# Application of Evolutionary Optimization Techniques in Reverse Engineering of Helical Gears: An Applied Study

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## Abstract

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## 1. Introduction

## 2. Theoretical Background of Gear Calculations

## 3. Materials and Methods

#### 3.1. Reverse Engineering of Cylindrical Helical Gear

#### 3.2. Problem Description and Solving Method

#### 3.2.1. Objective Function

#### 3.2.2. Methodology

#### 3.2.3. Evolutionary Optimization Algorithms

**Grey Wolf Optimization Algorithm:**GWO is a metaheuristic optimization algorithm that imitates the intelligent foraging behavior of grey wolves when hunting prey. Strictly following their social hierarchy, grey wolves are divided into 4 groups from top to bottom. The top group which makes decisions during hunting is called α. The next level belongs to $\mathsf{\beta}$ wolves who help as deputy chiefs in the group. The α wolves are replaced by $\mathsf{\beta}$ ones after dying or becoming inefficient as leaders. In the third level are δ wolves, which act as hunters and scouts of the group. The last level in the established hierarchy is occupied by the weakest members, titled $\omega $ wolves. When hunting, all types of wolves are organized according to decisions made by α wolves to identify, follow the prey, encircle it, and finally attack it. In this optimization algorithm, $\mathsf{\alpha}$, $\mathsf{\beta}$, and $\mathsf{\delta}$ indicate the members with fitness from high to low, in that order [33].

**Whale Optimization Algorithm:**Introduced by Mirjalili et al., WOA is a swarm optimization method inspired by the hunting behavior of humpback whales. The hunting process involves searching, encircling the prey, and finally attacking to catch it [35]. The relatively easy implementation of the algorithm and its configuration with fewer parameters are the advantages of this optimization technique. The prey is the answer to the optimization problem that must be determined during the WOA process [36]. The hunting of humpback whales is illustrated in Figure 5. The movement of randomly created whales toward the prey or leader whale is shown in Figure 5a. The encircling process of the prey by the whales and their spiral movement while emitting bubbles to surround the prey is illustrated in Figure 5b,c, respectively [37].

**Particle Swarm Optimization:**Nature-inspired metaheuristic optimization algorithms have recently been extensively utilized to optimize various manufacturing problems [40]. PSO has gained a special place among these algorithms because of its simplicity in programming and solving relatively complex functions [41].

**Genetic Algorithm:**As the most recognized subset of evolutionary algorithms, it mimics the biological evolution of organisms in nature. Having been utilized in various engineering problems, GA was initially developed and characterized by John Holland [44]. This naturally inspired optimization algorithm is structured on natural genetics and selection. It allows only the solutions to survive and produce successive generations with maximum fitness according to the defined optimization problem [45].

#### 3.2.4. Experimental Methodology

## 4. Results and Discussion

^{®}core TM i7, 4.00 GHz CPU, and 16 GB RAM was selected to run algorithms in MATLAB

^{®}software, version 2020b. For optimization with the GA, function ga in MATLAB was used. For optimization with the PSO algorithm, Particle Swarm Optimization MATLAB Toolbox Version 1.0.0.0 was selected. For optimization with the GWO algorithm, Grey Wolf Optimizer MATLAB Toolbox Version 1.0 was utilized, and for optimization with WOA, Whale Optimization Algorithm MATLAB Toolbox Version 1.0 was applied. The population size for all algorithms was equal to perform a fair comparison of the accuracy and convergence speed of the algorithms. The performance of all optimization algorithms was submitted after 20 runs, and the run with the best result was selected in comparison with the efficiency of the algorithms. The parameter configurations for GA and PSO were discussed in the corresponding sections. GWO and WOA do not need any specific parameters [13]. As mentioned before, the RE problem was approached with two different scenarios. Concerning the first one, the efficiency of the algorithms in terms of their convergence speed was assessed by selecting a definite number of function evaluations for each algorithm to run. In the second scenario, the minimum error was selected as the final goal of optimization. Overall, the fastest algorithm to find optimum input design parameters will be determined in this case.

## 5. Conclusions

- Based on the proposed methodology, accurate input parameters were reached. The validation of the obtained results was evaluated by the given equations.
- The performance of the algorithms was assessed in terms of the ability to reach the best solution, convergence speed, and stability. It was found that swarm-based optimization methods such as GWO and PSO are superior to the other considered algorithms such as GA.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 5.**Hunting process of humpback whales: (

**a**) movement of whales in a random direction or toward leader; (

**b**) encircling process after finding the prey; (

**c**) reaching the prey in spiral route.

**Figure 6.**Search mechanism and whales’ movement in WOA: (

**a**) shrinking encircling process; (

**b**) position update in the spiral way.

**Figure 10.**The algorithms’ convergent curves reaching the minimum defined objective function for GWO, WOA, PSO, and GA.

