Transmission Efficiency Optimal Design of Spiral Bevel Gear Based on Hybrid PSOGSA (Particle Swarm Optimization—Gravitational Search Algorithm) Method

Abstract

Transmission efficiency is a significant index of the transmission system. Even though much research has been carried out to calculate gear transmission efficiency, only a few of them studied spiral bevel gear due to its complexity. Moreover, spiral bevel gear does not have a “standard surface”, which means more complex coupling relations between different parameters and makes efficiency optimal design more difficult. Therefore, an instantaneous transmission efficiency computing model of a spiral bevel gear was set up based on loaded tooth contact analysis and hybrid elasto-hydrodynamic lubrication theory. Then, the particle swarm optimization–gravitational search algorithm (PSOGSA) optimal model was constructed to obtain the best parameters that maximize the average transmission efficiency of spiral bevel gears. Control parameters and machining parameters are optimized in sequence based on the proposed optimal model. The results showed that both optimal designs could help improve transmission efficiency, but the range of machining parameters is limited in a small interval because of the complex coupling relations. Therefore, the machining parameters optimization are conducted after control parameters optimization, which showed good results. Transmission efficiency was finally improved to 98.78%, which increased more than 4% at least. The proposed optimal model could also be applied into other gear design methods or even other fields.

1. Introduction

Transmission efficiency is one of the most important performance indexes of the transmission system that has a straight connection with the power losses, temperatures, noise, and wear. As the energy crisis intensifies, transmission efficiency improvement is becoming more significant and valuable. Therefore, many researchers have already studied transmission efficiency of gear pairs in order to find the rules and improve gear transmission efficiency.

For gear transmission efficiency, the research based on elasto-hydrodynamic lubrication (EHL) theory for both line contact and point contact has been conducted for many years. Z Yujing et al. [1] studied the instantaneous meshing efficiency at any contact point of spur gear, considering mixed EHL conditions. The influence of the working conditions and design parameters on the gear efficiency had also been analyzed to guide parameter design. Yimin Shao et al. [2] proposed a new method to calculate gear transmission efficiency considering the 3D micro-topography of the tooth surface. S. Baglioni et al. [3] explored the influences of addendum modification on spur gear efficiency and determined the best modification parameters to balance load capacity, safety against wear, vibration and efficiency. H Pei et al. [4] built an optimal model for the balance of gear mass, gear mesh efficiency, and transmission errors, in which the best values of the addendum and dedendum were determined. R Prabhu Sekar et al. [5] proposed a kind of non-standard gears to improve gear efficiency by minimizing sliding and rolling power losses, in which the influence of gear and drive parameters were discussed, including gear ratio, pressure angle, teeth number, addendum, etc. A. Vaidyanathan and A. Kahraman et al. [6,7] studied helical gear efficiency and spur gear efficiency by experimental method, respectively, in which the influences of load, oil, and rotational speed on transmission efficiency were discussed. Petrescu Florian Ion Tiberiu et al. [8] presented an algorithm to determine the transmission efficiency of a simple planetary mechanism based on numerical calculation. C Habermehl et al. [9] presented a new method for efficiency modeling of transmission efficiency and tested it on a six-speed dual-clutch transmission in a passenger car application, which could determine transient loss and the motion of a single part as well as the whole system. Cheng Wang et al. [10] derived the formula transmission efficiency formula of planetary gear/star gear and studied its effect on the closed differential double helical gear train, in which the results showed that the difference is 2%. D Li et al. [11] studied the influencing factors of gear train efficiency in GTF and built the efficiency calculation model of the gear train.

