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Macro geometry optimization of a helical gear pair for mass, efficiency, and transmission error
Mechanism and Machine Theory 144 (2020) 103634
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Research paper
Macro geometry optimization of a helical gear pair for mass, efficiency, and transmission error Su-chul Kim a, Sang-gon Moon a, Jong-hyeon Sohn a, Young-jun Park b,c, Chan-ho Choi d, Geun-ho Lee a,∗ a
Department of Smart Industrial Machinery, Korea Institute of Machinery & Materials, Daejeon 34103, South Korea Department of Biosystems & Biomaterials Science and Engineering, Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul 08826, South Korea c Research Institute of Agriculture and Life Sciences, Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul 08826, South Korea d Tractor Advanced Development Group, LSMtron, Anyang 14118, South Korea b
a r t i c l e
i n f o
Article history: Received 27 July 2019 Revised 24 September 2019 Accepted 25 September 2019 Available online xxx Keywords: Helical gear Macro geometry Optimization Transmission error NSGA-III
a b s t r a c t When designing helical gears, the goal is to optimize gear weight, efficiency, and noise while simultaneously achieving the required strength. In this study, the macro geometry of a helical gear pair was optimized for low weight, high efficiency, and low noise; further, trends of optimal solutions for five combinations of the three objectives were analyzed. The gear mass and efficiency were directly used as the design objectives. However, since the calculation of the gear noise is generally very complicated and time-consuming, the gear noise has not been directly used as the design objective, and the peak-to-peak static transmission error (PPSTE), which is the main source of the gear vibration, was selected as the design objective for the gear noise in the optimization. The objectives exhibited a trade-off relation between each other in the optimal space. If one of them was omitted, the objective considerably deteriorated. To analyze the results, the objectives were normalized and scored. As a result, most of the top ranks were from the optimal solutions considering the mass, efficiency, and PPSTE. Therefore, all three objectives should be considered in the gear optimization for low weight, high efficiency, and low noise. © 2019 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license. (http://creativecommons.org/licenses/by/4.0/)
1. Introduction Involute helical gear design is divided into macro and micro geometry designs. Most of the gear tooth geometry, such as the tip and root diameter and the tooth profile and thickness, is influenced by the macro design. Micro geometry design includes small modifications of the gear teeth in microscales such as crowning or relief. In the gear design process, to achieve satisfactory performance, the macro geometry is first designed, and the next is the micro geometry. Despite the performance improvement owing to the micro design, an optimal design of the macro geometry is very important when selecting the gear specifications because the gear performance is highly influenced by the initial macro design. ∗
Corresponding author. E-mail address: [email protected] (G.-h. Lee).
https://doi.org/10.1016/j.mechmachtheory.2019.103634 0094-114X/© 2019 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license. (http://creativecommons.org/licenses/by/4.0/)
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Nomenclature b bS bS ∗ d1 da1, 2 db1, 2 df1, 2 dFf dm1, 2 dNf di1, 2 ht hr i id k ka kb kh ks n mn san uf v m vt x x’ z1,2 E F Fbt Ft HV K KA KV KHβ KHα KFβ KFα M N PA PVZ Ra1, 2 Sf SHmin SFmin SR SR ∗ T V XR YF YS Yβ
face width, mm web thickness, mm web thickness coefficient reference diameter (pinion), mm tip diameter of pinion and wheel, mm base diameter of pinion and wheel, mm root diameter of pinion and wheel, mm root form diameter, mm mean diameter of pinion and wheel, mm Start of active profile diameter (SAP diameter), mm inner diameter of pinion and wheel, mm tooth depth, mm required service life, hr gear ratio allowable deviation of the gear ratio total equivalent mesh stiffness of a tooth pair axial compressive stiffness of a tooth pair bending stiffness of a tooth pair Hertzian contact stiffness shear stiffness of a tooth pair rotating speed, rpm normal module normal tooth thickness at tip circle, mm tooth height at the line of action average sum velocity over the engagement, m/s tangential velocity at pitch diameter, m/s original value of an objective normalized value of an objective number of teeth (pinion and wheel) Young’s modulus total mesh force, N nominal tooth normal force in the face section, N nominal tangential load, N tooth loss factor general total mesh stiffness of a gear pair application factor dynamic factor face load factor for contact stress transverse load factor for contact stress face load factor for tooth root stress transverse load factor for tooth root stress mass of a gear pair, kg the number of contact tooth pairs in the mesh drive power, kW load-dependent gear losses, kW arithmetic average roughness of pinion and gear wheel length of arc in the tooth root, mm minimum required safety factor for pitting minimum required safety factor for bending rim thickness, mm rim thickness coefficient transmitted torque, N·m volume of gears, mm3 roughness factor tooth form factor stress correction factor helix angle factor
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YB YDT YST YNT Yδ relT YRrelT YX ZB, D ZL ZR Zv ZW ZX ZH ZE ZNT Zε Zβ
α α wt β βb l εγ ρ ρm σH σ Hlim σ HP σF σ Flim σ FP ν μm εα ε1, 2 ηM ηVZ η40 θ ϕ ϕ1 ϕ2
3
rim thickness factor deep tooth factor stress correction factor (reference gear) life factor for tooth root stress relative notch sensitivity factor relative surface factor size factor single pair tooth contact factor lubricant factor roughness factor velocity factor work hardening factor size factor zone factor elasticity factor life factor contact ratio factor helix angle factor pressure angle, ° operating transverse pressure angle, ° helix angle at pitch diameter, ° base helix angle, ° sliced tooth width, mm total contact ratio gear density, kg/mm3 average radius of curvature in the normal section, mm contact stress, MPa allowable stress number (contact), MPa permissible contact stress, MPa bending stress, MPa allowable stress number (bending), MPa permissible bending stress, MPa Poisson’s ratio average gear friction coefficient transverse contact ratio transverse contact ratio of pinion and wheel lubricant viscosity, mPa·s gear mesh efficiency lubricant viscosity at 40 °C, mPa s lubricant temperature, °C roll angle roll angle from the tooth center to the tangent of the base circle roll angle from the start of involute to the tooth center
In general, the primary objective of optimal gear design is to define the appropriate macro geometry that can minimize weight or volume while achieving the required strength and conforming to the geometrical limitations. Various studies have been conducted to achieve this goal. Yokota et al. [1] optimized the weight of a gear pair using a genetic algorithm under constrained conditions such as bending strength of teeth, torsional strength of shaft, and gear dimensions. Thompson et al. [2] investigated the trade-off relationship between the volume and the surface fatigue life of multi-stage spur gear units. Mendi et al. [3] investigated the optimization of the module, shaft diameter and rolling bearing for spur gear to minimize the gearbox volume using a genetic algorithm. Savsani et al. [4] worked on the optimization algorithm for a spur gear train focused on particle swarm optimization and simulated annealing algorithms. Golabi et al. [5] presented practical graphs determined by the optimization process, which was conducted to minimize the volume/weight of the gearbox considering the number of stages and their ratio. The design parameters of the gearbox such as the number of stages, normal modules, and face-width of the gears could be derived from the graphs. The influence of profile shift on the spur gear optimization for volume of the gear pair was studied by Miler et al. [6]. The next design targets normally involve maximizing gear efficiency and minimizing gear noise. A study to simultaneously reduce the gear volume and loss of a spur gear pair was conducted by Miler et al. [7] using a non-dominated sorting genetic algorithm II (NSGA-II), which is a multi-objective optimization method. To achieve this goal, they used the gear
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Fig. 1. Gear shape for optimization.
