Light-Weighting of Planetary Gearbox Based on Multi-Strategy Optimization Sparrow Search Algorithm

Abstract

During the planetary gearbox design process, the issue of light-weighting has been studied using various methods. To get better results from light-weighting, it is necessary for new methods to be considered to deal with this problem. This study proposes the multi-strategy optimized sparrow search algorithm (MSOSSA) that employs five strategies to improve the ability to generate high-quality initial solutions, convergence ability and speed, and the quality of the solution. In the application, the volume of the gearbox is reduced by 6.39%, and the difference in ratio from the previous application is no more than 1.5%. The light-weighting problem of the gearbox is effectively solved. Compared to the other two algorithms, the MSOSSA is six times more likely to produce a high-quality initial solution in a set of 30 runs. The speed of convergence and the ability to converge and generate global optima are the best of the three.

1. Introduction

Planetary gearboxes play an important role in the industry. It is used in wind turbines, aircraft, cars, and electric screwdrivers [1,2]. Planetary gearboxes are usually composed of multi-stage planetary gear trains. The planetary gear train consists of four parts, including the sun gear, the planetary gear, the inner gear, and the planetary carrier [3]. Planetary gearbox has the following characteristics: (1) compared with fixed shaft transmission, it has less noise, longer life, and more compact structure; (2) compared with other gearboxes of the same volume, it can provide a larger transmission ratio; and (3) its input and output axes are concentric, so there is no bending moment or torque to generate radial force [4].

In the conventional design of the planetary gearbox, certain geometric and kinematic conditions are typically prioritized. The practical requirements of improving efficiency and lifespan, light-weighting, and energy loss of the gearbox are ignored in this design mode. Considering these factors, numerous recent studies have focused on optimizing the design of planetary gearboxes. To improve efficiency, Sedak [5] developed the theory of dynamic efficiency calculation and introduced a hybrid algorithm combining particle swarm optimization with differential evolution. Xu [2] established a multi-objective uncertainty optimization design framework for planetary gear trains and designed an improved non-dominated sorting genetic algorithm II (NSGA-II). To improve the lifespan and reduce the size, Jiao [6] used the exterior point penalty function method to optimize the design. The results show that this method can get ideal results. Yaw [7] proposed a hybrid algorithm combining the genetic algorithm with the three parents cross-transformation of the artificial immune system. This algorithm has been successfully applied to planetary gear trains, yielding results comparable to existing studies. For minimal weight and power loss, Parmar [8] proposed an optimization method for planetary gearboxes based on NSGA-II. The results demonstrate that the enhanced gearbox significantly reduces both weight and power loss. Sedak [9] proposed a multi-objective hybrid butterfly optimization and particle swarm optimization algorithm. The application results show that the method is effective.

Among these studies, the light-weighting of the gearbox becomes increasingly essential. It can improve dynamic performance and energy efficiency while reducing manufacturing and transportation costs. In addition, the smaller volume saves space and material resources, which is in line with the Sustainable Development Goals. Therefore, the light-weighting of gearboxes has been a subject of interest for many years. Prayoonrat and Walton [10] introduced an algorithm for optimizing multi-spindle gear trains, offering various optimization options including minimizing volume. Yokota [11] proposed a method to obtain the optimal weight of gear by using an improved genetic algorithm. Chong [12] proposed a design method combining a random search method and the simulated annealing (SA) algorithm to minimize the volume of the gearbox. With the development of optimization algorithms, numerous innovative methods have emerged in recent years. These new methods can address the shortcomings of the past methods, such as tedious steps and poor robustness. For the light-weighting design of the gearbox, Maputi [13] proposed a method to generate Pareto fronts using NSGA-II and investigate them using fuzzy decision tools. Hoseiniasl [14] innovatively applied the particle swarm optimization (PSO) algorithm to this issue. Tran [15] proposed a new method of applying Taguchi’s method and grey relational analysis (GRA). Top [16] tests the sensitivity of seven different metaheuristic algorithms to this problem.

However, the aforementioned methods still have limitations. The SA has the characteristics of a simple calculation process but a slow convergence speed. NSGA-II has a fast convergence rate, but its results are dependent on the selection of parameters. The PSO algorithm also has a fast convergence rate and low computational complexity, but it falls into a local optimum from time to time. Genetic algorithm has strong global search ability, but high time cost. The GRA is simple to calculate, but the process is subjective and relies on human experience, thus the results are somewhat random.

