# Statistical Analysis and Optimisation of Data for the Design and Evaluation of the Shear Spinning Process

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

^{n}). Often, an experiment design prepared in this way is impossible to implement due to the duration of the experiments, high costs or the number of available samples. Therefore, statistical planning of experiments is commonly used. Such an approach allows the number of required trials to be significantly reduced while obtaining sufficient information to build an accurate approximation of the dependent variables. The most frequently used designs are factorial, orthogonal and central composite [35]. An orthogonal design minimises the number of trials that are necessary, among which there is any combination of the values of any two factors. A method which uses orthogonal designs is the widely used Taguchi method of process optimisation [36], applied to research on the spinning process by Wang [37], and Sangkharata and Dechjarerena [38]. Analysis of the results obtained following this experimental design enables one to identify the main effects, their significance and interactions between the factors. The main effect is the influence of a specific factor on the dependent variable, independent of other factors. In contrast, in the case of factor interactions, the effect of one factor on the dependent variable varies depending on the value of the other factor. In the analysis of the shear spinning process, Kunert et al. [34] obtained interactions between the working radius of the roll and the distance between the roll and the forming block, as well as interactions between the working radius of the roll and the resulting tool path.

^{2}design), a simple way to identify the main effects and interactions is to plot the main effects. Figure 1 shows the results of theoretical experiments T

_{1}and T

_{2}, for factors A and B changing at two levels A

_{1}and A

_{2}, and B

_{1}and B

_{2}, respectively [36].

_{2}, there are no differences between the marginal means, however the level of factor A changes the influence of factor B on the results. This means that factors A and B interact. In the graph of effects, the main effect is visible as a parallel line for the results of experiments at individual levels (Figure 1a), while the crossing lines indicate the interaction of factors (Figure 1b).

## 2. Materials and Methods

_{0}were adopted as: R

_{0}= 220 rpm and S

_{0}= 20 mm/min, H

_{0}= 0. The experimental design was based on levels increased in relation to the reference values by 15% for the rotational speed, by 20% for the feed rate and by 50% for both rotational speed and feed rate. The differentiated increase for the values of the first level is a result of the need to examine the influence of the feed coefficient (f), defined as the ratio of feed rate to rotational speed. For equal percentage increments of the reference levels, the value of the feed coefficient would be the same in all cases. There are two levels of heating factor: the use of heating (1) or no heating (0). The levels of factors are shown in Table 3. The trial plan and the reference test conditions are shown in Table 4.

_{i}, which are quality indicators, it is possible to introduce their dependence from k factors (parameters) X

_{j}and an error ${\u03f5}_{\mathrm{i}}$:

_{ij}:

_{i}preference determines how much the value of $\hat{{\mathrm{Y}}_{\mathrm{i}}}$ is preferred, making it a value from 1 (most preferred) to 0 (not accepted). Based on the value of d

_{i}, the so-called global preference function G for the variables $\hat{{\text{}\mathrm{Y}}_{\mathrm{i}}}$. G function is defined as the weighted geometric mean of the preference value d

_{i}(Equation (3)):

_{i}is zero, G also becomes zero. In turn, G-values close to 1 indicate that values of d

_{i}are also close to 1 (most preferred). It means that the value of Y

_{i}is close to optimal. Subsequent maximisation of the G-function allows the process parameters that result in the best overall product quality to be found, while the individual indicators are maintained within the tolerance range.

_{i}increases (or decreases) with change invariable Y

_{i}, the goal of optimisation is to maximise (or minimise) the response Y

_{i}using a one-way preference transformation. This transform function is monotonous, reaching the value 0 below a certain fixed value T

_{min}and value 1 above a certain fixed value T

_{max}. In the interpretation, this means that it is most desirable to achieve at least the T

_{max}value, while values below T

_{min}are not accepted and the preference for them is 0. The one-sided transform function can be expressed by the following function (Figure 3a):

_{i}. For a certain response Y, its target value (the most preferred) T and the tolerance range $\left[\mathrm{T}-{\mathrm{t}}_{\mathrm{min}},\text{}\mathrm{T}+{\mathrm{t}}_{\mathrm{max}}\right]$, the transformation of the Y value to the preference value d is determined by the following function (Figure 3b):