Recently, with regards to spiral bevel gear (SBG) efficiency studies, researchers have also carried out some studies that focus on friction, lubrication, and efficiency. Yanzhong Wang et al. [12] built a thermo-elastic finite element model to study thermal characteristics and transient temperature fields and optimized the temperature field that would be beneficial for performance and life. Xiang Zhu et al. [13] established a quasi-analytical flow model and predicted windage power loss for spiral bevel gears. The influence of specific parameters of windage power loss was discussed and results were verified by experiment. Jiange Zhang et al. [14] proposed a coupling thermo-elasto-hydrodynamic model for the friction coefficient and wear of the spiral bevel gears and calculated the pressure, flash temperature, friction coefficient, and wear rate of spiral bevel gears. Pu Wei et al. [15] established a mixed EHL model and studied the transient friction, temperature, and contact fatigue behaviors for spiral bevel gears in different contact trajectories. Lars Bobach et al. [16] proposed a transient thermal elasto-hydrodynamic calculation model of spiral bevel gears built on the generalized Reynolds equation, considering mass-conserving cavitation, non-Newtonian flow, mixed friction, and micro-hydrodynamics. Jun Zhang et al. [17] studied a pair of circular arc spiral bevel gears in nutation drive system and analyzed influence of system parameters on transmission efficiency.

For the optimal design method of gears, more researchers are using intelligent optimization algorithms in gear design with the development of numerical methods. Cheng Wang [18] proposed a multi-objective modification optimal design method for helical gears to reduce vibration and average load. Sedak Milos et al. [19] established a hybrid metaheuristic algorithm considering the PSO algorithm to solve a multi-objective non-linear optimization problem for planetary gearbox. Based on the non-dominated sorting genetic algorithm III (NSGA-III), S Kim et al. [20] realized macro geometry optimal design of a helical gear pair by scoring the mass, efficiency, and peak-to-peak static transmission error. Daniel Miler et al. [21] used a genetic algorithm to conduct a multi-objective optimization of a spur gear pair to reduce both transmission volume and power losses. Da Cui et al. [22] proposed a simplified reliability Kriging model and optimized the reliability of a planetary gear by genetic algorithm (GA) and the discrete element method (DEM).

It could be easily concluded from former studies that gear transmission friction, efficiency, power loss, and so forth are becoming attractive research topics. The friction, mixed lubrication, and power loss of gears have been considered in-depth. While transmission efficiency and its optimization of spur gears and helical gears have been studied, transmission efficiency of spiral bevel gears are usually neglected. The friction coefficient and windage power loss are the main topics in recent research. There are two main reasons: (1) there is no “standard” tooth surface for spiral bevel gears and there are too many machining parameters, which means the interaction mechanism and the influence on efficiency between parameters are hard to determine; (2) due to the complexity of the spiral bevel gear tooth surface, different design methods and settings may conduct different tooth surfaces, which means it is hard to obtain an absolute best design. Due to the low complexity of cylindrical gears, a single optimal algorithm is usually used, which is not suitable for high complexity optimization in a spiral bevel gear design.

With the development of a modern optimal design method, there is a new way to solve such multi-objective and complex influence–mechanism questions. It could easily avoid multi-factor coupling analysis to combine particle swarm optimization and the gravitational search algorithm. Therefore, we could find a general approach to maximize the efficiency of SBG for certain design parameters or machining parameters. This paper is organized as follows: The basic theory of the efficiency model and PSOGSA optional theory is established in Section 2. Spiral bevel gear meshing theory and optimization settings are conducted in Section 3. Optimal simulation results considering control parameters and machining parameters are discussed in Section 4. Eventually, in Section 5, we draw the conclusions to summarize rules and characteristics of the research.

2. Basic Theories

To achieve efficiency optimization, the efficiency calculation model and optimal algorithm model should be established first. In this Section, the basic meshing coordinate system, tooth contact analysis, and loaded tooth contact analysis were carried out to obtain relevant parameters that would be useful for efficiency calculation. The PSOGSA optimal algorithm and optimized variables were also determined. The SBG design and modeling here is based on the local synthesis method proposed by Prof. Litvin, which could optimize the meshing behavior by controlling the second order contact parameters [23,24]. Even though this paper takes the local synthesis method as an example, it should be pointed out that other design methods, such as the ease-off flank modification method, could also combine with the PSOGSA algorithm to maximize efficiency. It should also be added that the present method focuses on tooth surface frictional power loss and the windage power loss has been ignored.