This study proposes a multi-strategy optimized sparrow search algorithm (MSOSSA). In recent years, there has been more and more research and applications on the sparrow search algorithms (SSA). Yue [17] introduces the application and development of SSA in power grid load prediction, image processing, path tracking, wireless sensor network routing performance optimization, wireless location, and fault diagnosis. Wang [18] uses sparrow search algorithms to predict the antenna’s performance and designs a compliant multi-target antenna. Zhang [19] uses an improved sparrow search algorithm to optimize the gain of a PID controller to mitigate the adverse effects on load oscillations and trolley positioning during bridge crane operation. The SSA and its improved algorithms are widely used in various fields. In this study, the light-weighting problem of planetary gearboxes will be investigated using the MSOSSA. The optimization strategies used include improved Circle mapping optimization strategy, simulated annealing optimization strategy, nonlinear weight factor optimization strategy, native alerters behavior optimization strategy, and magnification penalty function optimization strategy. The contribution of this study is to the following:

• A new multi-strategy optimized algorithm is proposed;
• The approach to the light-weighting problem of wind yaw gearboxes is improved;
• The light-weighting result is given and compared to the original result;
• The process and results of the MSOSSA are compared to other algorithms.

2. Light-Weighting Problem

The solution to the light-weighting problem was divided into two steps. The most suitable transmission ratios are determined in the first step, and then the critical parameters for each stage are identified. There are two reasons for this arrangement: Firstly, the ratios are very important parameters for gearboxes and are subject to specific conditions. Secondly, analyzing the gear parameters after the ratios for each stage have been determined helps to simplify the light-weighting problem and reduce the difficulty of the calculations.

2.1. Transmission Ratio Problem

2.1.1. Objective Function and Variables

The minimum volume of the gearbox is taken as the light-weighting design criterion. The volume of the gearbox is minimal when the radial dimension of the gearbox profile is minimal. The main determinant of the radial dimension of the profile is the pitch circle diameter of the inner gear. The space enclosed by the inner gear is the area in which the main components of the gearbox work. Therefore, the gearbox volume can be simplified as the sum of the volumes of multiple cylinders. The diameter of the cylinder is the diameter of the pitch circle of the inner gear, and the height of the cylinder is the face width of the inner gear. Minimizing the sum of the volume surrounded by the pitch circle of the inner gears of the multi-stage planetary transmission is taken as the goal. The transmission ratios at all levels are set as design variables.

The volume expression of the gearbox is given in Equation (1).

$${\mathrm{V}}_{sum}=\sum _{\xi =1}^n{V}_{\xi }=\sum _{\xi =1}^n{\displaystyle \frac{\pi {d_{r\xi }}^2{b}_{\xi }}{4}}$$

(1)

${\mathrm{V}}_{sum}$ represents the volume of the gearbox and $V_{\xi }$ represents the volume of the planetary transmission at each stage $\left(\xi =\mathrm{1,2},\cdots ,n\right)$. According to Figure 1, $d_r$ is the diameter of the pitch circle of the inner gear. b is the tooth width. The planetary gearbox consists of n stages of such transmission in series.

According to the literature [20], the relationship between the sun gear pitch circle and the inner gear pitch circle is shown in Equation (2).

$$d_r=d_s\left(I-1\right)$$

(2)

According to Figure 1, $d_s$ represents the sun gear pitch circle diameter. I represents the transmission ratio.

According to the calculation of the contact strength of the tooth surface [21], the pitch circle diameter of the sun gear, and the face width b can be expressed by Equations (3) and (4), respectively.

$$d_s=K_{td}\sqrt[3]{{\displaystyle \frac{T_a{K}_A{K}_{\gamma }{K}_{H\sum }}{{\left({\phi }_d\right)}_s{{\sigma }_{Hlim}}^2{n}_p}}·{\displaystyle \frac{I}{I-2}}}$$

(3)

$$b={\displaystyle \frac{{K_{td}}^3{T}_a{K}_A{K}_{\gamma }{K}_{H\sum }I}{{d_s}^2{{\sigma }_{Hlim}}^2{n}_p\left(I-2\right)}}$$

(4)

$K_{td}$ is the coefficient of calculation formula, for spur gears, $K_{td}$ = 768. $T_a$ is input torque, $K_A$ is application factor, $K_{\gamma }$ is the mesh load factor (takes into account the uneven distribution of the load between meshes for multiple transmission paths), $K_{H\sum }$ is the integration factor, $K_{H\sum }=1.8~2.4.$ ${\left({\phi }_d\right)}_s$ is the face width factor of sun gear, ${\sigma }_{Hlim}$ is the allowable stress number (contact), and $n_p$ is the number of wheels of planetary gears.

