formulated discrete version of non-dominated sorting genetic algorithm (NSGA-II). With the help of the genetic algorithm search technique the optimal design of the dynamic load sharing performance for an in-wheel motor planetary gear reducer is completed by Zhang et al. [42]. The research paper published by Chandrasekarana et al. [43] aims to optimize the design of a pair of spiral bevel gears, using NSGA-II, a nondominated sorting genetic algorithm for optimization of multiple objective functions. In no conformal contacts, such as is in gears, forces are transmitted through a thin film of lubricant, which separates the two solid mating components. However, under usual operating conditions (high load), the thin film of lubricant is not sufficient to completely separate the tooth surfaces, and the asperities on opposing surfaces come in contact. This is the regime of mixed elastohydrodynamic lubrication (mixed EHL) in which the applied load is shared by the asperities and the lubricant film. Predicting performance of gears operating in the mixed EHL regime is of significant importance because the asperity contacts give rise to high local pressures, which can significantly lower the fatigue life of the gears. At the same time, the asperity contact friction and lubricant shearing in mixed lubricated contacts generate heat, which results in extreme local temperature raise. All these factors have big influence on the efficiency of the gear pair and on the temperatures in the gear mesh. The early models of mixed lubrication were developed based on a stochastic approach [44]. In the last decades many theoretical and experimental research works were directed towards the more sophisticated deterministic model of mixed elastohydrodynamic lubrication. Only some of the related papers are referenced [45–63]. The first paper on the full thermo-elastohydrodynamic lubrication analysis of gears was published in 1981 [64]. Later, many research works were directed towards the EHL analysis in different types of gears [65–80]. The investigations have shown that in the case of full elastohydrodynamic lubrication, only a relatively small load (torque) can be transmitted. By applying a torque usually employed in gear pairs, mixed elastohydrodynamic lubrication appears. Recently, a considerable number of papers were published on mixed EHL in different types of gears. Some of them are referenced [81–100]. A multi-objective optimization method of hypoid gears correlating to the operating characteristics is presented. Optimal design of hypoid gears demands that multiple objectives be simultaneously achieved. Four objectives considered in this study are the minimization of the maximum tooth contact pressure, transmission error and the average temperature in the gear mesh, and the maximization of the mechanical efficiency of the gear pair. The goals of the optimization are achieved by the optimal modification of meshing teeth surfaces. In practice, these modifications are introduced by applying the appropriate machine tool settings for the manufacture of the pinion and the gear, and/or by using a tool with an optimized profile. The proposed optimization procedure relies heavily on the loaded tooth contact analysis for the prediction of tooth

V.V. Simon / Mechanism and Machine Theory 146 (2019) 103727

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Fig. 1. Machine tool setting for pinion teeth finishing.

contact pressure distribution and transmission errors, and on the mixed thermal elastohydrodynamic analysis of lubrication to determine temperature and efficiency. A fast elitist nondominated sorting genetic algorithm (NSGA-II) is applied to solve the model. 2. Machine tool settings for the manufacture of face-milled hypoid gears A face-milled hypoid gear pair with the generated pinion and the non-generated gear is considered. The pinion is the driving member. In order to improve the operating characteristics of the gear pair and to reduce the sensitivity of the gear pair to tooth errors and to the mutual position of the mating members appropriately chosen modifications are introduced into the teeth of the pinion. As a result of these modifications theoretically point contact of the meshed teeth surfaces appears instead of linear contact. The modifications are introduced by the variation in machine tool settings and in the tool geometry. The machine tool settings used for pinion tooth finishing are specified in Fig. 1: sliding base setting (c), basic radial (e), basic machine center to back increment (f), basic offset (g), tilt angle (β ), and swivel angle (δ ). The other manufacture parameters are the velocity ratio in the kinematical scheme of the machine tool for the generation of the pinion tooth surface (ig1 ), the radius of the tool (rt1 ), and the radii of the tool profile (r pro f 1 , r pro f 2 , Fig. 2). The tooth surface of the pinion is defined by the following system of equations:



r0(1 ) = M p0 · M p4 (ig1 ) · M p3 (c, f, g) · M p2 (e ) · M p1 (β , δ ) · rT(T11) rt1 , r pro f 1 , r pro f 2 v0(T 1,1) · e0(T 1) = 0



