## 4.2. Microstructure of gear

Analysis of the microstructure of the gear produced by flowforming helps to understand the deformation mechanism. In this research, samples from the preform and gear (in four passes) were prepared. Microstructures of the samples were obtained by means of a light microscope, and the results are shown in Fig. 10 **and** Fig. 11. As can be seen in Fig. 10, the microstructure of the preform consists of equiaxed grains, but as shown in Fig. 11, severe deformation and misaligned orientation of the grains in the gear are quite evident, so an inhomogeneous plastic deformation can be inferred. As shown in Fig. 11, the grains are oriented in the tangential and radial directions so that the mandrel grooves are snugly filled. In each pass, the amounts of elongation and deformation of the grains are increased until the fourth pass in which the maximum elongation of the grains is achieved.

## 4.3. Effective parameters and statistical optimization

Achieving a specific geometry is important in manufacturing industrial components; thus, the investigation of the influence of each parameter on teeth height is necessary. However, there is no concrete objective function to be used by statistical methods for optimizing the process parameters. Response surface method (RSM) was used to investigate the effect of each parameter on teeth height. Response surface method is a statistical method that is used to model and analyze processes that are affected by several parameters. The goal of this method is to model and optimize the response [14]. In this study, a central composite design (CCD) was applied. In this process, four parameters including roller diameter, thickness reduction percentage, feed rate and attack angle (as shown in Fig. 12) are more important than others [1]. The levels of these parameters is given in Table 2 According to the applied method, 31 experiments were considered with α = 2.

After doing the tests, the teeth height was obtained for each test, and the ANOVA results were obtained (Table 3). Figure 13 presents the residual distribution of the present study, and the normality of the distribution can be confirmed. A significance level of 95% was selected that the results are correct with a confidence level of 95%. Therefore, a parameter is significant if the P-value is less than 0.05.

According to Table 3, all parameters and interactions are significant and affect the teeth height. Pareto chart is shown in Fig. 14, which expresses the magnitude of the effect of each parameter on teeth height. According to Fig. 14, attack angle (α), thickness reduction percentage (T), interaction between roller diameter and attack angle (D × α), and interaction between roller diameter and feed rate (D × f) are, respectively, the most significant parameters affecting the teeth height. In this analysis, R-Sq = 99.99 and R-Sq (adj) = 99.98 that confirm ultra-high accuracy of the model developed using RSM. To investigate the influences of the parameters effective on the teeth height, the main effects and interactions should be investigated precisely. In this section, the influence of each parameter will be discussed. In the analysis of interactions, other parameters were considered in a balanced mode (central point) of tests.

Table 2

Effective parameters and their levels

parameter | Roller diameter(D) | Thickness reduction (T %) | Feed rate(f) | Attack angle(α) |

Low level | 20 mm | 15% | 0.05 mm/rev | 20º |

High level | 60 mm | 35% | 0.25 mm/rev | 60º |

Table 3

ANOVA table for teeth height.