2.1. Efficiency Calculation of SBG

2.1.1. Tooth Flank and Contact Model

Once the original pinion machine settings are defined, position vectors of the cutting cone r_pt and r_gt is calculated by cutter radius, cutter head parameters, and profile angle. The meshing coordinate system could be established based on geometry and movement relations [24]. Figure 1a,b show the machining coordinate systems of the pinion and gear. Coordinate systems S_m1-O_m1, x_m1, y_m1, z_m1 and S_m2-O_m2, x_m2, y_m2, z_m2 are the cradle coordinate systems for the pinion and gear, respectively. Coordinate systems S₁-O₁, x₁, y₁, z₁ and S₂-O₂, x₂, y₂, z₂ are the coordinate systems of the pinion and gear. Coordinate systems S_c1-O_c1, x_c1, y_c1, z_c1, S_d1-O_d1, x_d1, y_d1, z_d1, S_c2-O_c2, x_c2, y_c2, z_c2, and S_d2-O_d2, x_d2, y_d2, z_d2 are auxiliary coordinate systems. The meaning of the symbols are listed as follows: φ_p, φ_g are the cradle rotation angles of the pinion and gear; φ₁ and φ₂ are the rotation angle of the pinion and gear; q₁ and q₂ are the initial cradle angle settings of the pinion and gear; S_r1 and S_r2 are the cutter radial settings of the pinion and gear; X_b1 and X_b2 are the sliding base feed setting; E_m1 and E_m2 are the vertical offset; X_g1 and X_g2 are the increment of machine center to the back of the pinion and gear. All these machining parameters determine the shape of the tooth surface that means these parameters also directly affect meshing performance. According to the transformation of coordinates, the surface position vector r₁ and r₂ of the pinion and gear could be written as:

r₁ = M_1d1M_d1m1M_m1c1M_c1pr_pt

(1)

r₂ = M_2d2M_d2m2M_m2c2M_c2gr_gt

(2)

Similarly, the normal vector of tooth surface n₁ and n₂ could be obtained based on the position vectors of the cutting cone n_pt and n_gt:

n₁ = L_1d1L_d1m1L_m1c1L_c1pn_pt

(3)

n₂ = L_2d2L_d2m2L_m2c2L_c2gn_gt

(4)

where M_1d1, M_d1m1, M_m1c1, M_c1p, M_2d2, M_d2m2, M_m2c2, M_c2g, L_1d1, L_d1m1, L_m1c1, L_c1p, L_2d2, L_d2m2, L_m2c2, L_c2g are coordinate transformation matrixes.

Tooth surface contact of SBG is a typical point contact form. The contact ellipse parameters of spiral bevel gears are illustrated in Figure 2, which are obtained by tooth contact analysis (TCA). The contact ellipse parameters and curvature parameters are closely related to the machining parameters and tool parameters. As Figure 2 shows, a and b represent the major axis and minor axis of the contact ellipse, respectively. V and U are relative sliding velocity and entrainment velocity. θ_v and θ_u are angles between minor axis of the ellipse and V and U. Moreover, the radius of curvature in contact with minor and major axis direction r_x and r_y and coordinates of contact point M under the coordinate system of the pinion and gear R_mp and R_mg are obtained for the calculation of lubrication and efficiency, which has not been detailed here.

2.1.2. Loaded Tooth Contact Analysis

The loaded tooth contact analysis (LTCA) is a simulation method of gear contact performance under load, which has close proximity to actual working conditions. LTCA of SBG are mostly based on the finite element method (FEM) or its variants or modifications because of the complexity of SBG tooth surface [25,26]. A mixed finite element–mathematical programming method is used for LTCA simulation, which could obtain load distributions, deformations, transmission errors, stresses, and some other performance indexes at a fast computing speed. This method will support input parameters for lubrication and efficiency simulation and its computing speed means it would be suitable for iterative operation and optimal design. The main steps are as follows: (1) initial tooth surface clearance calculation; (2) flexibility matrix calculation; (3) mathematical programming model building and solving; (4) discrete point load and tooth deformation calculation; (5) postprocessing and outputs. The details could be found in reference [27,28] and is omitted here for brevity.