(1a) (1b)

(T ) where rT 1 is the radius vector of tool surface points, matrices M p0 , M p1 , M p2 , M p3 , and M p4 provide the coordinate trans1

formations from system KT 1 (rigidly connected to the cradle and tool T1 ) to the stationary coordinate system K0 . The second equation describes mathematically the generation of pinion tooth surface by the tool; v0(T 1,1) is the relative velocity vector

of the tool to the pinion and e0(T 1) is the unit normal vector of the tool surface. The position vector of the formed gear tooth surface points is obtained by a simple coordinate transformation of vector rT(T2 ) from system KT 2 (rigidly connected to the tool T2 ) into the to the stationary coordinate system K0 , as it follows 2

r0(2) = Mg0 · Mg1 · rT(T22)

(2)

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V.V. Simon / Mechanism and Machine Theory 146 (2019) 103727

Fig. 2. Curved tool profile.

The matrices and vectors of Eqs. (1) and (2) are defined in [101] and [102]. 3. Loaded tooth contact analysis The proposed optimization procedure relies heavily on the loaded tooth contact analysis for the prediction of the tooth contact pressure and the angular displacement error of the driven gear. The loaded tooth contact analysis model employed was developed by the author of this paper. In the applied loaded tooth contact analysis it is assumed that the point contact under load is spreading over a surface along the “potential” contact line, which line is made up of the points of the mating tooth surfaces in which the separations of these surfaces are minimal, instead of assuming the usually applied elliptical contact area. The separations of contacting tooth surfaces are calculated by applying the full theory of tooth surface generation in face-milled hypoid gears [64]. The bending and shearing deflections of gear teeth, the local contact deformations of mating surfaces, gear body bending and torsion, the deflections of supporting shafts, and the manufacturing and alignment errors of mating members are included. The method is fully described in [103]. 4. Model of mixed thermal elastohydrodynamic lubrication analysis in hypoid gears The modeling methodology of mixed gear lubrication consists of two major components. They are (i) a gear load distribution model to determine the load sharing among the tooth pairs instantaneously in contact and the normal tooth force distribution along the tooth surface, (ii) the mixed EHL model to predict the transient pressure p(x,y,t), temperature T(x,y,z,t), and shear τ (x,y,t) distributions along the contacting tooth surfaces. The load distribution model employed is described in the previous section. The governing equations of the mixed EHL analysis are the Reynolds, energy, Laplace’s and elasticity equations. 4.1. The Reynolds equation In the analysis of mixed lubrication of gears, the contact zone is divided into two different types of area: the hydrodynamic (EHL) regions where the two surfaces are separated by the lubricant film, and the asperity contact regions where the two surfaces are in direct contact. In the hydrodynamic region the pressure at the interface between the two tooth surfaces is governed by the transient generalized Reynolds equation for point contact. This type of equation takes into account the viscosity and density variations along and across the lubricant film for the purpose of thermal analysis. The general transient Reynolds equation for point contact is

     ∂ F  ∂ ∂p ∂ ∂p ∂  F3 ∂ (ρ · h ) 3 F2 + F2 =− (U1 − U2 ) − (V1 − V2 ) + ρ · (W1 − W2 ) + ∂x ∂x ∂y ∂y ∂ x F0 ∂ y F0 ∂t

(3)

The effective viscosity, η∗ , describes the non-Newtonian lubricant properties which can be calculated as follows:

η∗ =

η · ( τe / τ0 ) ; τe = sinh (τe /τ0 )



τx2 + τy2

(4)

The viscosity variation with respect to pressure and temperature and the density variation with respect to pressure are included:

η = η0 · eαη ·p−βη ·(T −Tη0 ) ; ρ = ρ0 · 1 +

α1 · p

1 + β1 · p

(5)