The same materials and heat treatment processes are used on each stage of the sun gear. The number of planetary gears in each stage is the same. The power lost during the transfer is ignored. By calculating Formulas (1)–(4), the volume of the gearbox can be represented by Equation (5).

$${\mathrm{V}}_{sum}={\displaystyle \frac{\pi {K_{td}}^3{T}_a}{4{{\sigma }_{Hlim}}^2{n}_p}}\times \sum _{\xi =1}^n\left(K_{A\xi }{K}_{\gamma \xi }{K}_{H\sum \xi }·{\displaystyle \frac{{\left(I_{\xi }-1\right)}^2}{I_{\xi }-2}}\prod _{\xi =1}^n{I}_{\xi }\right)$$

(5)

The objective function is established as shown in Equation (6).

$$F\left(x\right)=\sum _{\xi =1}^n\left(c_{\xi }·{\displaystyle \frac{{\left(x_{\xi }-1\right)}^2}{x_{\xi }-2}}\prod _{\xi =1}^n{x}_{\xi }\right)$$

(6)

$$c_{\xi }=K_{A\xi }{K}_{\gamma \xi }{K}_{H\sum \xi },\left(\xi =\mathrm{1,2},\cdots ,n\right)$$

(7)

Variables $X=\left(x_1,x_2,\cdots ,x_n\right)=\left(I_1,I_2,\cdots ,I_n\right)$.

${\mathrm{V}}_{sum}$ can be represented by Equation (8). ${\displaystyle \frac{\pi {K_{td}}^3{T}_a}{4{{\sigma }_{Hlim}}^2{n}_p}}$ is a constant and can be calculated. The problem of finding the extremum of ${\mathrm{V}}_{sum}$ is transformed into the problem of finding the extremum of $F\left(x\right)$.

$${\mathrm{V}}_{sum}={\displaystyle \frac{\pi {K_{td}}^3{T}_a}{4{{\sigma }_{Hlim}}^2{n}_p}}\times F\left(x\right)$$

(8)

2.1.2. Constraints

• The total transmission ratio deviation shall not exceed 1.5%. It can be represented by Equation (9).

$$\left(1-1.5\%\right)I_{sum}\le \prod _{\xi =1}^n{x}_{\xi }\le \left(1+1.5\%\right)I_{sum}$$

(9)

• The distribution of the transmission ratios should be uniform and reasonable. According to the literature [22], the transmission ratio of the low-speed stage should be between 4 and 5.6; the transmission ratio of the high-speed stage should be between 3.15 and 9; and the transmission ratio range of the intermediate stage should be between 5 and 8. It can be represented by Equations (10)–(12).

$$3.15\le x_1\le 9$$

(10)

$$4\le x_n\le 5.6$$

(11)

$$5\le x_{\xi }\le 8(1<\xi <n)$$

(12)

• As a rule of thumb, the radial diameter difference between two adjacent stages of transmission should not exceed 1.5 times. It can be represented by Equation (13).

$$1.0\le {\displaystyle \frac{{\mathrm{d}}_{r\left(\xi +1\right)}}{d_{r\xi }}}\le 1.5$$

(13)

2.2. Main Parameters Problem

2.2.1. Objective Function and Variables

The minimum volume of the gearbox is taken as the design criterion. Once the ratios have been determined, the size of the gearbox is minimized when the sum of the volumes of the gears is minimized. Therefore, the gearbox volume can be simplified as the sum of the volumes of the individual gears. The external gear can be regarded as a cylinder with the addendum circle diameter as its diameter and face width as its height. The inner gear can be regarded as a hollow cylinder, where the dedendum circle diameter serves as the outer diameter. The addendum circle diameter, on the other hand, constitutes the inner diameter, while the face width corresponds to the height of this cylinder.

As shown in Equation (14), ${\mathrm{V}}_{sum}$ is equal to the sum of the volumes f (y) of each planetary transmission stage. f (y) can be represented by Equation (15).

$${\mathrm{V}}_{sum}=\sum _{\xi =1}^n{f}_{\xi }\left(y\right)$$

(14)

$$f\left(\mathrm{y}\right)={\displaystyle \frac{\pi b}{4}}\left({d_{as}}^2+3{d_{ap}}^2+{d_{fr}}^2-{d_{ar}}^2\right)$$

(15)

According to Figure 1, $d_{as}$, $d_{ap}$, and $d_{ar}$ are the addendum circle diameters of sun gear, planet gear, and inner gear, respectively. $d_{fr}$ is the dedendum circle diameter of the inner gear. They can be represented by Equations (16)–(19).