Source | DF | Adj SS | Adj MS | F-Value | P-Value |

Model | 14 | 0.345173 | 0.024655 | 24023.37 | 0.000 |

Linear | 4 | 0.270140 | 0.067535 | 65804.28 | 0.000 |

D | 1 | 0.000032 | 0.000032 | 31.62 | 0.000 |

T | 1 | 0.050078 | 0.050078 | 48794.72 | 0.000 |

f | 1 | 0.002646 | 0.002646 | 2577.73 | 0.000 |

α | 1 | 0.221872 | 0.221872 | 216185.51 | 0.000 |

Square | 4 | 0.022202 | 0.005551 | 5408.29 | 0.000 |

D*D | 1 | 0.000402 | 0.000402 | 391.79 | 0.000 |

T*T | 1 | 0.008177 | 0.008177 | 7967.74 | 0.000 |

f*f | 1 | 0.000009 | 0.000009 | 8.52 | 0.011 |

α*α | 1 | 0.011106 | 0.011106 | 10821.19 | 0.000 |

2-Way Interaction | 6 | 0.044409 | 0.007401 | 7211.73 | 0.000 |

D*T | 1 | 0.008587 | 0.008587 | 8367.28 | 0.000 |

D*f | 1 | 0.011112 | 0.011112 | 10827.42 | 0.000 |

D*α | 1 | 0.012561 | 0.012561 | 12238.90 | 0.000 |

T*f | 1 | 0.000686 | 0.000686 | 668.81 | 0.000 |

T*α | 1 | 0.005527 | 0.005527 | 5385.69 | 0.000 |

f*α | 1 | 0.000396 | 0.000396 | 386.19 | 0.000 |

Error | 15 | 0.000015 | 0.000001 | | |

Lack-of-Fit | 9 | 0.000011 | 0.000001 | 1.81 | 0.242 |

Pure Error | 6 | 0.000004 | 0.000001 | | |

Total | 29 | 0.345189 | | | |

## 4.3.1. Influence of roller diameter

The influence of the roller diameter on teeth height is shown in Fig. 15, which indicates that the teeth height increases with increasing the roller diameter up to 40 mm and decreases with further increase. Additionally, according to the D × T interaction, which is shown in Fig. 16, the height increases with increasing the roller diameter at low thickness reduction percentages, but at values above 25%, the height decreases. As the roller diameter increases, the plastic deformation zone increases, and this leads to an increase in material flow beneath the roller and teeth height. However, as the roller diameter increases (more than 40 mm in Fig. 15 and at thickness reduction values above 25% in Fig. 16), the S/L ratio (circumferential contact length (S) to axial contact length (L)) increases, and due to the friction, the material flow increases in axial direction. However, to increase the gear height, the axial flow must be reduced. According to the D × f interaction in Fig. 17, the increase in feed rate decreases the teeth height because at high feed rates, the material does not remain beneath the roller and tends to escape from underneath it and flow in the opposite direction of the roller axial movement. However, this effect is reversed by increasing the roller diameter. As the roller diameter increases, the contact area becomes larger and the engagement of roller and the workpiece increases, so the material escape from the roller less frequently. Consequently, increasing the diameter of the roller results in a better flow of materials in the radial direction and an increased teeth height. According to Fig. 18, which shows the D × α interaction, the teeth height increases with increasing the roller diameter.

## 4.3.2. Influence of thickness reduction percentage

According to the main effect of thickness reduction percentage (Fig. 19) as well as the interactions of T × α (Fig. 20) and D × T (Fig. 16), the teeth height increases with increasing the thickness reduction percentage. As the thickness reduction percentage increases, the plastic deformation zone increases, and this causes an increase in material flow and teeth height. In addition, at high thickness reduction, the S/L ratio decreases and the axial flow is restricted.

## 4.3.3. Influence of feed rate

According to Fig. 21, increasing the feed rate reduces the teeth height, as described in Subsection 4.1. Increasing feed rate increases the S/L ratio, and due to the friction, the material flow increases in the axial direction and the teeth height decreases. The interaction of D × f was explained in Subsection 4.1.

## 4.3.4. Influence of attack angle

The influence of attack angle is shown in Fig. 22, which indicates that the teeth height decreases with increasing the attack angle. This effect can also be seen in Fig. 18 (interaction of D × α) and Fig. 20 (interaction of T × α). If the attack angle is zero, the flow of materials is in the radial direction and increases the gear height. As the attack angle increases, the axial flow of the material also increases, and the gear height decreases.

## 4.3.4. Response optimization

In the previous sections, the effective parameters on teeth height were found. In this section, we can find the situation for optimizing the response by using response optimization. In fact, in this method, from the selected levels, the best settings are set to achieve the desired goal, which is to achieve maximum teeth height. As shown in Fig. 23, a maximum teeth height of 0.7272 is obtained for D = 20 mm, T = 35%, f = 0.05 mm/rev and α = 20º.