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In the viscosity-pressure relationship Eq. (5) the exponent αη is constant in the case of Barus equation and it is pressure dependent in the Roelands’ expression:

αη =

ln (c1 · η0 ) · p



1+

p c2

z



−1

(6)

The unified lubrication-contact approach in mixed lubrication uses the same Reynolds equation in both hydrodynamic and contact regions. Within the contact regions, where h = ε (ε is a constant with very small value, usually ε = 2.5 nm), the Reynolds equation is reduced to the following form:

(U1 − U2 ) ·

∂h ∂h + (V1 − V2 ) · =0 ∂x ∂y

(7)

4.2. Film thickness equation The instantaneous lubricant film thickness, h, or the gap between two rough surfaces, is calculated by the geometric equation given below

h(x, y, t ) = h0 (t ) + s(x, y, t ) − δ1 (x, y, t ) − δ2 (x, y, t ) + d (x, y, t )

(8)

where h0 (t ) is the minimal distance of the surfaces, s(x, y, t ) is the geometrical separation of the contacting teeth surfaces (determined by the real shape of the pinion and gear tooth surfaces), δ1 (x, y, t ) and δ2 (x, y, t ) denote the roughness amplitude of surface 1 and 2, respectively. The tooth surface deformation, d, is calculated by ymax xmax  

d (x, y, t ) = Kd ·

xmin ymin

where Kd = π1 · (

1−μ21 E1

+

1−μ22 E2

p ( ξ , ς ) + pc ( ξ , ς ) · dξ · dς h (x − ξ )2 + (y − ς )2

(9)

), ph is the hydrodynamic pressure and pc the asperity contact pressure.

4.3. Temperature rise equation The tooth temperature is higher than that of the surrounding air-oil mixture because of the frictional heating in the oil film and at the asperity contacts. In mixed lubrication, the heat generates from shear film and asperities contacts. 4.3.1. The temperature rise in the EHL oil film - the energy equation The thermal analysis in the oil film is based on the solution of the energy equation in the hydrodynamic films and heat conduction equation in solids, simultaneously, along with other governing equations. The temperature and the viscosity are varying in the direction of the film thickness. The energy equation for the lubricant film is



  2  ∂T ∂T ∂T ∂ T ∂ 2T ∂ 2T cP · ρ · +u· +v· − k0 · + + ∂t ∂x ∂y ∂ x2 ∂ y2 ∂ z 2

   2  2  ∂p ∂p ∂p ∂u ∂v = αT · T · +u· +v· + η∗ · + ∂t ∂x ∂y ∂z ∂z

(10)

The equation governing the heat transfer in the pinion and gear teeth is Laplace’s equation

∂ 2 Tm ∂ 2 Tm ∂ 2 Tm + + =0 ∂ x2 ∂ y2 ∂ z2

(11)

where m = 1 for the pinion tooth, m = 2 for the gear tooth. 4.3.2. The temperature rise at asperity contacts The flash temperature rise at each contacting asperity spot can be calculated as

T f 1 =

2·q·l · k0



1

(12)

π · (1 + P e )

where q = μ · p · VS is the heat generated per unit area of contact, further μ is the coefficient of friction, p is the contact 2

2

pressure at the contact area, and VS is the relative sliding velocity, VS = (U1 − U2 ) + (V1 − V2 ) . k In Eq. (12) Pe = 2V··lκ is the Péclet number, where κ = ρ ·c0p is the thermal diffusivity of the solid, V is the velocity of the heat source relative to the contact area, l is the half length of the contact zone, k0 is the thermal conductivity of the solid, ρ is the density, and c p is the specific heat.