Input Parameters | Output Parameters | |
---|---|---|

Direct relation with involute form | Module (${m}_{n}$) Normal pressure angle (${\alpha}_{n}$) Helix angle ($\beta $) Addendum modification ($x$) | Span measurement (${s}_{m}$) Over-ball measurement (${d}_{m}$) Chordal thickness measurement (${s}_{j}$) |

No relation with involute form | Face width | Outside diameter Root diameter ... |

Population size | 40 |

Range of inertia weight | 0.4–0.8 |

Cognitive factor | 1.5 |

Social factor | 1.5 |

Stopping criteria | Minimum Specified Error |

Population size | 40 |

Length of chromosomes | 6 |

Selection operator | Roulette wheel |

Crossover operator | Single-point operator |

Crossover probability | 0.7 |

Mutation probability | 0.15 |

Fitness parameter | Operation time |

Input Parameter | Values of the Selected Gear Part |
---|---|

$z$ | 42 |

${m}_{n}$ | 4.233 |

${\alpha}_{n}$ | 25 |

$\beta $ | 17° 59′ 58″ |

$x$ | +0.0863 |

Output Parameter | Condition of Measurement | Measured Value |
---|---|---|

${d}_{m}^{1}$ | ${d}_{k}=6.5$ | 195.264 |

${d}_{m}^{2}$ | ${d}_{k}=7$ | 196.890 |

${d}_{m}^{3}$ | ${d}_{k}=7.5$ | 198.493 |

${s}_{m}$ | $k=7$ | 87.206 |

Parameter | Minimum | Maximum |
---|---|---|

${m}_{n}$ | 1 | 5 |

${\alpha}_{n}$ | 10 | 30 |

$\beta $ | 0 | 30 |

$x$ | −1 | 1 |

**Table 7.**Best solutions of the proposed algorithms for scenario I and relevant obtained input design parameters.

GWO | WOA | PSO | GA | |
---|---|---|---|---|

Normal module (${m}_{n}$) | 4.2333 | 4.0012 | 4.2122 | 5.2666 |

Normal pressure angle (${\alpha}_{n}$) | 25° | 25° | 25° | 21° |

Addendum modification ($x$) | 0.0863 | −0.0128 | 0.0055 | 0.1992 |

Helix angle ($\beta $) | 17° 59′ 58″ | 14° 36′ 32″ | 17° 02′ 11″ | 13° 52′ 20″ |

Function evaluations | 8824 | 10,000 | 10,000 | 10,000 |

Elapsed time | 549.8 | 736.4 | 1122.3 | 2364.1 |

GWO | WOA | PSO | GA | |
---|---|---|---|---|

Function evaluations (ave.) | 9502 | 10,000 | 10,000 | 10,000 |

Elapsed time (ave.) | 601.5 | 751.1 | 1206.7 | 2404 |

Standard deviation | 0.1421 | 4.0255 | 0.9054 | 29.1512 |

Function evaluations (ave.) | 9502 | 10,000 | 10,000 | 10,000 |

**Table 9.**Best solutions of the proposed algorithms for scenario II and relevant obtained input design parameters.

GWO | WOA | PSO | GA | |
---|---|---|---|---|

Normal module (${m}_{n}$) | 4.2333 | 4.2333 | 4.2333 | 4.0882 |

Normal pressure angle (${\alpha}_{n}$) | 25° | 25° | 25° | 25° |

Addendum modification ($x$) | 0.0863 | 0.0863 | 0.0863 | 0.0012 |

Helix angle ($\beta $) | 17° 59′ 58″ | 17° 59′ 58″ | 17° 59′ 58″ | 14° 12′ 10″ |

Function evaluations | 8824 | 24,541 | 11,142 | - |

Elapsed time | 549.8 | 1807.1 | 1250.5 | - |

GWO | WOA | PSO | GA | |
---|---|---|---|---|

Function evaluations (ave.) | 9816 | 34,101 | 12,294 | - |

Elapsed time (ave.) | 712.5 | 2511 | 1892.1 | - |

Standard deviation | 0.1902 | 8.9778 | 0.2252 | - |

T-Test for GWO-WOA | T-Test for GWO-PSO | |
---|---|---|

p value | <0.0001 | 0.0005 |

Significant difference (p < 0.05) | YES | YES |

t value | 12.80 | 4.211 |

df | 18 | 18 |

Mean iterations | 10,957/28,800 | 10,957/15,722 |

Difference between mean values | 17,843 ± 1394 | 4765 ± 1132 |

95% confidence interval | 14,913 to 20,773 | 2388 to 7143 |

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**MDPI and ACS Style**

Pourmostaghimi, V.; Heidari, F.; Khalilpourazary, S.; Qazani, M.R.C.
Application of Evolutionary Optimization Techniques in Reverse Engineering of Helical Gears: An Applied Study. *Axioms* **2023**, *12*, 252.
https://doi.org/10.3390/axioms12030252

**AMA Style**

Pourmostaghimi V, Heidari F, Khalilpourazary S, Qazani MRC.
Application of Evolutionary Optimization Techniques in Reverse Engineering of Helical Gears: An Applied Study. *Axioms*. 2023; 12(3):252.
https://doi.org/10.3390/axioms12030252

**Chicago/Turabian Style**

Pourmostaghimi, Vahid, Farshad Heidari, Saman Khalilpourazary, and Mohammad Reza Chalak Qazani.
2023. "Application of Evolutionary Optimization Techniques in Reverse Engineering of Helical Gears: An Applied Study" *Axioms* 12, no. 3: 252.
https://doi.org/10.3390/axioms12030252