macro geometry variables such as the module, face width, and profile shift coefficient as optimization variables; their results showed that a trade-off between gear volume and efficiency can be achieved in the optimal design space. Patil et al. [8] optimized two stage gearbox using NSGA-II to minimize the gearbox volume and power loss. The power loss in the optimized space decreased with increasing volume. They also showed in [9] showed that multi-objective optimization of volume and efficiency results in lower power losses and lower wear failures compared to similar approaches using single-objective optimization of volume. Wang et al. [10] minimized tooth surface flash temperature and vibration related to gear efficiency and noise by using tooth modification. The gear whine noise is closely related to the peak-to-peak static transmission error (PPSTE) on the line of action. The static transmission error is caused by the manufacturing error of the gear tooth, the gear misalignment, and the tooth deflection that occurs when the gear mesh stiffness varies with the number of tooth pairs in contact under load. Various studies have been carried out to minimize the PPSTE for low noise gear designs [11–14]. However, most studies have focused on micro geometry rather than the macro specifications of the gears. This is because the contact ratio of the gear is used as a design variable to reduce the mesh stiffness vibration in the macro design step [15–17]. This is a very useful way to design gears with low transmission errors. However, the transmission error is indirectly minimized, and it cannot be applied when the target contact ratio cannot be achieved owing to a few limited design conditions. The variation of the optimal solutions according to the combination of the objective functions when optimizing macro geometries was analyzed in this study, considering the mass, efficiency, and PPSTE of the involute helical gear pair. The macro design variables were the normal module, pressure angle, helix angle, teeth of pinion, and face width. Various design constraints were set so that abnormal tooth shape was not used in the optimization process and the gear strength was satisfied. The optimal design was carried out using a non-dominated sorting genetic algorithm III (NSGA-III), a genetic algorithm for multi-objective optimization, and ISO 6336:2006 was used to evaluate fatigue strength of the gear pair. 2. Method and materials 2.1. Mass calculation To simplify the gear mass calculation, the gear body shape is only expressed by the rim and web thickness, as shown in Fig. 1. It does not include the hub to connect between the gear and shaft, but it is sufficiently useful for estimating the gear mass and also includes the rim and web shapes for gear strength and stiffness calculation. To input the thicknesses as the same criteria in the optimization process, these are converted to coefficients using Eqs. (1) and (2).
SR∗ =
SR mn
(1)
b∗S =
bs b
(2)
Thin rim thickness could affect the gear bending strength. According to ISO 6336-3, the criteria of the rim thickness which does not influence the bending strength of external gears is SR / ht ≥ 1.2. If the tooth depth, ht , is 2.25 times the
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Table 1 Factors for gear strength calculation (for pinion and wheel). Description
Symbol
Value
Application factor Face load distribution factor for contact stress allowable stress number (contact), allowable stress number (bending), minimum required safety factor for pitting minimum required safety factor for bending
KA K Hβ
1.25 1.10 1500 MPa 430 MPa 1.00 1.40
σ Hlim σ Flim SHmin SFmin
normal module of the gears, the bending strength is not influenced when the rim thickness coefficient, SR ∗ , is greater than 2.7. The web thickness is related to the dynamic factor in ISO 6336-1 and the transmission error by gear body deflection. In ISO 6336-1, bS ∗ is presented as a boundary condition of 0.2–1.2 in the calculation of the gear blank factor. In the optimization process, SR ∗ and bS ∗ were set to 3.5 and 0.25, respectively, so that the gear mass was reduced to some extent but the thicknesses were not too thin. To calculate the exact mass of a gear pair, the tooth profile must be considered. However, because it is very complicated to calculate the mass considering the tooth profile, the mass of a gear pair was estimated by an approximate mass using the mean diameter of the tooth as follows:
V =
π dm2 1 + dm2 2 4
b−
π di21 + di22 4
( b − bs )
M =V ·ρ
dm 1,2 =
(3) (4)
da1,2 + d f 1,2
(5)
2
di1,2 = d f 1,2 − 2 · SR
(6)
If di was less than 0 and di was set to 0, ρ = 7830 × 10−9 . 2.2. Strength calculations There are various design methods to calculate the strength of helical gears. ISO 6336:2006 Method B is a widely used rating method for gear strength. This standard provides formulas for pitting resistance and bending failure of helical gears as follows [18]:
σH = ZB, D ZH ZE Zε Zβ σHP =
σHlim ZNT SHmin
Ft u + 1 KA KV KH β KH α d1 b u
ZL ZV ZR ZW ZX
SH =
σHP ·S σH Hmin
σF =
Ft YF YSYβ YBYDT KA KV KF β KF α bmn
σF P = SF =
σF limYST YNT SF min
σF P ·S σF F min
(7)
(8) (9)
Yδ relT YRrelT YX
(10)
(11) (12)
The pitting and bending failures are evaluated through safety factors, SH and SF . Most of the factors required for the calculations are based on Method B of ISO 6336; however, a few factors should be inputted directly by the design engineer. Table 1 lists the factors directly input for the gear rating in this study. The allowable stress numbers of contact and bending fatigue strength for the gear pair were based on 18CrNiMo7 which is case-hardened steel, and the lubricant viscosity was based on ISO VG 46.
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2.3. Gear efficiency calculation The total power losses of a gear pair consist of the no-load dependent losses such as churning and windage losses and the load-dependent loss during the sliding–rolling movement of tooth flanks [19,20]. In this study, only the load-dependent loss, which is dependent on the gear specifications, was considered in the optimization because the no-load dependent losses are more influenced by external factors such as oil level and input speed than gear specifications. There are many theoretical approaches to calculate the load-dependent loss [19–21], one of the most used method is according to Niemann [19]. The calculation is as follows:
PV Z = PA μm HV
(13)
where PA = Ft vt ; μm and HV are as obtained by the following equations.
μm = 0.045
KA Fbt /b
0 . 2
v m ρm
ηM−0.05 XR ≤ 0.2
(14)
Ft cos αt
(15)
vm = 2vt sin αwt
(16)
Fbt =
ρm = 0.5dw1 sin αwt ηM = η40 XR = 3.8
u cos βb (u + 1 )
(17)
40 2.85
(18)
θ
R + R 1 / 4 a1 a2
(19)
2d1
HV = π (u + 1 )/(z1 ucosβb ) · 1 − εα + ε12 + ε22
(20)
Thus, the gear efficiency is expressed as follows:
ηV Z =
PA − PV Z PA
(21)
2.4. Calculation of static transmission error Static transmission error of a gear pair can be divided into unloaded and loaded transmission errors. The unloaded static transmission error is caused by the tooth profile error and the gear misalignment on the line of action. Thus, it is theoretically zero if the gear pair has a perfect involute gear geometry and no misalignment. The loaded static transmission error (LSTE) is attributed to gear tooth deflections caused by the transmitted load [22]. The unloaded transmission error was excluded from the calculation to consider the influence of the gear specifications only. The LSTE is obtained by calculating the tooth deflection according to the tooth stiffness and acting force during the gear meshing cycle. If the transmission error is calculated in consideration of the dynamic characteristics of the gear, the speed term is required, however, since the static transmission error is used in the optimization process, the speed term is excluded from the calculation. The LSTE calculation process is time consuming because it requires several steps and iterations [23]. To reduce the calculation time, tooth stiffness was calculated using the analytical method by Wang et al. [24]. The LSTE can be expressed as follows:
LST E = F=
K=
F cos βb K
20 0 0T db1 N
j=1
kj
(22)
(23)
(24)
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Fig. 2. Geometrical parameters for gear mesh stiffness.
