# Optimization Design of RV Reducer Crankshaft Bearing
## 3 Mathematical Model of Optimization
### Crow Search Algorithm  
In this study, the optimization of the crankshaft bearing is a typical constraint nonlinear optimization problem. Metaheuristic algorithms have shown promising performance for solving real-world optimization problems that are extremely nonlinear. The crow search algorithm (CSA) is a novel metaheuristic optimization method proposed by [PERSON] [27]. The CSA has demonstrated its effectiveness and ability to solve many complex engineering optimization problems, such as environmental dispatch, renewable energy, and distribution networks [28]. Therefore, CSA is adopted to solve the optimization problem.  
CSA is a population-based metaheuristic algorithm. It is based on the intelligent behavior of crows and searches for the optimal solution through iteration. CSA is initialized with a certain number of crows (flock size) with a random position. The position of each crow represents a solution to the optimization problem. A vector is used to define the position of the crow. The vector dimension represents the number of design variables. During the iteration, the position of each crow is updated. There are two parameters to control the generating of the position. They are flight length and awareness probability respectively. The best solution of each crow is memorized during the iteration. The optimization process stops when the maximum number of iterations is reached. The best position of the crows is select as the optimal solution to the optimization problem.  
In Section 3.3, constraints on geometry, lubrication, contact strength of crankshaft bearings, and structure of the RV reducer are defined. To deal with the constraints, the penalty method is adopted in this study. The constrained problem is transformed into an unconstrained problem by the penalty function.  
The penalty function is defined as  
\[\Pi\big{(}X,\mu_{i},\
u_{i}\big{)}=f(X)-\sum_{i=1}^{M}\tau_{i}H_{i}[g_{i}(X)]g_{i} ^{2}(X)-\sum_{j=1}^{N}\
u_{j}F_{j}[\xi(X)]\xi_{j}^{2}(X) \tag{45}\]  
where \(\tau_{i}\gg 1\) and \(\upsilon_{j}\geq 0\). These penalty function parameters should be sufficiently large for this application, but the values depend on the required quality of the solution. \(M\) is the number of inequality constraints, and \(N\) is the number of equality constraints.  
\(H_{i}[g_{i}(X)]\) and \(F_{j}\big{[}\xi_{j}(X)\big{]}\) are index functions:  
\[\left\{\begin{array}{ll}H_{i}[g_{i}(X)]=1,&g_{i}(X)<0\\ H_{i}[g_{i}(X)]=0,&g_{i}(X)\geq 0\end{array}\right. \tag{46}\]  
\[\left\{\begin{array}{ll}F_{j}[\xi_{j}(X)]=1,&\xi_{j}(X)\
eq 0\\ F_{j}[\xi_{j}(X)]=0,&\xi_{j}(X)=0\end{array}\right. \tag{47}\]  
CSA is adopted to solve the optimization problem. Figure 6 shows the flowchart of the proposed optimization design of the bearing. First, the objective function, design variables, and constraints are defined. Second, the adjustable parameters of the CSA are specified. The parameter of the RV reducer and initial parameters of the bearing are given. The position and memory of the crows is initialized. Third, a new position is generated, and the feasibility of the solution is checked through constraints. The process is repeated until the maximum number of iterations is reached.  
Figure 6: Flowchart of the proposed optimization framework.