Designed in accordance with standard gear specifications. The pressure angle of the gear pitch circle is 20°, the coefficient of addendum is 1.0, and the coefficient of bottom is 0.25.

$$d_{as}=m\left(z_s+2\right)$$

(16)

$$d_{ap}=m\left(z_p+2\right)$$

(17)

$$d_{ar}=m\left(z_r-2\right)$$

(18)

$$d_{fr}=m\left(z_r+2.5\right)$$

(19)

m is the modulus. $z_s$, $z_p$, and $z_r$ are the number of teeth of the sun gear, the planet gear, and the inner gear, respectively.

Considering the requirements of the objective function and constraints, the number of teeth of the sun gear, the planet gear, and the inner gear, $z_s$, $z_p$, $z_r$, modulus m, and face width b are taken as parameters, as shown in Equation (20). Through the calculation of Equations (14) to (19), the objective function is expressed by Equation (21).

$$Y=\left(y_1,y_2,y_3,y_4,y_5\right)=\left(z_s,z_p,z_r,\mathrm{m},\mathrm{b}\right)$$

(20)

$$f\left(\mathrm{y}\right)={\displaystyle \frac{\pi }{4}}{y_4}^2{y}_5\left({y_1}^2+4y_1+3{y_2}^2+12y_2-9y_3+{\displaystyle \frac{55}{4}}\right)$$

(21)

2.2.2. Constraints

• It should be ensured that the number of gear teeth is selected to achieve the given transmission ratio. It can be expressed by Equation (22).

$$1+{\displaystyle \frac{y_3}{y_1}}=I$$

(22)

• The sum of the number of teeth of the internal gear and the sun gear must be an integral multiple of the number of teeth of the planetary gear. It can be expressed by Equation (23).

$${\displaystyle \frac{y_1+y_3}{n_p}}=\mathrm{N}$$

(23)

• To ensure that the individual planetary wheels do not collide, it is necessary to keep a distance between the tops of their teeth at all times. Therefore, the sum of the radii of the tooth top circles of two adjacent planetary gears is less than their center distance. It can be expressed by Equation (24).

$$2-y_1+0.134\times \left(y_1+y_2\right)<0$$

(24)

• To ensure that the rotation axes of sun gear, planet gear, and inner gear coincide, Equation (25) needs to be satisfied.

$$y_1+2·y_2=y_3$$

(25)

• The tooth breakage strength is met. It can be expressed by Equation (26).

$${\displaystyle \frac{F_t}{y_5{y}_4}}{K}_A{K_{\gamma }K}_v{K}_{F\beta }{K}_{F\alpha }{Y}_F{Y}_S{Y}_{DT}{S}_F\le {\sigma }_{FP}$$

(26)

$F_t$ is the transverse tangential load at reference cylinder per mesh, $K_{\gamma }$ is the mesh load factor, $K_v$ is the dynamic factor, $K_{F\beta }$ is the face load factor (root stress), $K_{F\alpha }$ is the transverse load factor (root stress), $Y_F$ is the tooth form factor, $Y_S$ is the stress correction factor, $Y_{DT}$ is the deep tooth factor, ${\sigma }_{FP}$ is the permissible bending stress, and $S_F$ is safety factor for tooth breakage.

• The tooth contact strength is met. It can be expressed by Equation (27).

$$Z_H{Z}_E{Z}_{\epsilon }\times \sqrt{{\displaystyle \frac{F_t}{y_5{y}_4{y}_1}}{\displaystyle \frac{u+1}{u}}{K}_A{K}_v{K}_{H\beta }{K}_{H\alpha }}\times S_H\le {\sigma }_{HP}$$

(27)

$Z_H$ is the zone factor, $Z_E$ is the elasticity factor, $Z_{\epsilon }$ is the contact ratio factor (pitting), u is the gear ratio, and $u={\displaystyle \frac{y_2}{y_1}}.K_{H\beta }$ is the face load factor (contact stress), $K_{H\alpha }$ is the transverse load factor (contact stress), ${\sigma }_{HP}$ is the permissible contact stress, and $S_H$ is safety factor for pitting.

• The gear is not undercut. It can be expressed by Equation (28).

$$y_1\ge 17,y_2\ge 17,y_3\ge 17$$

(28)

• According to experience, the module of the gear should be within an appropriate range. It can be expressed by Equation (29).

$$1.5\le y_4\le 12$$

(29)

• According to experience, the ratio of tooth width to module should be within an appropriate range. It can be expressed by Equation (30).

$$1.5\le y_4\le 12$$

(30)

3. Methodology

The new algorithm proposed in this study is based on multi-strategy optimization of the SSA. The characteristics of the SSA and the optimized contents of the new algorithm are described below.

