# Precision Obtained Using an Artificial Neural Network for Predicting the Material Removal Rate in Ultrasonic Machining

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## Abstract

**:**

## Featured Application

**The method employing the artificial neural network proposed in this study for predicting the material removal rate in ultrasonic machining can be considered as a guide for modelling complex general machining problems without explicit mathematical functions.**

## Abstract

## 1. Introduction

## 2. Review of Our Earlier Research Work

## 3. Neural Network Modelling Method

_{i}is the normalized element of any input value x

_{i}and x

_{max}and x

_{min}are the maximum and minimum values of x

_{i}, respectively.

^{®}Core™ i3-470 CPU. Overfitting is often the main problem associated with the training of ANNs [20,21]. To avoid overfitting of the BPANN, the point at which overfitting occurred was detected by conducting a comparison of the training and validation set errors every 100 epochs to determine when an increase in the validation set error occurred in conjunction with a decrease in the training set error. In addition, a specific search program comprising a series of loops, one in each sublayer, was proposed to obtain the optimum structure of the BPANN [20,21]. The number of neurons in each sub-layer was then successively increased automatically during training. Numerous methods can be employed to evaluate the performance of a model. The simplest method is based on the value of the correlation coefficient R for a plot of predicted value versus experimental output [1]. When all the loops have finished, the program considers the optimum structure to be that obtaining an absolute value of R that is closest to one for the validation set [20,21], where |R| = 1 corresponds to perfect correlation. In addition, the determination coefficient R

^{2}is another important value for evaluating the performance of a model, where an R

^{2}value closest to one implies an optimum model.

## 4. Results and Discussion

^{2}, of the three methods are presented in Table 2, where the ideal values are M = 1, B = 0, R = 1 and R

^{2}= 1. Here, the prediction precision increases as the obtained values approach the ideal values.

## 5. Conclusions

- (1)
- The BPANN model proposed in this study and the improved nonlinear regression model established in our earlier research are useful for predicting the MRR in the USM process, but the results of the present study demonstrate that the BPANN model provides the best results and requires no explicit mathematical function.
- (2)
- The method employing ANNs proposed in this study for predicting the MRR in USM can be considered as a guide for modelling complex general machining problems without explicit mathematical functions.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

Symbolic Definition of the BPANN | |||

Definition | Symbol | ||

Input vector of the BPANN | P_{k} = (a_{1}, a_{2}, …, a_{n}) | ||

where n represents the number of input vectors of the BPANN | |||

Target vector of the BPANN | T_{k} = (y_{1}, y_{2}, …, y_{q}) | ||

where q represents the number of target vectors of the BPANN | |||

Input vector of the middle layer unit | S_{k} = (s_{1}, s_{2}, …, s_{p}) | ||

Output vector of the middle layer unit | B_{k} = (b_{1}, b_{2}, …, b_{p}) | ||

where p represents the number of vectors of the middle layer unit | |||

Input vector of the output layer unit | L_{k} = (l_{1}, l_{2}, …, l_{q}) | ||

Output vector of the output layer unit | C_{k} = (c_{1}, c_{2}, …, c_{q}) | ||

where the number of vectors of the output layer unit is equal to that of the target vectors of the ANN | |||

Connection weight of the input layer to the middle layer | w_{ij}, i = 1, 2, …, n, j = 1, 2, …, p | ||

Connection weight of the middle layer to the output layer | v_{jt}, j = 1, 2, …, p, t = 1, 2, …, q | ||

Output threshold of each unit in the middle layer | θ_{j}, j = 1, 2, …, p | ||

Output threshold of each unit in the output layer | γ_{t}, t = 1, 2, …, q | ||

Parameter k | k = 1, 2, …, m | ||

where m represents the number of parameters k | |||

Learning Steps of the BPANN | |||

Step | Description | ||

1 | Initialize and subsequently assign a random value within the interval (−1, 1) to each w_{ij}, v_{jt}, θ_{j} and γ_{t}. | ||

2 | Randomly select and apply a set of input and target vectors (i.e., P_{k} and T_{k}) to the network. | ||

3 | Calculate the input and output value s_{j} and b_{j} of each unit in the middle layer by the transfer function according to the formula $\{\begin{array}{c}{s}_{j}={\displaystyle \sum _{i=1}^{n}{w}_{ij}{a}_{i}-{\theta}_{j}}\\ {b}_{j}=f({s}_{j})\end{array}\text{\hspace{1em}}j=1,2,\dots ,p$. | ||

4 | Calculate the input and output value l_{t} and c_{t} of each unit in the output layer by the transfer function according to the formula $\{\begin{array}{c}{l}_{t}={\displaystyle \sum _{j=1}^{p}{v}_{jt}{a}_{t}-{r}_{t}}\\ {c}_{t}=f({l}_{t})\end{array}\text{\hspace{1em}}t=1,2,\dots ,q$. | ||

5 | Calculate the generalized error of each unit of the output layer ${d}_{t}^{k}$ according to the formula ${d}_{t}^{k}=({y}_{t}^{k}-{c}_{t})\times {c}_{t}(1-{c}_{t})\text{\hspace{1em}}t=1,2,\dots ,q$. | ||

6 | Calculate the generalized error of each unit of the middle layer ${e}_{j}^{k}$ according to the formula ${e}_{j}^{k}=\left[{\displaystyle \sum _{t=1}^{q}{d}_{t}\times {v}_{jt}}\right]\times {b}_{j}(1-{b}_{j})\text{\hspace{1em}}t=1,2,\dots ,q$. | ||