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V.V. Simon / Mechanism and Machine Theory 146 (2019) 103727

4.4. Power losses The mechanical power loss at each gear tooth contact is caused by the viscous shear within the lubricated areas of the contact and by the contact friction due to any direct asperity interactions. The power loss due to viscous friction across the fluid film at position (x,y) and time t is determined by integrating the product of the local shear stress τEHL and sliding velocity along the film thickness as

PlossEHL (x, y, t ) = AEHL ·

h  0

 ∂u ∂v + τEHLy (x, y, z, t ) · · dz τEHLx (x, y, z, t ) · ∂z ∂z

(13)

Within the asperity contact regions, the shear stress due to sliding is defined as τasp (x, y, t ) = μs · p(x, y, t ), and the corresponding friction power loss is

Plossasp (x, y, t ) = Aasp · μs · p(x, y, t ) · (U2 − U1 + V2 − V1 )

(14)

In Eqs. (12) and (14), AEHL is the area of the hydrodynamic contact between the mating flanks and Aasp is the area of the asperity contact. The total instantaneous mechanical power loss over the entire contact area with both the fluid and asperity contact regions is then found as

Ploss (t ) =

N EHL 

PlossEHL (x, y, t )+

i=1

Nasp 

Plossasp (x, y, t )

(15)

j=1

where NEHL is the number of the discretized elements of the computational domain where the fluid film is maintained between the two surfaces and Nasp is the number of the discretized elements where actual asperity contacts take place. The efficiency of the gear pair is defined as

ηe f f =

Power − Ploss Power

(16)

where Power is the transmitted power by gear pair. The details of method for mixed elastohydrodynamic analysis of lubrication in hypoid gears developed by the author of this paper are presented in [104] and [105]. 5. Multi-objective optimization model A multi-objective optimization model is developed to systematically define optimal head-cutter geometry and machine tool settings simultaneously minimizing maximum tooth contact pressure, angular displacement error of the driven gear and average flash temperature and maximizing the efficiency of the gear pair. The proper manufacture variables, objective functions, and constraints are as follows. 5.1. Manufacture variables The following machine tool setting and tool geometry parameters are taken as the basis of the proposed optimization formulation (specified in Fig. 1): sliding base setting (c), basic radial (e), basic machine center to back increment (f), basic offset (g), tilt angle (β ), and swivel angle (δ ). The other parameters are the velocity ratio in the kinematics scheme of the machine tool for the generation of the pinion tooth surface (ig1 ), the radius of the tool (rt1 ), and the radii of the tool profile (r pro f 1 , r pro f 2 ). Therefore the vector of parameters is



mp = c, e, f, g, β , δ, ig1 , rt1 , r pro f 1 , r pro f 2



(17)

5.2. Objective functions and constraints The goal of the optimization is to minimize tooth contact pressure, transmission errors and fluid film average temperature and to maximize the efficiency of the gear pair while keeping the loaded contact pattern inside the physical tooth boundaries of the pinion and the gear. The applicable objective functions can be expressed as

f1 (mp) = pmax (mp);

f2 (mp) = φ2 max (mp);

f3 (mp) = Taverage (mp);

f4 (mp) = ηe f f (mp)

(18)

The objective functions are as follows f1 (mp) is the full (3) or reduced (7) Reynolds equation for maximum tooth contact pressure calculation, f2 (mp) is the equation of the angular displacement error of the driven gear from the loaded tooth contact analysis [103]:

φ2 max = φ2(d ) + φ2(k) =

yn |(r × a 0 ) · e| k · + φ2( ) rD |r|

(19)

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where φ2(k ) is the difference in the angular displacement of the driven gear caused by tooth surface modifications, yn is the composite displacement of contacting tooth surfaces in the direction of the unit tooth surface normal vector e, r is  0 is the unit vector in the direction of the gear axis and rD is the distance of the the position vector of the contact point, a contact point to the gear axis. f3 (mp) is the energy Eq. (10) in the EHL region and the flash temperature rise Eq. (12) in contacting asperity spots for the calculation of the average temperature, f4 (mp) is Eq. (16) for the calculation of the efficiency. The proper constraints are due to the requirements that the contact pattern remains inside the possible contact area defined by load distribution calculation and inside the physical tooth boundaries of the pinion and the gear. It leads to the requirement that the contact load outside the instantly possible contact area should be zero. As it was mentioned earlier, the load distribution calculation method applied is based on a new approach, it is assumed that the theoretical point contact of teeth surfaces under load spreads over a surface along the whole or part of the “potential” contact line made up of the points of the meshing teeth surfaces in which the geometrical separations of these surfaces are minimal. In every iteration cycle a search for the points of the “potential” contact lines that are in instantaneous contact is performed. For these points the following condition should be satisfied (based on the loaded tooth contact analysis [103])