_{i}, their transformation T

_{i}, and the set of parameters P is therefore as follows (Equation (6)):

**Z1**–

**Z5**) with different dimensional tolerances for specific nominal thicknesses of component (Figure 4) were designated.

**Z1**and

**Z6**(inner component surface) were designated which had different thickness and diameter tolerances. Then, comparisons of the 3D scan surfaces were also made with the CAD model and quality indicators were generated (SURF_Zn, where n is the number of the component zone): maximum positive surface deviation, maximum negative surface deviation, mean surface deviation, mean square deviation of the surface. All data were interpreted in the 2σ range. For overall indicators, TH_G and SURF_G, a positive (negative) tolerance was determined as the maximum (minimum) values from specific zone tolerances. For the TOTALFAIL indicators, the 50% threshold was adopted as the tolerance.

_{min}, t

_{max}) take the form (7):

_{i}sample, the following measure of the mean relative change in absolute deviation was introduced to the established reference sample P

_{r}for a certain subset of Q indicators (Equation (11)):

_{i}) is the value of the indicator q obtained in the trial P

_{i}. A function defined in this way makes it possible for the average and relative improvement of the indicator values for the sample P

_{i}, to be determined. The degree of improvement is considered to be the increase in the value of the indicators towards zero.

**Z1**,

**Z3**and

**Z5**(Figure 4).

## 3. Results and Discussion

**Z5**(40% deformation) is lower by 46.59% compared to the average grain size of the batch material, for

**Z3**(60% deformation) by 73.94%, while for

**Z1**(deformation 10%) by 39.47%. The approximate grain diameter for

**Z5**(40% deformation) is lower by 24% compared to the average grain diameter of the batch material, for

**Z3**(60% deformation) by 44%, and for Z1 (10% deformation) by 19.5%.

_{3}failed. It was impossible to uninstall a component from the spinning block. Components produced by trials V

_{1}and V

_{2}were sent for further research. After 3D scanning the results of an overall comparison of the components surface and thicknesses with the CAD model and measurement data was generated and analysed according to the experiment plan.

**Z1**–

**Z5**) and compliance of the surfaces with the tolerance in two zones (

**Z1**,

**Z6**) were analysed.

_{1}and V

_{2}were compared with regression models obtained in the optimisation process. The best compliance for the individual indicators was achieved for the second and third zones, which may result from the characteristics of those areas-they have simple geometry. For some models, the results obtained suggest the required correction of the model had occurred with the need to recognise certain experimental results as outliers.

_{1}trial with the estimates, (Figure 13a,b), with the best and worst results of the technological tests (Figure 14a,b), and with the absolute deviations of the results for the V

_{1}trial (Figure 15a,b). There is a good agreement found in the values and directions of the deviations (Figure 13a,b). The results of the validation tests were compared with the results of the P

_{3}test, for which the best results were found (eighteen indicators compliant with the tolerance limits). The mean, relative improvement of the absolute deviation (Figure 15a,b) for indicators outside the tolerance range compared to the P

_{3}trial was −10% for the V

_{1}trial.

_{2}trial with the estimates (Figure 16a,b), with the five best and worst results of the technological tests (Figure 17a,b), and with the absolute deviations of the results of the V

_{2}trial (Figure 18a,b). There is good agreement in the values and directions of the deviations (Figure 16a,b). The average, relative improvement of the absolute deviation (Figure 18a,b) for indicators outside the tolerance range compared to the P

_{3}trial was 6% for the V

_{2}trial.

_{1}and V

_{2}for the majority of indicators are correctly predicted by the models, keeping the direction and trend of the deviations (Figure 11a,b and Figure 16a,b). For the predicted number of indicators compliant with the tolerance limits, the prediction accuracy for trials V

_{1}and V

_{2}was 94% and 97%, respectively. For trials V

_{1}and V

_{2}, the number of indicators with values within the set tolerance increased by 11% (to twenty indicators within the desired range) for trial V

_{2}and decreased by 7% for trial V

_{1}(seventeen indicators within the tolerance range).