3.1. Sparrow Search Algorithm

The SSA [23], as an emerging bio-inspired optimization algorithm, draws its design inspiration from the behavior of sparrows foraging for food in the natural environment. Sparrows exhibit group cooperation, information sharing, and adaptive behaviors during foraging. These characteristics have been abstracted by the algorithm designers and applied to solve optimization problems.

The solution methodology involves several key steps: initially, a population of N sparrows is instantiated and their fitness is evaluated. Subsequently, the behavior of sparrows searching for food is simulated, with the algorithm iteratively guiding the population towards regions of higher fitness. Through multiple iterations, the sparrow population gradually converges towards the optimal solution.

In the algorithm, sparrows are classified into three types, including producers, scroungers, and alerters.

Producers are responsible for extensive exploration within the search space to discover new potential solutions. The location update formula is given in Equation (31).

$$X_{i,j}^{t+1}=\left\{\begin{array}{c}\begin{array}{cc}{X}_{i,j}^t·exp\left({\displaystyle \frac{-i}{\alpha ·{iter}_{max}}}\right)& if{R}_2<ST\end{array}\\ \begin{array}{cc}{X}_{i,j}^t+Q·L& if{R}_2\ge ST\end{array}\end{array}\right.$$

(31)

$X_{i,j}^{t+1}$ represents the jth dimension of the ith sparrow of generation t, ${iter}_{max}$ is the maximum number of iterations, and L is a d-dimensional row vector with each element being one.

Scroungers focus on further improvements to well-behaved solutions. The location update formula is given in Equation (32).

$$X_{i,j}^{t+1}=\left\{\begin{array}{c}\begin{array}{cc}Q·exp\left({\displaystyle \frac{X_{worst}^t-X_{i,j}^t}{i^2}}\right)& ifi>N/2\end{array}\\ \begin{array}{cc}{X}_P^{t+1}+\left|X_{i,j}^t-X_P^{t+1}\right|·A^{+}·L& otherwise\end{array}\end{array}\right.$$

(32)

$X_P$ represents the best position held by producers, $X_{worst}$ represents the worst position found so far, and A is a $1\times D$ matrix with elements randomly 1 or −1, $A^{+}=A^T{\left({AA}^T\right)}^{-1}$.

Alerters act as a monitor and balance in the search process. It ensures that the search process does not get stuck in a local optimal solution too early. The location update formula is given in Equation (33).

$$X_{i,j}^{t+1}=\left\{\begin{array}{c}\begin{array}{cc}{X}_{best}^t+\beta ·\left|X_{i,j}^t-X_{best}^t\right|& if{f}_i>f_g\end{array}\\ \begin{array}{cc}{X}_{i,j}^t+K·\left({\displaystyle \frac{\left|X_{i,j}^t-X_{worst}^t\right|}{\left(f_i-f_w\right)+\epsilon }}\right)& if{f}_i=f_g\end{array}\end{array}\right.$$

(33)

$X_{best}^t$ represents the current optimal position, $\beta$ is a normal distribution of random numbers, and $K$ is a random number, $K\in [-\mathrm{1,1}]$. $f_i$ is the fitness value of the current sparrow. $f_g$ and $f_w$ are the current global best and worst fitness values, respectively. $\epsilon$ is the smallest constant.

The specific process is shown in Figure 2.

3.2. Multi-Strategy Optimized Sparrow Search Algorithm

The SSA has the characteristics of high convergence accuracy and fast convergence speed. However, it has the disadvantages of uneven distribution of initialized populations, high frequency of falling into local optima and falling into the trap of invalid solutions in complex spaces. The improved Circle mapping produces a uniform initial population. It is introduced to correct the problem of uneven initialization population distribution of the SSA. The SA has the probability of accepting non-optimal solutions. Combined with the SA, it is beneficial for the producers of the SSA to jump out of the local optimal solution trap and improve the ability of global search. The scroungers that often fall into the local optima will also reduce the search ability of the SSA. The introduction of nonlinear weight factors in the generation process of scroungers can help improve the search efficiency of scroungers. In addition, the position update formula of the alerters is modified to make its behavior closer to the natural state. The logic of the penalty function is optimized in order to cope with the complexity of the original solution space.

These optimization strategies make the MSOSSA not only have the advantages of the SSA in terms of convergence accuracy and convergence speed but also have stronger global search capability and better robustness in complex spaces. The optimization strategies of the MSOSSA are described below.