φ2 − φ2(k(i)t ) yn(it ,iz ) ≤  |(r×a0 )·e|  rD ·|r|

(20)

(it ,iz )

where it is the identification number of contacting tooth pair, iz is the point of the instantaneous contact line. In the points of the potential contact lines that are not in instantaneous contact a single variable C is initialized to zero. Therefore, the constraint can be simply denoted by

C (mp) = 0

(21)

where C is the total of tooth surface points with instantaneously not existing contact loads. Therefore, it depends on the tooth surface topography through the manufacture parameters mp. The optimization problem to be solved is as follows:

min [ f1 (mp), f2 (mp) f3 (mp)] ; mp

max [ f4 (mp)] mp

(22)

subject to C (mp) = 0 6. Computational procedure This optimization process is an iterative process that involves constantly changing the manufacture parameters. The objective functions are built by loaded tooth contact analysis and mixed elastohydrodynamic lubrication analysis. Therefore, it is impossible to establish an exact analytical expression between the optimization variables and the objective functions. The often applied for solving a multi-objective optimization problem is to multiply the objectives by different weighting coefficients and add linearly, thus transforming it into a single optimization problem [43]. However, this approach has many disadvantages, including strong subjectivity, partial optimization, and the mutual restriction of each objective. In this paper, a fast elitist nondominated sorting genetic algorithm (NSGA-II) [101] is applied to solve the above optimization model. The optimization process can be mainly divided into the following steps: (a) The initial population is formed based on the randomly initialized manufacture parameters within a predesigned range, and the objective functions are calculated. This calculation consists of the following steps: In the first step for any instantaneous meshing position, defined by the position angle of the pinion, φ1 , the load distribution is calculated to determine the load sharing among the tooth pairs instantaneously in contact, the normal tooth force distribution along the contact area, and the angular displacement error of the driven gear. In the second step the mixed thermal EHL calculation is performed. The Reynolds and energy equations are coupled through changes in the lubricant rheological state (density and viscosity), dependent on the pressure and temperature. A full numerical solution using finite difference form of the Reynolds and energy equations is applied to calculate the pressure and temperature distributions in the oil film. Lubricant film temperature alters three-dimensionally, thus a three-dimension grid mesh is applied in the fluid film. The (x,y,z) reference frame is attached to the contact zone and moves with it as the gears roll in mesh. The x and y axes denotes the directions along the oil film and the z axis is in the direction of the oil film thickness. The computational domain is torque load dependent and defined by maximum Hertzian half width amax as of −2.5 · amax ≤ x ≤ 2.5 · amax and by the length of the contact zone l as of 0 ≤ y ≤ l. It is discretized into Nx × Ny × Nz grid elements. The x and y increments are kept constant and sufficiently small to capture the measured surface roughness variation accurately. An iterative procedure is used to make the combined load distribution and EHL calculations. The flowchart of the numerical procedure is presented in Fig. 3. As it can be seen, the calculation is performed for 31 instantaneous positions of the gear pair through a mesh cycle. The pressure distribution obtained by the loaded tooth contact analysis is used as the initial pressure distribution in the oil film. By applying the Reynolds and energy equations the pressure and temperature

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V.V. Simon / Mechanism and Machine Theory 146 (2019) 103727

Fig. 3. Flowchart of the algorithm of the computer program.