The k should be calculated for each meshing position represented by the roll angle, ϕ , as shown in Fig. 2, so that k varies with the roll angle. The k consists of bending stiffness (kb ), shear stiffness (ks ), axial compressive stiffness (ka ), gear foundation stiffness (kf ), and Hertzian contact stiffness (kh ) as follows:
k=
1 1 kh
1 = kb 1 = ks
+
+
1 ka1
+
1 kf1
+
1 kb2
+
1 k s2
+
1 ka2
+
(25)
1 kf2
2E l [(ϕ2 − ϕ ) cos ϕ + sin ϕ ]
−ϕ 1
−ϕ 1
dϕ
(26)
1.2(1 + υ )(ϕ2 − ϕ ) cos ϕ cos2 ϕ1 dϕ E l [(ϕ2 − ϕ ) cos ϕ + sin ϕ ]
(27)
(ϕ2 − ϕ ) cos ϕ sin2 ϕ1 dϕ 2E l [(ϕ2 − ϕ ) cos ϕ + sin ϕ ]
(28)
ϕ2 −ϕ 1
1 cos2 ϕ1 = kf E l kh =
1 k s1
ϕ2 3 1 + ϕ − ϕ sin ϕ cos ϕ − cos ϕ cos ϕ 2 ϕ − ϕ cos ϕ [ ( 2 ) ) 1 1] ( 2
ϕ2
1 = ka
1 kb1
+
L
∗
uf Sf
2
+M
∗
uf Sf
∗
∗
+ P 1 + Q tan
π E l 4 1 − υ2
2
ϕ1
(29)
(30)
The subscripts 1 and 2 in Eq. (25) denote the pinion and wheel, respectively. The uf and Sf of Eq. (29) are shown in Fig. 2. The uf can be simply calculated from Eq. (31). In the case of Sf , however, it was approximated as Eq. (32) because it requires a complex process to accurately calculate the starting point of the root round. The L∗ , M∗ , P∗ , and Q∗ are obtained by the polynomial functions presented by Sainsot et al. [25], as follows:
uf = Sf ≈
df db − 2 cos ϕ1 2
πdf
X∗ hf ,
z
A Ch D θ f = 2i + Bi h2f + i f + i + Ei h f + Fi θf θf θf
(31) (32) (33)
where hf = df / di . The constants, Ai , Bi , Ci , Di , Ei , and Fi , in Eq. (31) are given in Table 2. The LSTE is calculated from the beginning to the end of the engagement of a gear tooth. The peak-to-peak transmission error (PPSTE) is defined as the difference between the maximum and minimum values of the LSTE, as follows:
PPSTE = Max(LSTE ) − Min(LSTE )
(34)
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S.-c. Kim, S.-g. Moon and J.-h. Sohn et al. / Mechanism and Machine Theory 144 (2020) 103634 Table 2 Coefficient of the Eq. (31). Ai L (hf , θ f ) M∗ (hf , θ f ) P∗ (hf , θ f ) Q∗ (hf , θ f ) ∗
Bi −5
−5.574 × 10 60.111 × 10−5 −50.952 × 10−5 −6.2042 × 10−5
Ci −3
−1.9986 × 10 28.1 × 10−3 185.5 × 10−3 9.0889 × 10−3
Di −4
−2.3015 × 10 −83.431 × 10−4 0.0538 × 10−4 −4.0964 × 10−4
−3
4.7702 × 10 −9.9256 × 10−3 53.3 × 10−3 7.8297 × 10−3
Ei
Fi
0.0271 0.1624 0.2895 −0.1472
6.8045 0.9086 0.9236 0.6904
Table 3 Range and step of design variables. Variable
Min.
Max
Step
Normal module, mn (mm) pressure angle, α (degree) Helix angle, β (degree) number of teeth (pinion), z1 facewidth, b (mm)
1 20 0 10 5
5 25 25 100 60
0.1 2.5 1 1 1
Table 4 Gear specification in optimization process. Constant design parameters Profile shift coefficient Addendum coefficient Dedendum coefficient Root radius coefficient Quality (based on ISO 1328:1995)
Pinion x haP hfP
ρ fP Q
Wheel
0.00 1.00 1.25 0.30 6
2.5. Optimization 2.5.1. Optimization settings A non-dominated sorting genetic algorithm III (NSGA-III), which is a multi-objective optimization method, was used to optimize the helical gear pair. It is known to be more efficient than NSGA-II for the optimization problem with two or more objectives [26]. The number of populations and generations in the genetic algorithm were 500 and 500, respectively. The mutation rate and crossover rate were both 0.5. The number of divisions required to calculate the reference points in the NSGA-III was set to 20. In the optimization, the torque and speed of the driving gear were 300 N m and 1591.5 rpm, respectively. The required service life for the gear pair was 20,0 0 0 hr. The target gear ratio was set to 2.0, and allowable deviation of the gear ratio was set to ±1%. A total of five design variables were used to optimize the macro geometry of a helical gear pair: the normal module, pressure angle, helix angle, number of teeth of the pinion, and face-width. Each design variable was input to the optimization algorithm with the range and step, as listed in Table 3. The values were determined after several iterations to get well-distributed optimal solutions in a sufficiently wide range under the load condition. The design parameters, except for the five design variables and the parameters defined in the previous chapter, were defined as listed in Table 4. The design parameters were inputted equally for all gear pairs in the optimization process. The addendum and dedendum coefficients were set to 1.00 and 1.25, respectively, which were commonly used standard values for gears. Gear profile modifications, such as the tip relief and lead crowning, have not been applied to the calculation to consider only the effect of the macro geometry on the gear performances. The design objectives for helical gears are the mass, efficiency, and PPSTE. To optimize the objectives, the objective functions used in the optimization are as follows: Minimize mass: fmass (mn , α , β , z1 , b) = M Maximize efficiency: fe f f (mn , α , β , z1 , b) = ηV Z Minimize transmission error: fT E (mn , α , β , z1 , b) = P P ST E The optimal design is carried out with a total of five combinations of objective functions as follows: (1) (2) (3) (4) (5)
fmass fmass + feff fmass + fTE feff + fTE fmass + feff + fTE
The optimization was conducted by the procedure described in Fig. 3, using the previously defined conditions.
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Fig. 3. Procedure of the optimization for the gear macro geometry.
2.5.2. Design constraint The following design constraints were set so that the gear geometry meets the design requirements; the tip land, san , was set to be greater than 0.3·mn to ensure that it is not too sharp. The start of active profile diameter should be greater than the root form diameter. The minimum total contact ratio was set to 1.7, to be not too small. The ratio of the tooth width to the base circle diameter of the pinion for case-hardened gears is normally used to be less than 1.1 [19]. Thus, the maximum ratio was set to 1.1 so that the tooth width does not increase unexpectedly. The minimum normal module was set to b/20 by referring to the minimum normal module according to the gear quality of [19]. It was also configured to be eliminated cases that the gear geometry is not formed properly, such as the calculated value is infinity or NaN, according to the combination of design variables in the optimization process. SH > SHmin SF > SFmin san > 0.3·mn
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Fig. 4. Optimization results represented by mass, efficiency, and PPSTE axis.
Fig. 5. Optimization results represented by mass and efficiency axis.
dNf > dFf εγ > 1.7 b/d1 ≤ 1.1 mn ≥ b/20 3. Results Figs. 4–7 show the optimization results for the five combinations according to the coordinate systems. To identify the distribution of the optimization results for the three objectives, four zones, A, B, C, and D, were configured roughly, as shown in Fig. 4; The zone A is an area that is optimized for the efficiency and PPSTE, but has a large mass. The B has good performance for the mass and efficiency, but the PPSTE is high, and the C has low mass and PPSTE, but low efficiency. The D is a Pareto optimal space where the mass, efficiency, and PPSTE all show good performance. In the first combination, the selected objective for the optimization was only the mass. The solutions converged to a point where the minimum mass appeared. However, the efficiency and PPSTE were not optimal either. In the second combination of the mass and efficiency, the efficiency clearly had a trade-off relationship with the mass in the optimal space. The optimal solutions formed a curve on the optimal boundary of the mass and efficiency, as shown in Fig. 5. However, The PPSTE increased significantly in the middle region of the curve, as shown in zone B of Fig. 4. The mass and PPSTE were optimized in the same direction and concentrated on a small space with the minimum mass and PPSTE, as shown in Fig. 6. However, the solutions were in zone C with much lower efficiency than the other combinations. In the fourth combination, the efficiency and PPSTE were also optimized in the same direction, as shown in Fig. 7.
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Fig. 6. Optimization results represented by mass and PPSTE axis.
Fig. 7. Optimization results represented by efficiency and PPSTE axis. Table 5 Min., Max., and Avg. values of the objectives.
Max. Min. Avg.
Mass [kg]
Efficiency [%]
PPSTE [μm]
13.852 1.533 5.774
99.915 99.080 99.617
4.765 0.004 1.587
However, the solutions were concentrated on zone A had a relatively large mass compared with the results of the other combinations. When the mass, efficiency, and PPSTE were optimized simultaneously, the solutions appeared across all the optimal spaces of the three objectives including zone D. In the Pareto optimal space, there was a trade-off relationship in which the PPSTE was greatly reduced as the mass slightly increased or the efficiency slightly decreased.
4. Discussion The total number of the optimal solutions for all the five combinations was 500 × 5 = 2500, but when the overlapping solutions were removed, the total number was reduced to 214. Table 5 shows the maximum, minimum, and average values of each objective in the 214 total solutions.
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Table 6 Combinations and scores from ranking 1–10 in all optimal solutions. Rank
1 2 3 4 5 6 7 8 9 10
Combination number
5 5 5 5 5 5 5 5 5 5
Design variables
Objectives
x’
Score
mn [mm]
α
β
[°]
[°]
z1 [–]
z2 [–]
b [mm]
Mass [kg]
Eff [%]
PPSTE [μm]
Mass
Eff.
