# Light-Weighting of Planetary Gearbox Based on Multi-Strategy Optimization Sparrow Search Algorithm
## 2 Light-Weighting Problem
### 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).  
\[\mathrm{V}_{sum}=\sum_{\xi=1}^{n}V_{\xi}=\sum_{\xi=1}^{n}\frac{\pi d_{r\xi}{} ^{2}b_{\xi}}{4} \tag{1}\]  
\(\mathrm{V}_{sum}\) represents the volume of the gearbox and \(V_{\xi}\) represents the volume of the planetary transmission at each stage (\(\xi=1,2,\cdots,n\)). 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).  
\[d_{r}=d_{s}(I-1) \tag{2}\]  
According to Figure 1, \(d_{s}\) represents the sun gear pitch circle diameter. \(I\) represents the transmission ratio.  
Figure 1: Parameters of the gears in single-stage planetary gear transmission.  
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.  
\[d_{s}=K_{td}\sqrt[3]{\frac{T_{a}K_{A}K_{Y}K_{H\Sigma}}{(\sigma_{d})_{s}\sigma_{ Hlim}{}^{2}n_{p}}.\frac{I}{I-2}} \tag{3}\]  
\[b=\frac{K_{td}{}^{3}T_{a}K_{A}K_{Y}K_{H\Sigma}I}{d_{s}{}^{2}\sigma_{Hlim}{}^{2} n_{p}(I-2)} \tag{4}\]  
\(K_{td}\) 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), \(K_{H\Sigma}\) is the integration factor, \(K_{H\Sigma}=1.8\sim 2.4\). \((\varphi_{d})_{s}\) 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).  
\[\mathrm{V}_{sum}=\frac{\pi K_{td}{}^{3}T_{a}}{4\sigma_{Hlim}{}^{2}n_{p}}\times \sum_{\xi=1}^{n}\Bigg{(}K_{A\xi}K_{\gamma\xi}K_{H\Sigma}\xi\frac{(I_{\xi}-1)^{ 2}}{I_{\xi}-2}\prod_{\xi=1}^{n}I_{\xi}\Bigg{)} \tag{5}\]  
The objective function is established as shown in Equation (6).  
\[F(x)=\sum_{\xi=1}^{n}\Bigg{(}c_{\xi}.\frac{(x_{\xi}-1)^{2}}{x_{\xi}-2}\prod_{ \xi=1}^{n}x_{\xi}\Bigg{)} \tag{6}\]  
\[c_{\xi}=K_{A\xi}K_{\gamma\xi}K_{H\Sigma}\xi,(\xi=1,2,\cdots,n) \tag{7}\]  
Variables \(X=(x_{1},x_{2},\cdots,x_{n})=(I_{1},I_{2},\cdots,I_{n})\).  
\(\mathrm{V}_{sum}\) can be represented by Equation (8). \(\frac{\pi K_{td}{}^{3}T_{a}}{4\sigma_{Hlim}{}^{2}n_{p}}\) 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 \(F(x)\).  
\[\mathrm{V}_{sum}=\frac{\pi K_{td}{}^{3}T_{a}}{4\sigma_{Hlim}{}^{2}n_{p}}\times F (x) \tag{8}\]