distributions in the oil film are calculated. These equations are coupled by the pressure and temperature dependent oil density and viscosity. The iteration procedures are repeated until the prescribed precision of the solutions for the pressure and temperature distributions are achieved. The final pressure and temperature distributions are applied to calculate the friction forces in the oil and at the asperity contacts and the corresponding efficiency of the gear pair. (b) The child population is generated based on the parent population via the selection, the crossover and the mutation. Then the new population is generated by combining the parent and the child populations. In the next step the nondominated sorting of the objective functions is performed and the crowding distance is calculated. The new population is generated by selecting the better individuals, and then the next generation is set. If the maximum evolution generation is reached, the algorithm will be terminated, otherwise, the procedure will be repeated until the maximum evolution generation is reached. In this optimization problem, the objectives under consideration conflict with each other. Hence, optimizing the functions f1 (mp), f2 (mp), f3 (mp), f4 (mp) with respect to a single objective will cause unacceptable results with respect to the other objectives. Therefore, a perfect multi-objective solution that simultaneously optimizes each objective function is impossible. A reasonable solution is obtained by the investigation a set of solutions, each of which satisfies the objectives at an acceptable level without being dominated by any other solution. The set of such solutions constitutes the Pareto optimal set and the corresponding objective function values constitute the Pareto front. The goal of this multi-objective optimization algorithm is to identify solutions in the Pareto optimal set.

V.V. Simon / Mechanism and Machine Theory 146 (2019) 103727

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Table 1 Pinion and gear design data. Design parameters

Pinion

Gear

Number of teeth Module, mm Running offset, mm Blade angle, deg Mean spiral angle, deg Face width, mm Pitch diameter, mm Outside diameter, mm Pitch angle, deg

11 4.414 25.4 10 50.2597 31.911

41

77.585 18.5400

31 32.3007 27.762 180.975 181.859 70.5799

Table 2 Lubricant characteristics and operating parameters. Ambient lubricant viscosity, Pas Pressure viscosity exponent, Pa−1 Temperature viscosity exponent, K−1 Supplied oil temperature, C Transmitted torque, Nm Pinion’s revolution per minute, rpm

0.19361 0.14504·10−7 0.027 50 80 2000

Table 3 The number of tooth pairs instantaneously in contact through a mesh cycle influenced by the transmitted torque. T[Nm]

Nt

Steps

Nt

Steps

Nt

Steps

1 5 10 20 40 60 80 110 150

2 2 2 2 2 2 3 3 3

1–5 1–11 1–18 1–18 1–18 1–18 1–6 1–6 1–6

1 1 1 1 1 1 2 2 2

6–29 12–29 19–29 19–29 19–29 19–29 7–29 7–29 7–29

2 2 2 2 2 2 3 3 3

30–31 30–31 30–31 30–31 30–31 30–31 30–31 30–31 30–31

7. Results A computer program was developed to implement the formulation provided above. By using this program the multiobjective optimization is carried out for a face-milled hypoid gear pair. The main design data of the example gear pair used in this study are given in Table 1. The lubricant characteristics and operating parameters of the gear pair are presented in Table 2. By the use of multi-objective optimized manufacture parameters optimal tooth surface modifications are introduced causing theoretically point contact of the meshing tooth surfaces. From one up to three tooth pairs are instantaneously in contact through the mesh cycle. The number of tooth pairs instantaneously in contact depends on the instantaneous rotational position of the gear pair and on the transmitted torque (load). The number of tooth pairs instantaneously in contact (Nt ) for different transmitted torque values is shown in Table 3 when the mesh cycle is divided into 31 discrete steps of the pinion rotation angle. It can be observed that by transmitting torque of 80 Nm, three tooth pairs are instantaneously in mesh at the input and output part of the meshing cycle. In the middle of the mesh cycle, two tooth pairs are instantaneously engaged. The tooth contact pressure distribution and the values of the relevant operating characteristics are shown in Fig. 4, for the case when the usually used design method (calculation method) for hypoid gears is applied and no optimal tooth surface modifications are introduced. This figure is plotted for the rotational position of the gear pair when the transmitted load is shared with two tooth pairs and the instantaneous contact point of one of the pairs is in the middle point of the tooth surface. It was mentioned afore that optimizing the functions f 1 (mp), f2 (mp), f3 (mp), f4 (mp) with respect to a single objective will cause unacceptable results with respect to the other objectives. The individual optimal values of the functions and the corresponding values of the other three functions are presented in Table 4. It can be seen that the optimal values of individual functions are coupled with unacceptable values of the other three functions. In the case of the optimization of machine tool settings in regard to the other individual operating characteristics (φ2 max , Taverage , ηe f f ) edge contact occurs with very big maximum tooth contact pressures. The fast elitist nondominated sorting genetic algorithm (NSGA-II), described above, is applied to find the optimal solution.