PPSTE
1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8
22.5 25.0 22.5 22.5 22.5 22.5 22.5 22.5 22.5 22.5
22 22 22 22 22 22 22 22 22 22
76 82 77 78 77 79 77 78 79 80
152 164 153 155 154 157 155 157 158 159
15 15 15 15 15 15 15 15 15 15
3.46 3.96 3.51 3.59 3.54 3.67 3.58 3.66 3.71 3.76
99.83 99.87 99.83 99.84 99.83 99.84 99.83 99.84 99.84 99.84
0.004 0.067 0.059 0.060 0.059 0.060 0.059 0.061 0.061 0.060
0.843 0.803 0.839 0.833 0.837 0.826 0.834 0.828 0.824 0.820
0.896 0.941 0.901 0.905 0.901 0.910 0.902 0.907 0.910 0.914
1.000 0.987 0.988 0.988 0.988 0.988 0.988 0.988 0.988 0.988
2.739 2.731 2.728 2.726 2.726 2.725 2.724 2.722 2.722 2.722
Table 7 Highest rankings and scores for combinations in all optimal solutions. Combination number
1 2 3 4 5
Highest Ranking
198 103 144 134 1
Design variables
Objectives
x’
Score
mn [mm]
α
β
[°]
[°]
z1 [-]
z2 [-]
b [mm]
Mass [kg]
Eff. [%]
PPSTE [μm]
Mass
Eff.
PPSTE
2.3 1.7 2.4 1.9 1.8
25.0 25.0 20.0 25.0 22.5
24 25 19 20 22
28 100 27 95 76
55 198 54 191 152
21 23 23 35 15
1.53 7.89 1.67 13.11 3.46
99.31 99.91 99.08 99.91 99.83
2.296 1.241 0.009 0.024 0.004
1.000 0.484 0.989 0.060 0.843
0.269 0.999 0.000 0.992 0.896
0.519 0.740 0.999 0.996 1.000
1.788 2.223 1.988 2.047 2.739
To analyze the optimal solutions objectively, the objectives of the optimal solutions were rescaled to the range 0 to 1 using the min-max normalization. The rescaled values were calculated according to whether the optimization direction is maximum or minimum, as follows.
x =
x−min (x ) max (x )−min (x ) max (x )−x max (x )−min (x )
(for efficiency ) (for mass and PPSTE)
(35)
The sum of the rescaled data was used as a score to rank the optimal solutions. Table 6 lists the ranking from 1 to 10. The highest score was 2.739 out of 3 points, and the rankings 1–10 were all in the fifth combination considering the mass, efficiency and PPSTE. Even if the range was expanded from 1 to 100, the number of solutions from the fifth combination was 100, overwhelmingly higher than the other combinations. Table 7 shows the highest rankings and scores for each combination. The x’ for the objectives used in each combination is shown in bold. The order of the highest ranked combination was 5, 2, 4, 3, and 1. The above results show that the gear performance using the fifth combination can be better than the other combinations for all the objectives if there is no priority of the objectives. The solutions of the other combinations also can be a good solution according to priorities of the objectives because they were optimized for their own objectives. However, if any of the objectives were not included, its performances were normally poor than the others, so the three objectives should be considered simultaneously to optimize the gear mass, efficiency, and PPSTE. 5. Conclusion The gear mass, efficiency, and PPSTE were used as objectives to optimize a helical gear pair. The optimization was conducted with five objective combinations and the design variables chosen were the gear macro geometry variables; these included the normal module, pressure angle, helix angle, teeth of pinion, and face width. The three objectives exhibited a trade-off relationship with each other, and if one of them was omitted in the optimization, the corresponding performance deteriorated greatly. Therefore, the three objective functions must be considered for gear design when optimizing for mass, efficiency, and noise. The results of this study are summarized as follows. - The mass and efficiency showed a clear trade-off relationship in the optimal space; however, the combinations of the PPSTE and mass or efficiency were optimized in the same direction. - When the helical gear pair was optimized for the mass and efficiency, the PPSTE was very large in the middle region of the Pareto optimal set. - Optimizing the helical gear pair with the PPSTE and mass or efficiency, the omitted objectives were normally deteriorated.
S.-c. Kim, S.-g. Moon and J.-h. Sohn et al. / Mechanism and Machine Theory 144 (2020) 103634
13
- In the optimization by considering the mass, efficiency, and PPSTE, the solutions were distributed over a relatively wide range in the optimal space including the Pareto optimal space where the mass, efficiency, and PPSTE all show good performance. - When the optimal solutions were ranked by using min-max normalization, most of the top ranks were from the optimal solutions considering the mass, efficiency, and PPSTE. Declaration of Competing Interest None Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.mechmachtheory. 2019.103634. References [1] T. Yokota, T. Taguchi, M. Gen, A solution method for optimal weight design problem of the gear using genetic algorithms, Comput. Ind. Eng. 35 (1998) 523–526, doi:10.1016/S0360-8352(98) 00149-1. [2] D.F. Thompson, S. Gupta, A. Shukla, Tradeoff analysis in minimum volume design of multi-stage spur gear reduction units, Mech. Mach. Theory 35 (20 0 0) 609–627, doi:10.1016/S0 094-114X(99)0 0 036-1. [3] F. Mendi, T. Bas¸ kal, K. Boran, F.E. Boran, Optimization of module, shaft diameter and rolling bearing for spur gear through genetic algorithm, Expert Syst. Appl. 37 (2010) 8058–8064, doi:10.1016/j.eswa.2010.05.082. [4] V. Savsani, R.V. Rao, D.P. Vakharia, Optimal weight design of a gear train using particle swarm optimization and simulated annealing algorithms, Mech. Mach. Theory 45 (2010) 531–541, doi:10.1016/j.mechmachtheory.2009.10.010. [5] S. Golabi, J.J. Fesharaki, M. Yazdipoor, Gear train optimization based on minimum volume/weight design, Mech. Mach. Theory 73 (2014) 197–217, doi:10.1016/j.mechmachtheory.2013.11.002. [6] D. Miler, A. Loncˇ ar, D. Žeželj, Z. Domitran, Influence of profile shift on the spur gear pair optimization, Mech. Mach. Theory 117 (2017) 189–197, doi:10.1016/j.mechmachtheory.2017.07.001. ´ Multi-objective spur gear pair optimization focused on volume and efficiency, Mech. Mach. Theory 125 [7] D. Miler, D. Žeželj, A. Loncˇ ar, K. Vucˇ kovic, (2018) 185–195, doi:10.1016/j.mechmachtheory.2018.03.012. [8] M. Patil, P. Ramkumar, S. Krishnapillai, Multi-Objective Optimization of Two Stage Spur Gearbox Using NSGA-II, 2017, pp. 1–6. SAE Tech. Pap. 2017–July, doi:10.4271/2017- 28- 1939. [9] M. Patil, P. Ramkumar, K. Shankar, Multi-objective optimization of the two-stage helical gearbox with tribological constraints, Mech. Mach. Theory 138 (2019) 38–57, doi:10.1016/j.mechmachtheory.2019.03.037. [10] X. Wang, Y. Wang, Y. Liu, X. Zhang, D. Zheng, Multi-objective optimization modification of a tooth surface with minimum of flash temperature and vibration acceleration RMS, j, Mech. Sci. Technol. 32 (2018) 3097–3106, doi:10.1007/s12206- 018- 0612- z. [11] D.R. Houser, J. Harianto, The Effect of Micro-Geometry and Load on Helical Gear Noise Excitations, 2005, pp. 16–19. SAE Tech. Pap. 2005–May. [12] G. Bonori, M. Barbieri, F. Pellicano, Optimum profile modifications of spur gears by means of genetic algorithms, J. Sound Vib. 313 (2008) 603–616, doi:10.1016/j.jsv.2007.12.013. [13] C. Jia, Z. Fang, Y. Zhang, Topography of modified surfaces based on compensated conjugation for the minimization of transmission errors of cylindrical gears, Mech. Mach. Theory 116 (2017) 145–161, doi:10.1016/j.mechmachtheory.2017.05.017. [14] J.A. Korta, D. Mundo, Multi-objective micro-geometry optimization of gear tooth supported by response surface methodology, Mech. Mach. Theory 109 (2017) 278–296, doi:10.1016/j.mechmachtheory.2016.11.015. [15] K. Umezawa, T. Suzuki, H. Houjoh, T. Sato, Vibration of power transmission helical gears: the effect of contact ratio on the vibration, Bull. JSME 28 (1985) 694–700, doi:10.1299/jsme1958.28.694. [16] M. Umeyama, M. Kato, K. Inoue, Effects of gear dimensions and tooth surface modifications on the loaded transmission error of a helical gear pair, J. Mech. Mar. 120 (1998) 119. [17] A. Kahraman, G.W. Blankenship, Effect of involute contact ratio on spur gear dynamics, J. Mech. Des. 121 (1999) 112, doi:10.1115/1.2829411. [18] ISO Standard 6336:2006, Calculation of Load Capacity of Spur and Helical Gears, International Organization for Standardization, Geneva, Switzerland, 2006. [19] G. Niemann, H. Winter, Maschinenelemente, Band 2, Springer, Berlin, 1983. [20] ISO/TR 14179-2:2001, Gears – Thermal Capacity Part 2: Thermal Load-Carring Capacity, International Organization for Standardization, Geneva, Switzerland, 2001. [21] H. Xai, Development of a Generalized Mechanical Efficiency Prediction Methodology for Gear Pairs, The Ohio State University, 2005. [22] D.R. Houser, J. Harianto, Y. Ueda, Determining the source of gear whine noise, Gear Solutions (2004) 21–39 Feb.. [23] U. Kissling, Effects of profile corrections on peak-to-peak transmission error, Gear Technology (2010) 52–61 July. [24] Q. Wang, Y. Zhang, A model for analyzing stiffness and stress in a helical gear pair with tooth profile errors, JVC/J. Vib. Control. 23 (2017) 272–289, doi:10.1177/1077546315576828. [25] P. Sainsot, P. Velex, O. Duverger, Contribution of gear body to tooth deflections—a new bidimensional analytical formula, J. Mech. Des. 126 (2004) 748, doi:10.1115/1.1758252. [26] G. Campos Ciro, F. Dugardin, F. Yalaoui, R. Kelly, A NSGA-II and NSGA-III comparison for solving an open shop scheduling problem with resource constraints, IFAC-PapersOnLine 49 (2016) 1272–1277, doi:10.1016/j.ifacol.2016.07.690.15/1.1758252.