2.1.3. Lubrication Model

Contact behavior of SBG is typical point contact (shown in Figure 2) and the pressure distribution in elliptical contact area could be calculated by the Reynolds equation considering the angle between movement direction and contact elliptical axis direction [29,30]:

$$\frac{\partial }{\partial x}(\frac{\rho }{12\overline{\eta }}{h}^3\frac{\partial p}{\partial x})+\frac{\partial }{\partial y}(\frac{\rho }{12\overline{\eta }}{h}^3\frac{\partial p}{\partial y})=U\cos ({\theta }_u)\frac{\partial (ph)}{\partial x}+U\sin ({\theta }_u)\frac{\partial (ph)}{\partial y}+\frac{\partial (ph)}{\partial t}$$

(5)

where h, p, ρ, $\overline{\eta }$ represent lubricant film thickness, contact pressure, lubricating oil density, and lubricating oil viscosity, respectively.

The equation of lubricant film thickness could be written as:

$$h(t)=h_0(t)+\frac{x^2}{2r_x}+\frac{y^2}{2r_y}+\upsilon (x,y,t)+{\delta }_1(x,y,t)+{\delta }_2(x,y,t)$$

(6)

where h₀(t) and t are initial oil film thickness and time variable; r_x and r_y are the radius of curvature along the minor and major axis direction of contact ellipse; ${\delta }_1$ and ${\delta }_2$ are the surface roughness of the pinion and gear; $\upsilon (x,y,t)$ is the local contact deformation that could be obtained by Boussines integration:

$$\upsilon (x,y,t)=\frac{2}{\pi E^{\prime }}{\displaystyle \underset{\Omega }{\iint }\frac{p(\xi ,\zeta )}{\sqrt{{(x-\xi )}^2+{(y-\zeta )}^2}}}d\xi d\zeta$$

(7)

where E’ represents the equivalent modulus of elasticity.

According to previous studies, the viscous pressure equation and density pressure equation could be written as:

$$\eta ={\eta }_0{e}^{\alpha p}$$

(8)

$$\rho ={\rho }_0\left(1+\frac{0.6\times {10}^{-9}p}{1+1.7\times {10}^{-9}p}\right)$$

(9)

where ρ₀ and η₀ are the initial density and viscous, respectively, α is pressure viscosity exponent.

The equations could be solved after dimensionless analysis based on the finite difference method and progressive mesh densification (PMD) method, which is detailed in reference [29,30,31,32].

2.1.4. Friction and Efficiency Calculation Model

The friction coefficient μ is usually defined by fluid shear stress τ and load W:

$$\mu =\frac{{\displaystyle \iint \tau dxdy}}{W}$$

(10)

To balance computing speed and accuracy in optimal design, the fitting formula of friction coefficient μ proposed by Xu and Kahraman based on friction calculation and multiple linear regression [33], which contains many key parameters including maximum contact pressure p_h, radius of curvature along minor axis direction of contact ellipse r_x, slide-roll ratio SR, viscosity of lubricating oil η, relative sliding velocity and entrainment velocity V and U, root-mean-square roughness (RMS) σ:

$$\mu =e^{f(SR,p_h,{\eta }_0,\sigma )}{p}_h^{b_2}{\left|SR\right|}^{b_3}{U}^{b_6}{\eta }_0^{b_7}{r}_x^{b_8}$$

(11)

where:

$$f(SR,p_h,{\eta }_0,\sigma )=b_1+b_4\left|SR\right|p_h{\log }_{10}({\eta }_0)+b_5{e}^{-\left|SR\right|p_h{\log }_{10}({\eta }_0)}+b_9{e}^{\sigma }$$

(12)

where b₁ = −8.916465, b₂ = 1.03303, b₃ = 1.036077, b₄ = −0.354068, b₅ = 2.812084, b₆ = −0.100601, b₇ = 0.752755, b₈ = −0.390958, b₉ = 0.620305 are fitting coefficients.