7 | Modify the connection weights v_{jt} and the thresholds γ_{t} according to formulas $\{\begin{array}{c}{v}_{jt}(N+1)={v}_{jt}(N)+\alpha \times {d}_{t}^{k}\times {b}_{j}\\ {\gamma}_{t}(N+1)={\gamma}_{t}(N)+\alpha \times {d}_{t}^{k}\end{array}\text{\hspace{1em}}t=1,2,\dots ,q,\text{\hspace{1em}}j=1,2,\dots ,p,\text{\hspace{1em}}0\prec \alpha \prec 1$. | ||

8 | Modify the connection weights w_{ij} and the thresholds θ_{j} according to formulas $\{\begin{array}{c}{w}_{ij}(N+1)={w}_{ij}(N)+\beta \times {e}_{j}^{k}\times {a}_{i}^{k}\\ {\theta}_{j}(N+1)={\theta}_{j}(N)+\beta \times {e}_{j}^{k}\end{array}\text{\hspace{1em}}i=1,2,\dots ,n,\text{\hspace{1em}}j=1,2,\dots ,p,\text{\hspace{1em}}0\prec \beta \prec 1$. | ||

9 | Randomly select and apply the next learning sample vectors to the network and return to Step 3 until the completion of m training samples. | ||

10 | Re-select a set of input and target sample vectors from m study samples and return to Step 3 until the network global error E is less than a pre-set minimum value, i.e., the network converges. Go to Step 11. If the number of training steps is greater than the pre-set value, the network cannot converge. | ||

11 | End. |

## Appendix B

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**Figure 2.**Correlation between the predicted values and the experimental values along with the best linear fit to the data: (

**a**) the conventional linear regression model; (

**b**) the improved nonlinear regression model; and (

**c**) the BPANN model.

**Figure 3.**Comparison of measured MRR values and the corresponding prediction values of the three models for the sixteen experimental datasets.

**Table 1.**Main details and results of our earlier research work regarding regression models [5].

Research Detail | Description | ||||||||
---|---|---|---|---|---|---|---|---|---|

Experimental USM apparatus | Primary components consist of an ultrasonic machine tool, multi-axis computer numerical control (CNC) system, ultrasonic generator and data recorder. The unit was employed to perform a series of experiments based on the Taguchi L_{16} orthogonal array. To reduce the measurement error, each experiment was conducted twice. | ||||||||

Processed workpiece | Workpiece material: soda-lime glass Shape: round head slot Plane dimension: 15 mm × 6 mm Number of reciprocating processing: 20 times. | ||||||||

Fixed factors | Ultrasonic frequency | 20.12 kHz | |||||||

Ultrasonic amplitude | 12 μm | ||||||||

Oscillating current | 300 mA | ||||||||

Abrasive slurry ingredients and concentration | Silicon carbide mixed with water in a 1:3 ratio | ||||||||

Length of ultrasonic horn | 150 mm | ||||||||

Diameter of the tool | 6 mm | ||||||||

Control factors | Symbol | Parameters | Levels | ||||||

1 | 2 | 3 | 4 | ||||||

G | Abrasive granularity (mesh) | 320 | 240 | 120 | 80 | ||||

P | Feed pressure (N) | 20 | 30 | 40 | 50 | ||||

FS | Feed speed (mm/s) | 1.32 | 2.08 | 2.94 | 3.62 | ||||

Process response | Material removal rate (MRR) | Because the plane dimension of the workpiece is certain, the MRR is calculated as follows: $MRR=D/T\text{\hspace{1em}}(\mathsf{\mu}\mathrm{m}/\mathrm{s})$ where D (um) is the processing depth, measured using a digital display depth micrometre with a resolution of 0.001 mm, and T (s) is the processing time, recorded by the data recorder. | |||||||

Intuitive analysis | The value of G exhibits an obvious and nonlinear behaviour. | ||||||||

ANOVA analysis | The effect of G was significant. | ||||||||

Regression models | Linear and nonlinear regression models given by Equations (3) and (4), respectively, in this study. | ||||||||

Comparison of the modelling precision with 8 new experimental conditions | The MRR values obtained from the nonlinear regression equation are in good agreement with the measured MRR values. |

**Table 2.**Prediction precision indicators of the three modelling methods, where the ideal values are M = 1, B = 0, R = 1 and R

^{2}= 1.

Three Modelling Methods | M | B | R | R^{2} |
---|---|---|---|---|

(a) Conventional linear regression model | 0.149 | 3.57 | 0.387 | 0.150 |

(b) Improved nonlinear regression model | 0.859 | 0.603 | 0.927 | 0.859 |

(c) Back-propagation artificial neural network model | 0.884 | 0.371 | 0.948 | 0.899 |

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**MDPI and ACS Style**

Zhong, G.; Kang, M.; Yang, S. Precision Obtained Using an Artificial Neural Network for Predicting the Material Removal Rate in Ultrasonic Machining. *Appl. Sci.* **2017**, *7*, 1268.
https://doi.org/10.3390/app7121268

**AMA Style**

Zhong G, Kang M, Yang S. Precision Obtained Using an Artificial Neural Network for Predicting the Material Removal Rate in Ultrasonic Machining. *Applied Sciences*. 2017; 7(12):1268.
https://doi.org/10.3390/app7121268

**Chicago/Turabian Style**

Zhong, Gaoyan, Min Kang, and Shoufeng Yang. 2017. "Precision Obtained Using an Artificial Neural Network for Predicting the Material Removal Rate in Ultrasonic Machining" *Applied Sciences* 7, no. 12: 1268.
https://doi.org/10.3390/app7121268