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Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmachtheory
Research paper
Macro geometry optimization of a helical gear pair for mass, efficiency, and transmission error Su-chul Kim a, Sang-gon Moon a, Jong-hyeon Sohn a, Young-jun Park b,c, Chan-ho Choi d, Geun-ho Lee a,∗ a
Department of Smart Industrial Machinery, Korea Institute of Machinery & Materials, Daejeon 34103, South Korea Department of Biosystems & Biomaterials Science and Engineering, Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul 08826, South Korea c Research Institute of Agriculture and Life Sciences, Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul 08826, South Korea d Tractor Advanced Development Group, LSMtron, Anyang 14118, South Korea b
a r t i c l e
i n f o
Article history: Received 27 July 2019 Revised 24 September 2019 Accepted 25 September 2019 Available online xxx Keywords: Helical gear Macro geometry Optimization Transmission error NSGA-III
a b s t r a c t When designing helical gears, the goal is to optimize gear weight, efficiency, and noise while simultaneously achieving the required strength. In this study, the macro geometry of a helical gear pair was optimized for low weight, high efficiency, and low noise; further, trends of optimal solutions for five combinations of the three objectives were analyzed. The gear mass and efficiency were directly used as the design objectives. However, since the calculation of the gear noise is generally very complicated and time-consuming, the gear noise has not been directly used as the design objective, and the peak-to-peak static transmission error (PPSTE), which is the main source of the gear vibration, was selected as the design objective for the gear noise in the optimization. The objectives exhibited a trade-off relation between each other in the optimal space. If one of them was omitted, the objective considerably deteriorated. To analyze the results, the objectives were normalized and scored. As a result, most of the top ranks were from the optimal solutions considering the mass, efficiency, and PPSTE. Therefore, all three objectives should be considered in the gear optimization for low weight, high efficiency, and low noise. © 2019 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license. (http://creativecommons.org/licenses/by/4.0/)
1. Introduction Involute helical gear design is divided into macro and micro geometry designs. Most of the gear tooth geometry, such as the tip and root diameter and the tooth profile and thickness, is influenced by the macro design. Micro geometry design includes small modifications of the gear teeth in microscales such as crowning or relief. In the gear design process, to achieve satisfactory performance, the macro geometry is first designed, and the next is the micro geometry. Despite the performance improvement owing to the micro design, an optimal design of the macro geometry is very important when selecting the gear specifications because the gear performance is highly influenced by the initial macro design. ∗
Corresponding author. E-mail address: [email protected] (G.-h. Lee).
https://doi.org/10.1016/j.mechmachtheory.2019.103634 0094-114X/© 2019 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license. (http://creativecommons.org/licenses/by/4.0/)
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S.-c. Kim, S.-g. Moon and J.-h. Sohn et al. / Mechanism and Machine Theory 144 (2020) 103634
Nomenclature b bS bS ∗ d1 da1, 2 db1, 2 df1, 2 dFf dm1, 2 dNf di1, 2 ht hr i id k ka kb kh ks n mn san uf v m vt x x’ z1,2 E F Fbt Ft HV K KA KV KHβ KHα KFβ KFα M N PA PVZ Ra1, 2 Sf SHmin SFmin SR SR ∗ T V XR YF YS Yβ
face width, mm web thickness, mm web thickness coefficient reference diameter (pinion), mm tip diameter of pinion and wheel, mm base diameter of pinion and wheel, mm root diameter of pinion and wheel, mm root form diameter, mm mean diameter of pinion and wheel, mm Start of active profile diameter (SAP diameter), mm inner diameter of pinion and wheel, mm tooth depth, mm required service life, hr gear ratio allowable deviation of the gear ratio total equivalent mesh stiffness of a tooth pair axial compressive stiffness of a tooth pair bending stiffness of a tooth pair Hertzian contact stiffness shear stiffness of a tooth pair rotating speed, rpm normal module normal tooth thickness at tip circle, mm tooth height at the line of action average sum velocity over the engagement, m/s tangential velocity at pitch diameter, m/s original value of an objective normalized value of an objective number of teeth (pinion and wheel) Young’s modulus total mesh force, N nominal tooth normal force in the face section, N nominal tangential load, N tooth loss factor general total mesh stiffness of a gear pair application factor dynamic factor face load factor for contact stress transverse load factor for contact stress face load factor for tooth root stress transverse load factor for tooth root stress mass of a gear pair, kg the number of contact tooth pairs in the mesh drive power, kW load-dependent gear losses, kW arithmetic average roughness of pinion and gear wheel length of arc in the tooth root, mm minimum required safety factor for pitting minimum required safety factor for bending rim thickness, mm rim thickness coefficient transmitted torque, N·m volume of gears, mm3 roughness factor tooth form factor stress correction factor helix angle factor
S.-c. Kim, S.-g. Moon and J.-h. Sohn et al. / Mechanism and Machine Theory 144 (2020) 103634
YB YDT YST YNT Yδ relT YRrelT YX ZB, D ZL ZR Zv ZW ZX ZH ZE ZNT Zε Zβ
α α wt β βb l εγ ρ ρm σH σ Hlim σ HP σF σ Flim σ FP ν μm εα ε1, 2 ηM ηVZ η40 θ ϕ ϕ1 ϕ2
3
rim thickness factor deep tooth factor stress correction factor (reference gear) life factor for tooth root stress relative notch sensitivity factor relative surface factor size factor single pair tooth contact factor lubricant factor roughness factor velocity factor work hardening factor size factor zone factor elasticity factor life factor contact ratio factor helix angle factor pressure angle, ° operating transverse pressure angle, ° helix angle at pitch diameter, ° base helix angle, ° sliced tooth width, mm total contact ratio gear density, kg/mm3 average radius of curvature in the normal section, mm contact stress, MPa allowable stress number (contact), MPa permissible contact stress, MPa bending stress, MPa allowable stress number (bending), MPa permissible bending stress, MPa Poisson’s ratio average gear friction coefficient transverse contact ratio transverse contact ratio of pinion and wheel lubricant viscosity, mPa·s gear mesh efficiency lubricant viscosity at 40 °C, mPa s lubricant temperature, °C roll angle roll angle from the tooth center to the tangent of the base circle roll angle from the start of involute to the tooth center
In general, the primary objective of optimal gear design is to define the appropriate macro geometry that can minimize weight or volume while achieving the required strength and conforming to the geometrical limitations. Various studies have been conducted to achieve this goal. Yokota et al. [1] optimized the weight of a gear pair using a genetic algorithm under constrained conditions such as bending strength of teeth, torsional strength of shaft, and gear dimensions. Thompson et al. [2] investigated the trade-off relationship between the volume and the surface fatigue life of multi-stage spur gear units. Mendi et al. [3] investigated the optimization of the module, shaft diameter and rolling bearing for spur gear to minimize the gearbox volume using a genetic algorithm. Savsani et al. [4] worked on the optimization algorithm for a spur gear train focused on particle swarm optimization and simulated annealing algorithms. Golabi et al. [5] presented practical graphs determined by the optimization process, which was conducted to minimize the volume/weight of the gearbox considering the number of stages and their ratio. The design parameters of the gearbox such as the number of stages, normal modules, and face-width of the gears could be derived from the graphs. The influence of profile shift on the spur gear optimization for volume of the gear pair was studied by Miler et al. [6]. The next design targets normally involve maximizing gear efficiency and minimizing gear noise. A study to simultaneously reduce the gear volume and loss of a spur gear pair was conducted by Miler et al. [7] using a non-dominated sorting genetic algorithm II (NSGA-II), which is a multi-objective optimization method. To achieve this goal, they used the gear
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S.-c. Kim, S.-g. Moon and J.-h. Sohn et al. / Mechanism and Machine Theory 144 (2020) 103634
Fig. 1. Gear shape for optimization.