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V.V. Simon / Mechanism and Machine Theory 146 (2019) 103727

Fig. 4. Tooth contact pressure distribution and the values of operating characteristics when no optimal tooth surface modifications are introduced. Table 4 The single optimal values of the functions. pmax [MPa]

φ2 max [arcsec]

Taverage [0C]

ηe f f

667(min.) 8109 5024 5049

25.49 5.61(min.) 23.45 21.54

101.49 71.60 66.79(min) 99.34

0.9281 0.9044 0.9222 0.9734(max)

Fig. 5. Pareto solution P1.

Fig. 6. Pareto solution P2.

The individual influence of the ten design variables on the load distribution and mixed EHL parameters is investigated to make the proper selection of the parameters of the genetic algorithm. On the basis of the obtained results the following bound constrains were set:

15.0 ≤ c [mm] ≤ 18.5, 70.2 ≤ e [mm] ≤ 73.5, −1.1 ≤ f [mm] ≤ −0.5, 23.0 ≤ g [mm] ≤ 27.4, 73.9 ≤ rt1 [mm] ≤ 77.0, 17.2 ≤ β [deg] ≤ 29.0, −37.0 ≤ δ [deg] ≤ −29.5, 3.35 ≤ ig1 ≤ 3.65, 500 ≤ r pro f 1 [mm] ≤ ∞ (straight ) 500 ≤ r pro f 2 [mm] ≤ ∞ (straight ). The parameters of the NSGA-II algorithm were: The population size 100, crossover probability 0.9, and mutation rate 0.4. The algorithm is stopped if maximum number of iterations exceeded 100. A single LTCA analysis takes a relatively small CPU time of 46 s by applying the method presented in [103], in which the tooth deformations are calculated by equations obtained by regression analysis based on FEM results. The total CPU time of the multi-objective optimization takes 26 h.

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Fig. 7. Pareto solution P3.

Fig. 8. Pareto solution P4.

Fig. 9. Pareto solution P5.

Fig. 10. Pareto solution P6.

A reasonable solution (Pareto solution) is obtained by the investigation a set of solutions, which satisfies the objectives at an acceptable level without being dominated by any other solution. The set of these solutions constitutes the Pareto optimal set. The solutions belonging to the corresponding Pareto optimal front are shown in Figs. 5–10. Therefore, a perfect multi-objective solution that simultaneously optimizes each objective function is impossible. The selected optimal solution from the Pareto front depends on the most important operating characteristic of the considered gear pair. The improvements in the individual operating characteristics of the gear pair for the Pareto solutions are presented in Table 5. These improvements are calculated regard to the case when no optimal tooth surface modifications are introduced (Fig. 4). The corresponding manufacture parameters for the basic example and the Pareto solutions are presented in Table 6.

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V.V. Simon / Mechanism and Machine Theory 146 (2019) 103727 Table 5 Improvements in operating characteristics of the gear pair for Pareto solutions. Functions

P1

P2

P3

P4

P5

P6

pmax [%]

−14.02 −35.69 −24.74 +1.17

−20.51 −36.14 −19.62 +1.61

−18.07 −30.71 −23.03 +2.39

−22.00 −38.09 −18.11 +2.45

−17.43 −19.88 −21.02 3.37

−18.60 −35.31 −14.06 +2.11

φ2 max [%] Taverage [%]

ηe f f [%]

Table 6 Machine tool settings for Pareto solutions. Parameters

Basic

P1

P2

P3

P4

P5

P6

c [mm] e [mm] f [mm] g [mm] β [deg] δ [deg] ig 1 rt1 [mm] rprof1 [mm] rprof2 [mm]