3.2.1. Improved Circle Mapping Optimization Strategy

The SSA employs a randomly initialized population, which can lead to significant clustering of sparrow individuals and uneven distribution. Chaotic mapping addresses this issue by enhancing population diversity. Common chaotic mappings include Logistic mapping, Cubic mapping, and Circle mapping. The MSOSSA utilizes the Circle map, which is known for its good order and existence. However, the Circle map also has the drawback of uneven chaotic value distribution. To address this issue, we have improved the Circle map to enhance its randomness and mitigate the uneven distribution problem inherent in the original Circle mapping.

The original formula of the Circle map is given in Equation (34).

$$x_{n+1}=mod\left(x_n+0.2-{\displaystyle \frac{0.5}{2\pi }}\sin \left(2\pi \times x_n\right),1\right)$$

(34)

The initial population distribution diagram is shown in Figure 3. The distribution diagrams of sequences generated in 3000 iterations in different intervals are shown in Figure 4.

The improved Circle mapping formula is given by Equation (35).

$$x_{n+1}=mod\left(2x_n+0.2-{\displaystyle \frac{0.7}{2\pi }}\cos \left(2\pi \times x_n+0.5\pi \right)+random\left(\mathrm{0,1}\right),1\right)$$

(35)

The initial population distribution diagram is shown in Figure 5. The distribution diagrams of sequences generated in 3000 iterations in different intervals are shown in Figure 6.

According to Figure 4, the original Circle map produces values concentrated between 0.2 and 0.6. This phenomenon will lead to a lack of population diversity, which may bring poor global search results. The improved Circle map produces a more uniform distribution of individuals. This can enable the algorithm to explore the search space more fully in the early stage and avoid the search blind area caused by the uneven distribution of the initial population.

3.2.2. Simulated Annealing Optimization Strategy

In this study, we have enhanced the SSA by incorporating the concept of simulated annealing into the position update process of the producers. SA is an optimization algorithm based on the physical annealing process. Its core idea is to introduce random perturbations and gradually reduce the “temperature” during the optimization process. This allows the algorithm to accept not only better solutions but also inferior ones with a certain probability.

By doing so, the algorithm effectively avoids getting trapped in local optima, thereby enhancing its global search capability. In the modified MSOSSA, producers update their positions based not only on the original update rules but also on the acceptance probability function of the simulated annealing algorithm. This function helps decide whether to accept the current non-optimal solution. This improvement ensures high diversity during the optimization process and increases the algorithm’s ability to escape local optima, ultimately enhancing overall optimization performance.

The probability P that the current solution is accepted can be expressed by Equation (36).

$$P=\left\{\begin{array}{c}\begin{array}{cc}1& iff\left(t_{new}\right)<\end{array}f\left(t_{curr}\right)\\ \begin{array}{cc}\mathrm{m}\mathrm{i}\mathrm{n}[\exp \left(-{\displaystyle \frac{f\left(t_{new}\right)-f\left(t_{curr}\right)}{T_t}}\right),1]& iff\left(t_{new}\right)>f\left(t_{curr}\right)\end{array}\end{array}\right.$$

(36)

$f\left(t_{curr}\right)$ is the fitness value of the current solution, $f\left(t_{new}\right)$ is the fitness value of the newly generated solution, and $T_t$ is the current temperature.

3.2.3. Nonlinear Weight Factor Optimization Strategy

It is pointed out that the nonlinear weight factor plays an important role in enhancing global search ability and escaping local optimum traps. In the MSOSSA, this strategy is incorporated into the position update method of scroungers, aiming to improve the overall performance of the algorithm. Specifically, in the early iterations of the algorithm, the nonlinear weight factor w is small and increases gradually. During this phase, scroungers eagerly pursue the food in the search space, demonstrating strong exploratory behavior. This behavior aids the algorithm in swiftly escaping from local optima traps, thereby enhancing the convergence speed of the algorithm. As iterations proceed, the weight factor w continues to increase. At this point, scroungers gradually increase their consideration of the distance between themselves and the food when navigating to high-quality food locations. A larger weight factor w leads scroungers to behave in a more exploitative manner. This enhanced local search capability enables the algorithm to conduct a more detailed search when approaching the optimal solution region. Thus, the solution’s precision and the algorithm’s overall optimization effectiveness are improved. Through this dynamic adjustment mechanism, the nonlinear weight factor effectively balances global search and local search capabilities, allowing the MSOSSA to perform optimally in each iteration stage.