After the calculation of the friction coefficient μ, the instantaneous sliding friction could be written as:

$$F_{\mu }({\phi }_i)=\mu ({\phi }_i)W({\phi }_i)$$

(13)

where φ_i represents each rotational angle (i = 1,2,3…). μ(φ_i) and W(φ_i) are the friction coefficient and normal load at contact position i.

The rolling friction force F_r is also needed before efficiency calculation, which could be obtained by [34,35]:

$$F_r({\phi }_i)={\Phi }_T({\phi }_i)F_{ro}({\phi }_i)$$

(14)

where Φ_T is heat impact factor and defined as:

$${\Phi }_T({\phi }_i)=\frac{1-13.2(\frac{p_h({\phi }_i)}{E’})L_s^{0.42}}{1+0.213(1+2.23S{R}^{0.83})L_s^{0.64}}$$

(15)

$$L_s=(\frac{d\eta }{dT}|T_0)\frac{U^2}{K_f}$$

(16)

where K_f is the coefficient of heat conduction, while the function of the isothermal rolling friction force F_ro is:

$$F_{ro}=4.318{(G\tilde{U})}^{0.658}{\tilde{W}}^{0.0126}{R}_x/\alpha$$

(17)

where G = αE’ is the nondimensional material parameter; $\tilde{U}=\frac{{\eta }_{0}U}{E’R_x}$ is the nondimensional entrainment velocity; $\tilde{W}=\frac{W}{E’R_x}$ is the nondimensional load.

Based on the calculation of sliding and rolling friction, the instantaneous transmission efficiency η_e(φ_i) could be written as:

$${\eta }_e({\phi }_i)=1-\frac{1}{T_g{\omega }_g}(\left|F_{\mu }({\phi }_i)\cdot V({\phi }_i)\right|+\left|F_r({\phi }_i)\cdot U({\phi }_i)\right|)$$

(18)

where T_g and ω_g are the torque and rotational speed of gear. The average transmission efficiency η_e could be obtained by averaging the instantaneous transmission efficiency η_e(φ_i).

2.2. Hybrid PSOGSA Optimization Method

The basic idea of the particle swarm optimization (PSO) algorithm is to simulate the predatory behavior of birds randomly searching for food. The birds adjust their search path through their own experience and the exchange between the groups so as to find the place with food. The position or path of each bird is a combination of independent variables. Each search will adjust direction and speed according to its own experience and population exchange (the optimal location of historical search), until the optimal solution has been reached. For the PSO algorithm, the typical steps are as follows [36,37]:

(1) the position and velocity of the particle swarm are randomly initialized, while best position p_best and g_best are defined by the initialized position;

(2) the particle fitness should be calculated;

(3) p_best and g_best are updated based on particle fitness value;

(4) velocity and position are iterated and updated by:

$$v_{ji}^{k+1}=\varpi \cdot v_{ji}^k+c_1\cdot r_1(p_{best}{}_{ji}^k-x_{ji}^k)+c_1\cdot r_2(g_{best}{}_{ji}^k-x_{ji}^k)$$

(19)

$$x_{ji}^{k+1}=x_{ji}^k+v_{ji}^{k+1}$$

(20)

(5) step 2 to step 4 are iterated and cycle calculated until the convergent conditions are met:

• where x and v are position and velocity of particles; p_best is the historical best fitness position; g_best is the global best fitness position; ϖ is weight coefficient; c₁ and c₂ are learning factors; r₁ and r₂ are random numbers between 0 and 1; subscript j and i represent dimension and particle swarm number.