macro geometry variables such as the module, face width, and profile shift coefficient as optimization variables; their results showed that a trade-off between gear volume and efficiency can be achieved in the optimal design space. Patil et al. [8] optimized two stage gearbox using NSGA-II to minimize the gearbox volume and power loss. The power loss in the optimized space decreased with increasing volume. They also showed in [9] showed that multi-objective optimization of volume and efficiency results in lower power losses and lower wear failures compared to similar approaches using single-objective optimization of volume. Wang et al. [10] minimized tooth surface flash temperature and vibration related to gear efficiency and noise by using tooth modification. The gear whine noise is closely related to the peak-to-peak static transmission error (PPSTE) on the line of action. The static transmission error is caused by the manufacturing error of the gear tooth, the gear misalignment, and the tooth deflection that occurs when the gear mesh stiffness varies with the number of tooth pairs in contact under load. Various studies have been carried out to minimize the PPSTE for low noise gear designs [11–14]. However, most studies have focused on micro geometry rather than the macro specifications of the gears. This is because the contact ratio of the gear is used as a design variable to reduce the mesh stiffness vibration in the macro design step [15–17]. This is a very useful way to design gears with low transmission errors. However, the transmission error is indirectly minimized, and it cannot be applied when the target contact ratio cannot be achieved owing to a few limited design conditions. The variation of the optimal solutions according to the combination of the objective functions when optimizing macro geometries was analyzed in this study, considering the mass, efficiency, and PPSTE of the involute helical gear pair. The macro design variables were the normal module, pressure angle, helix angle, teeth of pinion, and face width. Various design constraints were set so that abnormal tooth shape was not used in the optimization process and the gear strength was satisfied. The optimal design was carried out using a non-dominated sorting genetic algorithm III (NSGA-III), a genetic algorithm for multi-objective optimization, and ISO 6336:2006 was used to evaluate fatigue strength of the gear pair. 2. Method and materials 2.1. Mass calculation To simplify the gear mass calculation, the gear body shape is only expressed by the rim and web thickness, as shown in Fig. 1. It does not include the hub to connect between the gear and shaft, but it is sufficiently useful for estimating the gear mass and also includes the rim and web shapes for gear strength and stiffness calculation. To input the thicknesses as the same criteria in the optimization process, these are converted to coefficients using Eqs. (1) and (2).
SR∗ =
SR mn
(1)
b∗S =
bs b
(2)
Thin rim thickness could affect the gear bending strength. According to ISO 6336-3, the criteria of the rim thickness which does not influence the bending strength of external gears is SR / ht ≥ 1.2. If the tooth depth, ht , is 2.25 times the
S.-c. Kim, S.-g. Moon and J.-h. Sohn et al. / Mechanism and Machine Theory 144 (2020) 103634
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Table 1 Factors for gear strength calculation (for pinion and wheel). Description
Symbol
Value
Application factor Face load distribution factor for contact stress allowable stress number (contact), allowable stress number (bending), minimum required safety factor for pitting minimum required safety factor for bending
KA K Hβ
1.25 1.10 1500 MPa 430 MPa 1.00 1.40
σ Hlim σ Flim SHmin SFmin
normal module of the gears, the bending strength is not influenced when the rim thickness coefficient, SR ∗ , is greater than 2.7. The web thickness is related to the dynamic factor in ISO 6336-1 and the transmission error by gear body deflection. In ISO 6336-1, bS ∗ is presented as a boundary condition of 0.2–1.2 in the calculation of the gear blank factor. In the optimization process, SR ∗ and bS ∗ were set to 3.5 and 0.25, respectively, so that the gear mass was reduced to some extent but the thicknesses were not too thin. To calculate the exact mass of a gear pair, the tooth profile must be considered. However, because it is very complicated to calculate the mass considering the tooth profile, the mass of a gear pair was estimated by an approximate mass using the mean diameter of the tooth as follows:
V =
π dm2 1 + dm2 2 4
b−
π di21 + di22 4
( b − bs )
M =V ·ρ
dm 1,2 =
(3) (4)
da1,2 + d f 1,2
(5)
2
di1,2 = d f 1,2 − 2 · SR
(6)
If di was less than 0 and di was set to 0, ρ = 7830 × 10−9 . 2.2. Strength calculations There are various design methods to calculate the strength of helical gears. ISO 6336:2006 Method B is a widely used rating method for gear strength. This standard provides formulas for pitting resistance and bending failure of helical gears as follows [18]:
σH = ZB, D ZH ZE Zε Zβ σHP =
σHlim ZNT SHmin
Ft u + 1 KA KV KH β KH α d1 b u
ZL ZV ZR ZW ZX
SH =
σHP ·S σH Hmin
σF =
Ft YF YSYβ YBYDT KA KV KF β KF α bmn
σF P = SF =
σF limYST YNT SF min
σF P ·S σF F min
(7)
(8) (9)
Yδ relT YRrelT YX
(10)
(11) (12)
The pitting and bending failures are evaluated through safety factors, SH and SF . Most of the factors required for the calculations are based on Method B of ISO 6336; however, a few factors should be inputted directly by the design engineer. Table 1 lists the factors directly input for the gear rating in this study. The allowable stress numbers of contact and bending fatigue strength for the gear pair were based on 18CrNiMo7 which is case-hardened steel, and the lubricant viscosity was based on ISO VG 46.
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2.3. Gear efficiency calculation The total power losses of a gear pair consist of the no-load dependent losses such as churning and windage losses and the load-dependent loss during the sliding–rolling movement of tooth flanks [19,20]. In this study, only the load-dependent loss, which is dependent on the gear specifications, was considered in the optimization because the no-load dependent losses are more influenced by external factors such as oil level and input speed than gear specifications. There are many theoretical approaches to calculate the load-dependent loss [19–21], one of the most used method is according to Niemann [19]. The calculation is as follows:
PV Z = PA μm HV
(13)
where PA = Ft vt ; μm and HV are as obtained by the following equations.