The beggar position update formula in MSOSSA is given in Equations (37) and (38).

$$X_{i,j}^{t+1}=\left\{\begin{array}{c}\begin{array}{cc}Q·exp\left({\displaystyle \frac{X_{worst}^t-X_{i,j}^t}{i^2}}\right)& ifi>N/2\end{array}\\ \begin{array}{cc}{X}_P^{t+1}+w·\left|X_{i,j}^t-X_P^{t+1}\right|·A^{+}·L& otherwise\end{array}\end{array}\right.$$

(37)

$$w=\mathsf{\lambda }\left[\tan \left({\displaystyle \frac{\pi t}{3{iter}_{max}}}·{\displaystyle \frac{t^2}{{{iter}_{max}}^2}}\right)+1\right]$$

(38)

$\mathsf{\lambda }$ is the adjustment coefficient. $\mathsf{\lambda }=0.35.$

3.2.4. Native Alerters Behavior Optimization Strategy

In the original SSA, the alerters’ location update principle was to move to the worst position as soon as they hit the current best position. This strategy does help to escape the trap of local optima. However, this strategy is too conservative and unnatural. To further improve search efficiency and make sparrow behavior more natural, the MSOSSA improves the location update strategy of alerters. Specifically, upon attaining the current best position, the alerters no longer immediately relocate to the worst position. Instead, they adopt a more strategic approach, moving aggressively to a randomly selected location with a predetermined regularity. This modification allows alerters to move more intelligently, thus improving search efficiency. The location update formula is given in Equation (39).

$$X_{i,j}^{t+1}=\left\{\begin{array}{c}\begin{array}{cc}{X}_{best}^t+\beta ·\left|X_{i,j}^t-X_{best}^t\right|& if{f}_i>f_g\end{array}\\ \begin{array}{cc}\left(1+\beta \right)·\left(lb+ub-X_{i,j}^t\right)& if{f}_i=f_g\end{array}\end{array}\right.$$

(39)

$\beta$ is a normal distribution of random numbers. lb is the upper bound of the search range and ub is the lower bound.

This improvement leads to significant search efficiency improvement. Firstly, the random location selection enables alerters to explore the solution space more broadly and avoid getting stuck in local optima. Secondly, by choosing locations regularly, the search capability of alerters is utilized more effectively, which improves search efficiency and global search capability.

3.2.5. Magnification Penalty Function Optimization Strategy

The space consisting of all solutions conforming to the constraints can be referred to as the original solution space. Because there are many constraints, the original solution space of planetary gearbox optimization design is very complex. To avoid producing too many invalid solutions, the magnification penalty function is introduced into the programming of the fitness function. When the resulting solution is outside the original solution space, instead of assigning the fitness of that solution to infinity or a constant, it is assigned a value with a magnification. The magnification depends on the number of constraints that are not satisfied. This strategy can help sparrows tend to jump into the original solution space and avoid being trapped in invalid solutions outside the original solution space. Thus, this strategy improves the convergence efficiency.

This strategy can be expressed by Equation (40).

$$f\left(t_{new}\right)=\left\{\begin{array}{c}\begin{array}{cc}f\left(t_{curr}\right)& ifnum=0\end{array}\\ \begin{array}{cc}{\displaystyle \frac{m}{num}}\times f\left(t_{curr}\right)& ifnum>0\end{array}\end{array}\right.$$

(40)

${\displaystyle \frac{m}{num}}$ is the magnification penalty function. $m$ is a coefficient, $m={10}^{10}.$ $num$ is the number of constraints that are not satisfied.

The specific process is shown in Figure 7.

4. Application, Results, and Discussion

4.1. Application

The transmission schematic of the yaw gearbox produced by a Chinese company is shown in Figure 8. The gearbox is a 4-stage planetary gear drive. It operates with an input power of 0.6 kW, an input speed of 700 r/min, a total transmission ratio of 1600, and is designed for a service life of 20 years. In this gearbox, each stage of the planetary transmission has three planetary gears. The material of the sun gear and the planetary gear are 17CrNiMo6 treated by carburizing and quenching. The material of the inner gear is 42CrMoA treated with gas nitriding technique. The light-weight objective is to reduce the volume of the gearbox while satisfying the strength condition, and the change in the total transmission ratio must not exceed 1.5%.

4.2. Application Result

In order to achieve the light-weight goal, a computer program based on the MSOSSA is created to conduct simulation experiments.

Table 1. The light-weight solution of transmission ratio optimization.

stage	1	2	3	4
optimal solution	9	6.8	5.6	4.6

shows the light-weight solution of the transmission ratio obtained by the simulation experiment (the values have been rounded).

Table 2. The light-weight solution of main parameters optimization.

	${\mathit{z}}_{\mathit{s}}$	${\mathit{z}}_{\mathit{p}}$	${\mathit{z}}_{\mathit{r}}$	$\mathbf{m}$	$\mathbf{b}$
stage 1	18	63	144	1.5	20
stage 2	20	49	118	1.5	20
stage 3	17	31	79	1.5	20
stage 4	17	22	61	1.5	20

is the main parameter light-weight solution (the values have been rounded). Figure 9 is the volume comparison before and after light-weighting.