For the gravitational search algorithm (GSA), the core idea is: with the circulation of the algorithm, the particles move continuously in the search space by the gravitational force between them to finally find the best solution. It should be pointed out that the particle movement strategy of GSA is different from that of the PSO algorithm. For GSA, the typical steps are as follows [38,39]:

(1) the particle swarm are randomly initialized, while best position p_best and g_best are defined by the initialized position;

(2) the particle fitness should be calculated;

(3) G, p_best, and g_best should be updated;

(4) resultant force, acceleration, velocity, and position should be calculated and updated by:

$$F_i^d(k)={\displaystyle \sum _{j\in M_{qbest},j\ne i}^N{r}_j}{F}_{ij}^d(k)$$

(21)

$$a_i^d(k)=\frac{F_i^d(k)}{M_{ii}(k)}$$

(22)

$$v_i^d(k+1)=r_i\cdot v_i^d(k)+a_i^d(k)$$

(23)

$$x_i^d(k+1)=x_i^d(k)+v_i^d(k+1)$$

(24)

(5) step 2 to step 4 are iterated and cycle calculated until the convergent conditions are met:

• where G is the gravitational constant, which is used for gravity calculation by $F_{ij}^d(k)=G(k)\frac{M_{pi}(k)\cdot M_{aj}(k)}{R_{ij}(k)+\epsilon }(x_j^d(k)-x_i^d(k))$; M_aj is the active gravitational mass related to object j; M_pi is the passive gravitational mass related to object i; ε is a small constant to prevent the denominator from being zero; R_ij is the Euclidian distance between i and j; d represents the dimension; r_i is random numbers between 0 and 1; M_qbest is the top q in descending order of individual qualities; a is acceleration; M_ii is the mass of object i.

Since the PSO algorithm usually easily runs into the local optimization solution and GSA always shows low convergence speed in the later period of the optimization, researchers tried to combine the two methods to guarantee both convergence speed and global convergence [40,41], which showed good performance in complex optimal problems, especially for multivariable coupling and a non-linear complicated system. Therefore, the hybrid PSOGSA algorithm is suitable for efficiency optimization of SBG, and the basic optimal steps are as follows:

(1) the initial population is randomly initialized, while best position p_best and g_best are defined;

(2) the particle fitness should be calculated;

(3) M, G, p_best, and g_best are updated;

(4) the resultant force F is calculated;

(5) the accelerations, velocities, and positions are updated, in which the velocity formula could be updated as:

$$v_i^d(k+1)=r_i\cdot v_i^d(k)+c_1\cdot r_k(p_{best}^d-x_i^d(k))+c_2\cdot r_m(g_{best}^d-x_i^d(k))+a_i^d(k)$$

(25)

(6) step 2 to step 5 are iterated and cycle calculated until the convergent conditions are met.

It could easily be seen that the values of constant c₁ and c₂ would affect the effects of PSO on GSA. The main computational formulas are listed from Equation 19 to Equation 25. A series of recent literature has also been summarized for reference. Considering the simulation procedure of SBG efficiency, the flow chart for SBG efficiency optimization based on the hybrid PSOGSA algorithm is shown in Figure 3.

3. Basic Simulation Model and Settings

Taking a specific SBG pair as an example, the basic parameters are shown in

Table 1. Basic parameters.

Items	Pinion	Gear
Teeth number	27	79
Modulus (mm)	3.15	3.15
Pressure angle (°)	20	20
Mean spiral angle (°)	30	30
Face width (mm)	30	30
Shaft angle (°)	90	90
Mean cone distance (mm)	116.49	116.49
Hand of spiral	Right	Left
Pitch angle (°)	18.87	71.13
Root angle (°)	17.9	72.1
Addendum (mm)	3.34	1.32
Dedendum (mm)	1.91	3.94

. Since the final machining parameters are determined by optimization, there is uncertainty here.

