Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 7834621, 11 pages

https://doi.org/10.1155/2017/7834621

## Bending Angle Prediction Model Based on BPNN-Spline in Air Bending Springback Process

Department of Mechanical Engineering, Southeast University, Nanjing 211189, China

Correspondence should be addressed to Wencheng Tang; nc.ude.ues@cwgnat

Received 16 October 2016; Revised 16 January 2017; Accepted 7 February 2017; Published 27 February 2017

Academic Editor: Marek Lefik

Copyright © 2017 Zhefeng Guo and Wencheng Tang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In order to rapidly and accurately predict the springback bending angle in V-die air bending process, a springback bending angle prediction model on the combination of error back propagation neural network and spline function (BPNN-Spline) is presented in this study. An orthogonal experimental sample set for training BPNN-Spline is obtained by finite element simulation. Through the analysis of network structure, the BPNN-Spline black box function of bending angle prediction is established, and the advantage of BPNN-Spline is discussed in comparison with traditional BPNN. The results show a close agreement with simulated and experimental results by application examples, which means that the BPNN-Spline model in this study has higher prediction accuracy and better applicable ability. Therefore, it could be adopted in a numerical control bending machine system.

#### 1. Introduction

The sheet metal bending is an essential metalworking process in many industries [1], wherein V-die air bending is a main topic in the stamping field. The flat sheet is bent to a curved sheet by the pressing on the bending machine. Because of the existence of elastic deformation in curved sheet, springback will occur during the unloaded process of curved sheet in the bending machine. As a result, the final bending angle of curved sheet will be obtained.

Mohammadi et al. [2] have developed a springback prediction analytical formulation for sandwich sheets based on a wrap-around assumption and primary bending theories. Numerical simulation and experiments have been carried out to verify the effect of punch radius, die opening, and punch stroke on springback. Results showed that the springback is increased with an increase in punch stroke, punch radius, and die opening. De Vin [3] has presented an air bending geometric model. This model has been found to be convenient to calculate the bending angle.

Finite element (FE) simulation is a popular method at present [4], due to low time costs and simple operation. Vorkov et al. [5] have studied the springback of different types of high-strength steels using FE model. The material properties and a correct material model for high-strength steels have been obtained and built, respectively. Fu et al. [6] have developed a material constitutive model based on Hill’s yielding criterion under plane strain conditions. The multiple step incremental air bending forming and springback processes have been simulated. Jamli et al. [7] have developed an artificial neural network based on constitutive model which can be used in the FE code through user defined material subroutine. FE analysis procedures have been carried out for the springback prediction of sheet metal based on an L-bending experiment. The FE analysis presented an improvement in the prediction accuracy in comparison with the measured data.

The approximate model can be chosen to replace the FE simulation. When the approximate model is established, the response of research parameters is obtained immediately. Approximate model is an efficient research method to reduce the time consumption. Many kinds of approximate model are able to achieve the approximation of bending springback process, such as artificial neural network [8–11], radial basis function [12, 13], and response surface method [14, 15]. The error backpropagation neural network (BPNN) is one of the widely used and successful networks at present. It is particularly useful in process modeling and has been used in diverse applications, such as control, robotics, pattern recognition, forecasting, and manufacturing optimization [16]. In general, the approximate model needs a lot of samples. Orthogonal experiment [17–19], Latin hypercube design [20], and so forth can reduce the number of samples under the same approximate calculation accuracy.

FE simulation and theory calculation method are mainly used to research the V-die air bending springback process at present. Theory calculation method has better general property and faster prediction speed. However, there are many inevitable assumptions in calculation model which will result in low calculation accuracy. In terms of the FE simulation, at least to the authors’ knowledge, one FE model only simulates one group of bending data. Although it generates more accurate result, it is not speedy enough for bending angle prediction. In addition, the technical requirement for the user on FE software is higher. Therefore, there is a need towards an accurate, general, speedier, and more maneuverable means to predict the bending angle.

In this study, the authors propose a springback bending angle prediction model which is the combination of BPNN and spline function (BPNN-Spline). The BPNN-Spline combined model is expressed by simple mathematical formulas. The complex numerical simulation process will be turned into a simple mathematical calculation procedure. It enables a faster and maneuverable bending angle prediction in comparison with FE simulation. Besides, the BPNN-Spline combined model is developed by the accurate FE simulation results, where the prediction accuracy is relatively high. It will offset the disadvantages of FE simulation and theory calculation method at the same time. Due to the full utilization of better generalization ability of BPNN and smooth feature of the spline function, the model outperforms traditional BPNN model. Also, through the test of application examples, the general and precise properties of BPNN-Spline are also fully demonstrated. During the bending process, input the parameters to the BPNN-Spline black box; the springback bending angle is obtained immediately. Consequently, as a computation method, the model can be applied properly in a bending machine numerical control system.

#### 2. Introduction of Air Bending Springback Process

The flat metal sheet is bent into a curved sheet with angle on the bending machine, as shown in Figure 1(a). Due to the existence of material elasticity, when the punch is unloaded, the springback will occur, and the curved sheet will become angle (hereinafter uniformly referred to as bending angle) from , as shown in Figure 1(b), where , , , , , and are the nominal width of die notch, die corner radius, punch corner radius, sheet thickness, die angle, and punch displacement, respectively.