16.595 71.861 −1.131 23.399 21.2247 −34.0750 3.54672 75.000 ∞ (straight) ∞ (straight)

16.483 72.306 −1.122 23.562 21.2478 −34.5341 3.55567 75.134 ∞ (straight) ∞ (straight)

16.451 72.337 −1.108 23.433 21.3152 −34.3212 3.55582 75.087 8035 ∞ (straight)

16.392 72.111 −1.112 23.511 21.2813 −34.4513 3.54969 75.112 7856 ∞ (straight)

16.437 72.374 −1.128 23.487 21.3256 −34.2236 3.55496 75.067 8122 ∞ (straight)

16.412 73.489 −1.125 23.456 21.4610 −34.1346 3.54882 75.095 8345 ∞ (straight)

16.385 72.695 −1.121 23.575 21.2856 −34.3287 3.55136 75.101 7956 ∞ (straight)

8. Conclusion A multi-objective optimization method to improve the operating characteristics of hypoid gears is developed. Four objectives considered in this study are the minimization of the maximum tooth contact pressure, transmission error and average temperature in the gear mesh, and the maximization of the mechanical efficiency of the gear pair. The goals of the optimization are achieved by the optimal modification of meshing tooth surfaces introduced by the optimized machine tool settings for the manufacture of pinion teeth. A fast elitist nondominated sorting genetic algorithm (NSGA-II) [106] is applied to solve the model. The effectiveness of the method is demonstrated by using hypoid gear examples. By applying the optimized head-cutter geometry and machine tool settings considerable reductions in the maximum tooth contact pressure, transmission errors and average temperature in the gear mesh can be achieved. In percentage, the smallest improvements in the efficiency of the gear pair are achieved, but considering the absolute value of the efficiency’s increase, it is of big importance for the reduction of power losses in the gear pair. Declaration of Competing Interest None. Acknowledgment The research reported in this paper was supported by the Higher Education Excellence Program of the Ministry of Human Capacities in the frame of Artificial intelligence research area of Budapest University of Technology and Economics (BME FIKP-MI). References [1] Q. Fan, R.S. DaFoe, J.W. Swanger, Higher-order tooth flank form error correction for face-milled spiral bevel and hypoid gears, ASME J. Mech. Des. 130 (7) (2008) 1–7 Art. no. 072601. [2] A. Artoni, M. Gabiccini, M. Kolivand, Ease-off based compensation of tooth surface deviations for spiral bevel and hypoid gears: only the pinion needs corrections, Mech. Mach. Theory 61 (2013) 84–101. [3] K. Kawasaki, T. Isamu, H. Gunbara, H. Houjoh, Method for remanufacturing large-sized skew bevel gears using CNC machining center, Mech. Mach. Theory 92 (2015) 213–229. [4] S. Mo, Y. Zhang, Spiral bevel gear true tooth surface precise modeling and experiments studies based on machining adjustment parameters, Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci. 229 (14) (2015) 2524–2533. [5] I.G. Perez, A. Fuentes, R.R. Orzaez, An approach for determination of basic machine-tool settings from blank data in face-hobbed and face-milled hypoid gears, ASME J. Mech. Des. 137 (2015) 1–10 Art. no. 093303. [6] R. Tan, B. Chen, C. Peng, General mathematical model of spiral bevel gears of pure-rolling contact, Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci. 229 (15) (2015) 2810–2826. [7] W. Zhang, B. Cheng, X. Guo, M. Zhang, Y. Xing, A motion control method for face hobbing on CNC hypoid generator, Mech. Mach. Theory 92 (2015) 127–143. [8] Y. Gao, B. Chen, D. Liang, Mathematical models of hobs for conjugate-curve gears having three contact points, Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci. 229 (13) (2015) 2011–2402. [9] H. Chen, X. Zhang, X. Cai, Z. Ju, C. Qu, D. Shi, Computerized design, generation and simulation of meshing and contact of hyperboloidal-type normal circular-arc gears, Mech. Mach. Theory 96 (2016) 127–143.

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