## 4. Conclusions

_{1}and V

_{2}were 94% and 97%, respectively. The values obtained in the trials V

_{1}and V

_{2}for most indicators are correctly predicted by the models, keeping the direction and trend of the deviations. Inaccuracies in the results could be the result of the approximation methods used, the quality of model fit, and the measurements carried out by the operator.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Interaction graph for experiment T

_{1}: (

**a**) no interaction between factors A and B and (

**b**) interaction of factors A and B.

**Figure 4.**Characteristic zones of the component with specific thicknesses and variations in toler-ance, (

**a**)

**Z1**–

**Z5**, (

**b**)

**Z6**.

**Figure 5.**Graphs of (

**a**) the preference function for the indicators MIN, MAX, AVG; (

**b**) the minimising preference function for the indicators MIN, MAX, AVG; (

**c**) the preference function for the SIGMA and TOTALFAIL indicators; (

**d**) the graph of the minimising preference function for the SIGMA and TOTALFAIL indicators.

**Figure 7.**SEM images of the test material microstructure (

**a**) cross section, (

**b**) longitudinal section. SEM images of the microstructure of the material after deformation taken (

**c**) from zone

**Z1**, (

**d**) zone

**Z3**, and (

**e**) zone

**Z5**.

**Figure 8.**Areas of components after technological tests within given tolerances: P

_{0}(

**a**), P

_{1}(

**b**), P

_{2}(

**c**), P

_{3}(

**d**), P

_{4}(

**e**), P

_{5}(

**f**), and P

_{6}(

**g**).

**Figure 9.**Graphs of the global preference function for (

**a**) the variant CV and weights W_ALL and (

**b**) CMV and weights W_Z; with H = 1 fixed.

**Figure 10.**Histogram of the most common sets of parameters in the 5% of the results that were the best (the darker the colour, the higher the frequency of occurrences).

**Figure 11.**Histogram of (

**a**) the most common sets of parameters for the best 5% of the CMV results and the W_ALL weight set and of (

**b**) the best 5% of the CMV results and the TH weight set (the darker the colour, the higher the frequency of occurrences).

**Figure 12.**Areas of components after technological tests within given tolerances: ${\mathrm{V}}_{1}$ (

**a**); ${\mathrm{V}}_{2}$ (

**b**).

**Figure 13.**Comparison of the estimate of the results of the V

_{1}trial (based on regression models) with the actual results of the trial for the indicators of (

**a**) deviations and (

**b**) tolerances.

**Figure 14.**Comparison of the results of the V

_{1}trial with the best and worst results of the experiments P

_{3}and P

_{1}for the indicators of (

**a**) deviation and of (

**b**) tolerance.

**Figure 15.**Comparison of the absolute deviations of the results of the V1 trial with the absolute best and average results of the experiment for the indicators of (

**a**) deviation and of (

**b**) tolerance.

**Figure 16.**Comparison of the estimates of the results of the V2 trial (based on regression models) with the actual test results for the indicators of (

**a**) deviations and of (

**b**) tolerance.

**Figure 17.**Comparison of the results of the V2 trial with the best and worst results of the experimental trials P3 and P1 for the indicators of (

**a**) deviation and of (

**b**) tolerance.

**Figure 18.**Comparison of the absolute deviations of the results of the V2 trial with the absolute best and average results of the experimental trials for the indicators of (

**a**) deviation and of (

**b**) tolerance.

T_{1} | T_{2} | ||||||
---|---|---|---|---|---|---|---|

A1 | A2 | Marginal mean | A1 | A2 | Marginal mean | ||

B1 | 10 | 10 | 10 | B1 | 10 | 20 | 15 |

B2 | 20 | 20 | 20 | B2 | 20 | 10 | 15 |

Marginal mean | 15 | 15 | Marginal mean | 15 | 15 |

No | X1 | X2 | X3 |
---|---|---|---|

1 | 1 | 1 | 1 |

2 | 1 | −1 | −1 |

3 | −1 | 1 | 1 |

4 | −1 | −1 | −1 |

Factor | Level 1 | Level 2 |
---|---|---|

S | ${\mathrm{S}}_{1}=1.2{\mathrm{S}}_{0}=36\text{}\left(\mathrm{mm}/\mathrm{min}\right)$ | ${\mathrm{S}}_{2}=1.5{\mathrm{S}}_{0}=45\text{}\left(\mathrm{mm}/\mathrm{min}\right)$ |