μm = 0.045
KA Fbt /b
0 . 2
v m ρm
ηM−0.05 XR ≤ 0.2
(14)
Ft cos αt
(15)
vm = 2vt sin αwt
(16)
Fbt =
ρm = 0.5dw1 sin αwt ηM = η40 XR = 3.8
u cos βb (u + 1 )
(17)
40 2.85
(18)
θ
R + R 1 / 4 a1 a2
(19)
2d1
HV = π (u + 1 )/(z1 ucosβb ) · 1 − εα + ε12 + ε22
(20)
Thus, the gear efficiency is expressed as follows:
ηV Z =
PA − PV Z PA
(21)
2.4. Calculation of static transmission error Static transmission error of a gear pair can be divided into unloaded and loaded transmission errors. The unloaded static transmission error is caused by the tooth profile error and the gear misalignment on the line of action. Thus, it is theoretically zero if the gear pair has a perfect involute gear geometry and no misalignment. The loaded static transmission error (LSTE) is attributed to gear tooth deflections caused by the transmitted load [22]. The unloaded transmission error was excluded from the calculation to consider the influence of the gear specifications only. The LSTE is obtained by calculating the tooth deflection according to the tooth stiffness and acting force during the gear meshing cycle. If the transmission error is calculated in consideration of the dynamic characteristics of the gear, the speed term is required, however, since the static transmission error is used in the optimization process, the speed term is excluded from the calculation. The LSTE calculation process is time consuming because it requires several steps and iterations [23]. To reduce the calculation time, tooth stiffness was calculated using the analytical method by Wang et al. [24]. The LSTE can be expressed as follows:
LST E = F=
K=
F cos βb K
20 0 0T db1 N
j=1
kj
(22)
(23)
(24)
S.-c. Kim, S.-g. Moon and J.-h. Sohn et al. / Mechanism and Machine Theory 144 (2020) 103634
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Fig. 2. Geometrical parameters for gear mesh stiffness.
The k should be calculated for each meshing position represented by the roll angle, ϕ , as shown in Fig. 2, so that k varies with the roll angle. The k consists of bending stiffness (kb ), shear stiffness (ks ), axial compressive stiffness (ka ), gear foundation stiffness (kf ), and Hertzian contact stiffness (kh ) as follows:
k=
1 1 kh
1 = kb 1 = ks
+
+
1 ka1
+
1 kf1
+
1 kb2
+
1 k s2
+
1 ka2
+
(25)
1 kf2
2E l [(ϕ2 − ϕ ) cos ϕ + sin ϕ ]
−ϕ 1
−ϕ 1
dϕ
(26)
1.2(1 + υ )(ϕ2 − ϕ ) cos ϕ cos2 ϕ1 dϕ E l [(ϕ2 − ϕ ) cos ϕ + sin ϕ ]
(27)
(ϕ2 − ϕ ) cos ϕ sin2 ϕ1 dϕ 2E l [(ϕ2 − ϕ ) cos ϕ + sin ϕ ]
(28)
ϕ2 −ϕ 1
1 cos2 ϕ1 = kf E l kh =
1 k s1
ϕ2 3 1 + ϕ − ϕ sin ϕ cos ϕ − cos ϕ cos ϕ 2 ϕ − ϕ cos ϕ [ ( 2 ) ) 1 1] ( 2
ϕ2
1 = ka
1 kb1
+
L
∗
uf Sf
2
+M
∗
uf Sf
∗
∗
+ P 1 + Q tan
π E l 4 1 − υ2
2
ϕ1
(29)
(30)
The subscripts 1 and 2 in Eq. (25) denote the pinion and wheel, respectively. The uf and Sf of Eq. (29) are shown in Fig. 2. The uf can be simply calculated from Eq. (31). In the case of Sf , however, it was approximated as Eq. (32) because it requires a complex process to accurately calculate the starting point of the root round. The L∗ , M∗ , P∗ , and Q∗ are obtained by the polynomial functions presented by Sainsot et al. [25], as follows:
uf = Sf ≈
df db − 2 cos ϕ1 2
πdf
X∗ hf ,
z
A Ch D θ f = 2i + Bi h2f + i f + i + Ei h f + Fi θf θf θf
(31) (32) (33)
where hf = df / di . The constants, Ai , Bi , Ci , Di , Ei , and Fi , in Eq. (31) are given in Table 2. The LSTE is calculated from the beginning to the end of the engagement of a gear tooth. The peak-to-peak transmission error (PPSTE) is defined as the difference between the maximum and minimum values of the LSTE, as follows:
PPSTE = Max(LSTE ) − Min(LSTE )
(34)
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S.-c. Kim, S.-g. Moon and J.-h. Sohn et al. / Mechanism and Machine Theory 144 (2020) 103634 Table 2 Coefficient of the Eq. (31). Ai L (hf , θ f ) M∗ (hf , θ f ) P∗ (hf , θ f ) Q∗ (hf , θ f ) ∗
Bi −5
−5.574 × 10 60.111 × 10−5 −50.952 × 10−5 −6.2042 × 10−5
Ci −3
−1.9986 × 10 28.1 × 10−3 185.5 × 10−3 9.0889 × 10−3
Di −4
−2.3015 × 10 −83.431 × 10−4 0.0538 × 10−4 −4.0964 × 10−4
−3
4.7702 × 10 −9.9256 × 10−3 53.3 × 10−3 7.8297 × 10−3
Ei
Fi
0.0271 0.1624 0.2895 −0.1472
6.8045 0.9086 0.9236 0.6904
Table 3 Range and step of design variables. Variable
Min.
Max
Step
Normal module, mn (mm) pressure angle, α (degree) Helix angle, β (degree) number of teeth (pinion), z1 facewidth, b (mm)
1 20 0 10 5
5 25 25 100 60
0.1 2.5 1 1 1
Table 4 Gear specification in optimization process. Constant design parameters Profile shift coefficient Addendum coefficient Dedendum coefficient Root radius coefficient Quality (based on ISO 1328:1995)
Pinion x haP hfP
ρ fP Q
Wheel
0.00 1.00 1.25 0.30 6
2.5. Optimization 2.5.1. Optimization settings A non-dominated sorting genetic algorithm III (NSGA-III), which is a multi-objective optimization method, was used to optimize the helical gear pair. It is known to be more efficient than NSGA-II for the optimization problem with two or more objectives [26]. The number of populations and generations in the genetic algorithm were 500 and 500, respectively. The mutation rate and crossover rate were both 0.5. The number of divisions required to calculate the reference points in the NSGA-III was set to 20. In the optimization, the torque and speed of the driving gear were 300 N m and 1591.5 rpm, respectively. The required service life for the gear pair was 20,0 0 0 hr. The target gear ratio was set to 2.0, and allowable deviation of the gear ratio was set to ±1%. A total of five design variables were used to optimize the macro geometry of a helical gear pair: the normal module, pressure angle, helix angle, number of teeth of the pinion, and face-width. Each design variable was input to the optimization algorithm with the range and step, as listed in Table 3. The values were determined after several iterations to get well-distributed optimal solutions in a sufficiently wide range under the load condition. The design parameters, except for the five design variables and the parameters defined in the previous chapter, were defined as listed in Table 4. The design parameters were inputted equally for all gear pairs in the optimization process. The addendum and dedendum coefficients were set to 1.00 and 1.25, respectively, which were commonly used standard values for gears. Gear profile modifications, such as the tip relief and lead crowning, have not been applied to the calculation to consider only the effect of the macro geometry on the gear performances. The design objectives for helical gears are the mass, efficiency, and PPSTE. To optimize the objectives, the objective functions used in the optimization are as follows: Minimize mass: fmass (mn , α , β , z1 , b) = M Maximize efficiency: fe f f (mn , α , β , z1 , b) = ηV Z Minimize transmission error: fT E (mn , α , β , z1 , b) = P P ST E The optimal design is carried out with a total of five combinations of objective functions as follows: (1) (2) (3) (4) (5)
fmass fmass + feff fmass + fTE feff + fTE fmass + feff + fTE
The optimization was conducted by the procedure described in Fig. 3, using the previously defined conditions.
S.-c. Kim, S.-g. Moon and J.-h. Sohn et al. / Mechanism and Machine Theory 144 (2020) 103634
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Fig. 3. Procedure of the optimization for the gear macro geometry.