After light-weighting, the volume of the gearbox is reduced by 6.39%. By setting the penalty function, it ensures that the light-weight solution satisfies the requirements of the intensity condition and other constraints. The light-weight total transmission ratio is 1576.512. It is 1.46% different from the transmission ratio before light-weighting, which meets the requirements.

4.3. Comparative Analysis and Discussion

The MSOSSA is compared with the SSA and the Grey Wolf algorithm (GWO). The main parameters of algorithms are set as shown in

Table 3. The main parameters of algorithms.

Algorithm	Main Parameters
MSOSSA	N = 400, n1:n2:n3 = 7:1:2; Max iter = 6
SSA	N = 400, n1:n2:n3 = 7:1:2; Max iter = 6
GWO	N = 400; Max iter = 6

Where N is the number of populations, n1:n2:n3 is the ratio of the three types of sparrows, and Max iter is the maximum number of iterations.

. Each of the three algorithms is run 30 times to complete the transmission ratio light-weighting.

Every algorithm has some dependence on the initial solution. A high-quality initial solution can better help the algorithm to find the global optimal solution. When the initial solution is far away from the optimal value neighborhood, the global optimal solution may not be computed in a finite number of iterations. The program may be limited to the neighborhood of the current solution during the search process. Therefore, generating uniform and reliable initial solutions is an important step. The best initial solution produced by each run of the three algorithms was recorded. Figure 10a–c shows the fitness values or objective function values of the best initial solutions over 30 runs. From Figure 10a, it can be seen that because of the introduction of the improved Circle mapping optimization strategy, the MSOSSA improves the probability of generating initial solutions with low fitness values in a complex solution space. The MSOSSA is six times more likely than the SSA and GWO to produce initial solutions with low fitness values in this set of 30 runs. This helps to help the procedure to compute the global optimal solution.

The speed of convergence is related to the number of iterations to converge. Algorithms are expected to give relatively accurate values using a small number of iterations. The average number of iterations to reach convergence is counted. The solution space of the planetary gearbox light-weighting problem is very complex. The procedure has the possibility of falling into a local optimum solution. So, the number of convergence to local optimal solutions is also counted, as shown in Figure 11. Obviously, GWO has the slowest convergence rate. The convergence speed of SSA is faster than that of GWO, but it often converges to the local optimal solution. The MSOSSA inherits the characteristic of fast convergence of the SSA. Moreover, the convergence speed is improved again by introducing the nonlinear weight factor optimization strategy and native alerter behavior optimization strategy. The introduction of the simulated annealing optimization strategy significantly improves the ability of the MSOSSA to jump out of local optima.

The solutions obtained by the metaheuristics are often not the true optimal solution of the problem, but close to the original optimal solution. From the experimental results, all three algorithms can produce suitable solutions. Figure 12 shows that the MSOSSA has the best ability among them to produce global optima and produces the highest quality solutions. The magnification penalty function optimization strategy improves the ability of the MSOSSA to find the global optimal solution. It makes the result easy to distinguish whether it is a local optima.

From the above results and analyses, it can be concluded that the optimization strategies have a positive effect on improving the ability to generate high-quality initial solutions, the speed of convergence, the ability to converge, and the quality of the solution.

5. Conclusions

In this study, the MSOSSA is proposed for the light-weighting of the gearbox. The problem of light-weighting the gearbox has been divided into two steps. The first step is to find the most suitable ratio for each stage, and the second step is to find the optimum gear main parameters. Distinguishing from the traditional SSA, the MSOSSA employs improved circle mapping to generate initial solutions. A simulated annealing optimization strategy is incorporated into the position update strategy of producers. The nonlinear weight factor strategy is used to improve the position update strategy of scroungers. Similarly, the behavior of the alerters is made natural by modifying the position update strategy. The magnification penalty function optimization strategy is also used to improve the convergence efficiency.

The application results show that gearboxes can be light-weighted in this way. Compared to before the application, the volume of the gearbox was reduced by 6.39% and the requirement of the ratio variation within ±1.5% was met. Furthermore, the performance of the MSOSSA on the gearbox light-weighting problem is compared with the SSA and GWO. The comparative analysis shows that the multi-strategy optimization gives it an advantage in several aspects.

This study provides an idea and steps for the light-weighting problem of gearboxes in various fields. It also provides ideas for optimization strategies for other meta-heuristic algorithms.