The working conditions and material properties are defined as follows: elastic modulus E = 2.06 × 10⁵ MPa, Poisson’s ratio μ = 0.29; applied torque on gear T = 300 Nm; input rotational speed ω_p = 1000 rpm; initial temperature T₀ = 80 °F; initial lubricating oil viscosity η₀ = 0.05 Pa·s; pressure viscosity exponent α = 12.5 GPa⁻¹; RMS roughness σ = 0.30μm; Reynolds equation solution domain 5 ≤ X ≤ 5, 5 ≤ Y ≤ 5; thermal conductivity of lubricating oil K_f = 0.25.

There are two different optimal design examples of SBG transmission efficiency. One takes second-order design parameters as optimization variables, and the other takes machining tool parameters as optimization variables. Therefore, the numbers of the optimization variables are different in these two simulation examples. For the two different simulations, the control parameters could be defined as the same before optimization of SBG efficiency: c₁ = 1.2 and c₂ = 1.5 are determined to balance global optimization and convergence rate. The dimension of generation is defined as D = 10, the maximum number of iterations NI = 8, and convergence criteria ξ < 0.00005. The same optimal parameter settings would help to compare the differences between these two different settings, which would be in favor of the design and optimization of SBG.

It should be pointed out that the present method could also be used in different design methods or modeling processes, such as higher-order design, high-contact ratio design, etc. [42,43].

4. Simulation and Discussion

As mentioned in Section 3, there are two simulation examples in this section. One is for second-order design parameters, and another is for machining tool parameters. Due to the complex coupling relationships between different machining tool parameters, the optimal parameter ranges are usually constricted in a narrow interval, which means a globally optimal solution would hardly be solved. Therefore, this paper conducted the optimal design of control parameters first in a wild range and then optimized machining tool parameters based on former optimal results.

For the first example, design contact ratio, the radius of the semi-major axis of the contact ellipse, and the amplitude of the meshing transition point are the three control parameters, as well as the optimization variables. However, for the second example, the gear processing parameters remain unchanged while the machining parameters of the pinion are optimized, including the cutter radial settings S_r1, the sliding base feed setting X_b1, the vertical offset E_m1, the increment of machine center to back X_g1, the second-order coefficients c_md, the third-order coefficients c_me, and the fourth-order coefficients c_mf,. Therefore, there are three optimization variables in example 1 and seven optimization variables in example 2.

4.1. Example 1: Efficiency Optimization Based on Control Parameters

According to the design parameters and TCA analysis, the range of the main three control parameters could be determined: the values of the design contact ratio are between 1.0 and 2.0; the values of the radius of the semi-major axis of the contact ellipse are between 2 mm and 10 mm; the values of amplitudes of the meshing transition point are between 4′′ and 10′′ (second of arc). The optimization parameters are defined as shown in Section 3.

After the optimization calculation based on Figure 3, the best solutions and detailed machining parameters are obtained (shown in

Table 2. Optimal machining parameters based on control parameter optimization.

Items	Pinion (Concave)	Gear (Convex)
Cradle angle (°)	47.90	−32.98
Radial distance (mm)	84.41	101.02
Vertical offset (mm)	−5.67	0
Machine center to back (mm)	−21.04	0
Sliding base (mm)	5.04	−3.55
Machine root angle (°)	18.66	70.77
Roll ratio	0.405	0.946
Second-order coefficient	0.121	-
Third-order coefficient	−0.386	-

). The transmission efficiency could achieve 97.10% and the optimal parameters are as follows: the design contact ratio value is 1.19, the radius of the semi-major axis of the contact ellipse is 2.01 mm, the amplitude of meshing transition point is 8.26′′.

According to the simulation, the changing values of key performance indexes during the gear meshing process are obtained, including the entertainment speed, the relative sliding velocity, the load distribution, the friction coefficient, the transmission efficiency, and fitness values, as shown in Figure 4. The entertainment speed would increase during the meshing process while the relative sliding velocity will decrease first and then increase (as shown in Figure 4a,b). Both entertainment speed and relative sliding velocity are significantly affected by control parameters. The curve of transmission efficiency (Figure 4d) showed that transient efficiency in a different meshing position would be totally different, and the changing range could be up to a maximum of nearly 7%, which is affected by load, the entertainment speed, the relative sliding velocity, lubrication conditions, and many other parameters. Figure 4e showed that the PSOGSA algorithm has a good convergence speed, in which the third step made transmission efficiency more than 97%. It should be pointed out that a minus sign was put on the transmission efficiency value in order to calculate the minimum more conveniently.