R | ${\mathrm{R}}_{1}=1.15{\mathrm{R}}_{0}=253\text{}\left(\mathrm{rpm}\right)$ | ${\mathrm{R}}_{1}=1.5{\mathrm{R}}_{0}=330\text{}\left(\mathrm{rpm}\right)$ |

H | 1 | 0 |

Trial | $\mathbf{S}\left(\mathbf{m}\mathbf{m}/\mathbf{m}\mathbf{i}\mathbf{n}\right)$ | $\mathbf{R}\left(\mathbf{r}\mathbf{p}\mathbf{m}\right)$ | $\mathbf{H}$ | $\mathbf{f}\left(\mathbf{S}/\mathbf{R}\right)\left(\mathbf{m}\mathbf{m}/\mathbf{o}\mathbf{b}\mathbf{r}\right)$ |
---|---|---|---|---|

${\mathrm{P}}_{0}$ | 30 | 220 | 0 | 0.136 |

${\mathrm{P}}_{1}$ | 36 | 253 | 0 | 0.142 |

${\mathrm{P}}_{2}$ | 36 | 330 | 1 | 0.109 |

${\mathrm{P}}_{3}$ | 45 | 253 | 1 | 0.177 |

${\mathrm{P}}_{4}$ | 45 | 330 | 0 | 0.136 |

${\mathrm{P}}_{5}$ | 24 | 253 | 0 | 0.095 |

${\mathrm{P}}_{6}$ | 20 | 253 | 0 | 0.080 |

Direction | Lankford Coefficient |
---|---|

0° | 0.68 |

45° | 1.25 |

90° | 1.12 |

Out-of-Tolerance Value of Preference: 0 | Preference Value out of Tolerance: 0.01 | |
---|---|---|

Regular tolerances | UV | CV |

Minimisation tolerances | UMV | CMV |

Set of Weights | Preferred Indicators (Validity 10) | Other Indicators (Validity 1) |
---|---|---|

W_ALL (all indicators) | TH_G, TH_Z, SURF_G, SURF_Z | |

W_TH (thickness indicators) | TH_G, TH_Z | SURF_G, SURF_Z |

W_SURF (surface indicators) | SURF_G, SURF_Z | TH_G, TH_Z |

W_Z (indicators of zones Z1, Z3, Z6) | TH_Z1, TH_Z3, TH_Z6, SURF_Z1, SURF_Z6 | TH_G, TH_Z2, TH_Z4, TH_Z5, SURF_G |

W_ALL | W_TH | W_SURF | W_Z | |
---|---|---|---|---|

UV | ||||

CV | [1, 39, 225] | [1, 39, 225] | [39, 225, 1] | [39, 225, 1] |

UMV | [45, 266, 1] | [45, 266, 1] | [45, 266, 1] | [45, 266, 1] |

CMV | [37, 330, 1] | [37, 330, 1] | [37, 330, 1] | [37, 330, 1] |

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**MDPI and ACS Style**

Puchlerska, S.; Żaba, K.; Pyzik, J.; Pieja, T.; Trzepieciński, T.
Statistical Analysis and Optimisation of Data for the Design and Evaluation of the Shear Spinning Process. *Materials* **2021**, *14*, 6099.
https://doi.org/10.3390/ma14206099

**AMA Style**

Puchlerska S, Żaba K, Pyzik J, Pieja T, Trzepieciński T.
Statistical Analysis and Optimisation of Data for the Design and Evaluation of the Shear Spinning Process. *Materials*. 2021; 14(20):6099.
https://doi.org/10.3390/ma14206099

**Chicago/Turabian Style**

Puchlerska, Sandra, Krzysztof Żaba, Jarosław Pyzik, Tomasz Pieja, and Tomasz Trzepieciński.
2021. "Statistical Analysis and Optimisation of Data for the Design and Evaluation of the Shear Spinning Process" *Materials* 14, no. 20: 6099.
https://doi.org/10.3390/ma14206099