2.5.2. Design constraint The following design constraints were set so that the gear geometry meets the design requirements; the tip land, san , was set to be greater than 0.3·mn to ensure that it is not too sharp. The start of active profile diameter should be greater than the root form diameter. The minimum total contact ratio was set to 1.7, to be not too small. The ratio of the tooth width to the base circle diameter of the pinion for case-hardened gears is normally used to be less than 1.1 [19]. Thus, the maximum ratio was set to 1.1 so that the tooth width does not increase unexpectedly. The minimum normal module was set to b/20 by referring to the minimum normal module according to the gear quality of [19]. It was also configured to be eliminated cases that the gear geometry is not formed properly, such as the calculated value is infinity or NaN, according to the combination of design variables in the optimization process. SH > SHmin SF > SFmin san > 0.3·mn
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Fig. 4. Optimization results represented by mass, efficiency, and PPSTE axis.
Fig. 5. Optimization results represented by mass and efficiency axis.
dNf > dFf εγ > 1.7 b/d1 ≤ 1.1 mn ≥ b/20 3. Results Figs. 4–7 show the optimization results for the five combinations according to the coordinate systems. To identify the distribution of the optimization results for the three objectives, four zones, A, B, C, and D, were configured roughly, as shown in Fig. 4; The zone A is an area that is optimized for the efficiency and PPSTE, but has a large mass. The B has good performance for the mass and efficiency, but the PPSTE is high, and the C has low mass and PPSTE, but low efficiency. The D is a Pareto optimal space where the mass, efficiency, and PPSTE all show good performance. In the first combination, the selected objective for the optimization was only the mass. The solutions converged to a point where the minimum mass appeared. However, the efficiency and PPSTE were not optimal either. In the second combination of the mass and efficiency, the efficiency clearly had a trade-off relationship with the mass in the optimal space. The optimal solutions formed a curve on the optimal boundary of the mass and efficiency, as shown in Fig. 5. However, The PPSTE increased significantly in the middle region of the curve, as shown in zone B of Fig. 4. The mass and PPSTE were optimized in the same direction and concentrated on a small space with the minimum mass and PPSTE, as shown in Fig. 6. However, the solutions were in zone C with much lower efficiency than the other combinations. In the fourth combination, the efficiency and PPSTE were also optimized in the same direction, as shown in Fig. 7.
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Fig. 6. Optimization results represented by mass and PPSTE axis.
Fig. 7. Optimization results represented by efficiency and PPSTE axis. Table 5 Min., Max., and Avg. values of the objectives.
Max. Min. Avg.
Mass [kg]
Efficiency [%]
PPSTE [μm]
13.852 1.533 5.774
99.915 99.080 99.617
4.765 0.004 1.587
However, the solutions were concentrated on zone A had a relatively large mass compared with the results of the other combinations. When the mass, efficiency, and PPSTE were optimized simultaneously, the solutions appeared across all the optimal spaces of the three objectives including zone D. In the Pareto optimal space, there was a trade-off relationship in which the PPSTE was greatly reduced as the mass slightly increased or the efficiency slightly decreased.
4. Discussion The total number of the optimal solutions for all the five combinations was 500 × 5 = 2500, but when the overlapping solutions were removed, the total number was reduced to 214. Table 5 shows the maximum, minimum, and average values of each objective in the 214 total solutions.
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Table 6 Combinations and scores from ranking 1–10 in all optimal solutions. Rank
1 2 3 4 5 6 7 8 9 10
Combination number
5 5 5 5 5 5 5 5 5 5
Design variables
Objectives
x’
Score
mn [mm]
α
β
[°]
[°]
z1 [–]
z2 [–]
b [mm]
Mass [kg]
Eff [%]
PPSTE [μm]
Mass
Eff.
PPSTE
1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8
22.5 25.0 22.5 22.5 22.5 22.5 22.5 22.5 22.5 22.5
22 22 22 22 22 22 22 22 22 22
76 82 77 78 77 79 77 78 79 80
152 164 153 155 154 157 155 157 158 159
15 15 15 15 15 15 15 15 15 15
3.46 3.96 3.51 3.59 3.54 3.67 3.58 3.66 3.71 3.76
99.83 99.87 99.83 99.84 99.83 99.84 99.83 99.84 99.84 99.84
0.004 0.067 0.059 0.060 0.059 0.060 0.059 0.061 0.061 0.060
0.843 0.803 0.839 0.833 0.837 0.826 0.834 0.828 0.824 0.820
0.896 0.941 0.901 0.905 0.901 0.910 0.902 0.907 0.910 0.914
1.000 0.987 0.988 0.988 0.988 0.988 0.988 0.988 0.988 0.988
2.739 2.731 2.728 2.726 2.726 2.725 2.724 2.722 2.722 2.722
Table 7 Highest rankings and scores for combinations in all optimal solutions. Combination number
1 2 3 4 5
Highest Ranking
198 103 144 134 1
Design variables
Objectives
x’
Score
mn [mm]
α
β
[°]
[°]
z1 [-]
z2 [-]
b [mm]
Mass [kg]
Eff. [%]
PPSTE [μm]
Mass
Eff.
PPSTE
2.3 1.7 2.4 1.9 1.8
25.0 25.0 20.0 25.0 22.5
24 25 19 20 22
28 100 27 95 76
55 198 54 191 152
21 23 23 35 15
1.53 7.89 1.67 13.11 3.46
99.31 99.91 99.08 99.91 99.83
2.296 1.241 0.009 0.024 0.004
1.000 0.484 0.989 0.060 0.843
0.269 0.999 0.000 0.992 0.896
0.519 0.740 0.999 0.996 1.000
1.788 2.223 1.988 2.047 2.739
To analyze the optimal solutions objectively, the objectives of the optimal solutions were rescaled to the range 0 to 1 using the min-max normalization. The rescaled values were calculated according to whether the optimization direction is maximum or minimum, as follows.
x =
x−min (x ) max (x )−min (x ) max (x )−x max (x )−min (x )
(for efficiency ) (for mass and PPSTE)
(35)
The sum of the rescaled data was used as a score to rank the optimal solutions. Table 6 lists the ranking from 1 to 10. The highest score was 2.739 out of 3 points, and the rankings 1–10 were all in the fifth combination considering the mass, efficiency and PPSTE. Even if the range was expanded from 1 to 100, the number of solutions from the fifth combination was 100, overwhelmingly higher than the other combinations. Table 7 shows the highest rankings and scores for each combination. The x’ for the objectives used in each combination is shown in bold. The order of the highest ranked combination was 5, 2, 4, 3, and 1. The above results show that the gear performance using the fifth combination can be better than the other combinations for all the objectives if there is no priority of the objectives. The solutions of the other combinations also can be a good solution according to priorities of the objectives because they were optimized for their own objectives. However, if any of the objectives were not included, its performances were normally poor than the others, so the three objectives should be considered simultaneously to optimize the gear mass, efficiency, and PPSTE. 5. Conclusion The gear mass, efficiency, and PPSTE were used as objectives to optimize a helical gear pair. The optimization was conducted with five objective combinations and the design variables chosen were the gear macro geometry variables; these included the normal module, pressure angle, helix angle, teeth of pinion, and face width. The three objectives exhibited a trade-off relationship with each other, and if one of them was omitted in the optimization, the corresponding performance deteriorated greatly. Therefore, the three objective functions must be considered for gear design when optimizing for mass, efficiency, and noise. The results of this study are summarized as follows. - The mass and efficiency showed a clear trade-off relationship in the optimal space; however, the combinations of the PPSTE and mass or efficiency were optimized in the same direction. - When the helical gear pair was optimized for the mass and efficiency, the PPSTE was very large in the middle region of the Pareto optimal set. - Optimizing the helical gear pair with the PPSTE and mass or efficiency, the omitted objectives were normally deteriorated.
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- In the optimization by considering the mass, efficiency, and PPSTE, the solutions were distributed over a relatively wide range in the optimal space including the Pareto optimal space where the mass, efficiency, and PPSTE all show good performance. - When the optimal solutions were ranked by using min-max normalization, most of the top ranks were from the optimal solutions considering the mass, efficiency, and PPSTE. Declaration of Competing Interest None Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.mechmachtheory. 2019.103634. References [1] T. Yokota, T. Taguchi, M. Gen, A solution method for optimal weight design problem of the gear using genetic algorithms, Comput. Ind. Eng. 35 (1998) 523–526, doi:10.1016/S0360-8352(98) 00149-1. [2] D.F. Thompson, S. Gupta, A. Shukla, Tradeoff analysis in minimum volume design of multi-stage spur gear reduction units, Mech. Mach. 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