4.2. Example 2: Efficiency Optimization Based on Machining Parameters

The optimal design of machining parameters is based on the control parameter optimization results (as shown in Table 2) to maximize optimal efficiency. The range of the main seven control parameters could be determined according to Table 2: the optimal range of radial distance is from 84.2 to 84.4 mm; the optimal range of vertical distance is -from −5.8 to −5.6 mm; the optimal range of the machine center to back is from −22.0 to −19.0 mm; the optimal range of the sliding base is from 4.8 to 5.2 mm; the optimal range of the second-order coefficient is from −0.5 to 0.5; the optimal range of the third-order coefficient is from −0.5 to 0.5; the optimal range of the fourth-order coefficient is from −3 to 3.

After the optimization calculation based on Figure 3, the best solutions and detailed machining parameters are obtained (as shown in

Table 3. Optimal machining parameters based on machining parameter optimization.

Items	Pinion (Concave)
Radial distance (mm)	84.60
Vertical offset (mm)	−5.60
Machine center to back (mm)	−21.11
Sliding base (mm)	5.189
Second-order coefficient	−0.50
Third-order coefficient	0.48
Fourth-order coefficient	−2.737

). The transmission efficiency could achieve 98.78% and the changing values of key performance indexes during the gear meshing process are shown in Figure 5.

According to the comparisons between Figure 4 and Figure 5, the entertainment speed increased, and the relative sliding velocity decreased after optimization (as shown in Figure 5a,b), which would help to increase transmission efficiency. The changing of entertainment speed and relative sliding velocity directly leads to the decrease in the friction coefficient (as shown in Figure 5c). Therefore, the transmission efficiency increased from 97.10% to 98.78%, an increase of 1.68 percent. The fitness value curve shows that the PSOGSA algorithm has a good convergence speed with the sixth step close to the convergence value, as shown in Figure 5e.

According to Figure 4 and Figure 5, the hybrid PSOGSA algorithm shows good convergence, in which the third and the sixth step nearly achieve the best solutions, respectively. Actually, the good performance of the PSOGSA algorithm has already been proven in former studies [44,45] in other fields. The good convergence and stability of the hybrid PSOGSA algorithm has been confirmed again in mechanical engineering problems. Moreover, the friction coefficient and transmission efficiency also obey distribution rules and value ranges proposed by former studies, as in references [12,13,14,15,16] etc.

5. Conclusions

This paper proposed a novel model and method for spiral bevel gear transmission efficiency optimization. Two optimal steps are conducted, and several conclusions could be drawn according to the simulation and research:

(1) A hybrid PSOGSA algorithm for SBG transmission efficiency was proposed, considering the loaded contact analysis and lubrication conditions. The proposed model combined geometric analysis, mechanical analysis, and lubrication analysis in an efficient way.

(2) Transmission efficiency optimization was realized by two steps. One is control parameter optimization, which is of high speed and has a flexible wide range. Another is machining parameter optimization, which contains more machining parameters and higher accuracy. These two steps could achieve fast convergence and global optimal search.

(3) According to the numerical example, the transmission efficiency of spiral bevel gear pairs could increase to 98.78%, an increase of more than 4%, based on two-step optimization. The factors with influence on transient effects are definitely improved, including entertainment speed, relative sliding velocity, friction coefficient, etc.

The present method and model could also be applied into other gear design processes or even other fields. However, it should be pointed out that the determination of the optimization range should also take load bearing strength and other performance indexes into account to guarantee the rationality of the optimal design. The results also need verifications by experiments in further studies in the future.