A New Method for Incremental Sheet Metal Bending Based on Minimum Energy Principle *

Transcription

1 Proceedings of the IEEE International Conference on Information and Automation Ningbo, China, August 16 A New Method for Incremental Sheet Metal Bending Based on Minimum Energy Principle * Xiaobing Dang and Ruxu Du Institute of Precision Engineering Department of Mechanical and Automation Engineering The Chinese University of Hong Kong Shatin, N.T., Hong Kong {xbdang & rdu}@mae.cuhk.edu.hk Qingyin Liu Taian Hualu Metalforming Machine Tool Co., Ltd. Taian, Shandong, China hldy-1@vip.163.com Kai He and Wei Li Shenzhen Key Laboratory of Precision Engineering Shenzhen Institutes of Advanced Technology Chinese Academy of Sciences Shenzhen, Guangdong, China {kai.he, wei.li}@siat.ac.cn Abstract - Sheet Metal Bending is a common process used in many heavy industry sectors such as shipbuilding. The traditional method is the so-called line heating, which is not only labor intensive but also inefficient and error-prone. This paper presents a new incremental bending method based on minimum energy principle. In this method, the steel blank is supported by an array of hydraulic cylinders with rotary head. This ensures the blank being properly lifted regardless of its shape and size. First, the incremental punching trajectory is generated based on the minimum energy principle. Assuming that the blank can be approximated by a number of strips and each strip is supported at its end. Then the strip can be described by a simply supported beam. According to the minimum energy principle, the punch location will be at where the error between the current shape and desired shape is maximum. Upon obtaining the punch location, the bending of the strip is also computed completing one incremental step. The new method is tested in a custom build incremental bending machine. Based on the experiment results, the method is successfully used in sheet metal bending. Comparing to the existing methods, the new method has a number of advantages, including simple, fast and highly energy efficient. Index Terms - Metal forming, Incremental forming, Sheet metal bending, Forming trajectory. I. INTRODUCTION Hull forming is a fundamental process in ship building. The shape of the hull surface is designed to satisfy fluid dynamics requirement and reduce the resistance of water flow as much as possible. But the manufacturing of hull surface is not very easy. The hull surface is normally welded by pieces of complicated curved sheet metals. All of the sheet metals are of different shape. Traditional method such as stamping isn t suitable here because very large capacity of press is required to provide the forming load and different dies are also required to form different sheet metals. Both of these constrains traditional stamping applied in ship building industry. Therefore, it is necessary to develop proper process to form hull surface. Line heating has been the most widely used process to manufacture hull surface. In line heating, thermal stress is mainly utilized to make the sheet metal deform plastically. The process does not require complex equipment. It is mainly composed of a flame gun to heat the metal and a water-jet to cool the metal simultaneously. But the process is deeply dependent on worker s experience. An apprentice always needs to spend several years to grasp the technique. The environment is also badly affected by the noise and pollution emitted from the heating tools. In addition, the efficiency of the process is very low. It always takes days of time to complete a single part. Several other techniques, therefore, have been proposed to overcome the defects of line heating such as multi-point forming (MPF), roll forming (RF) and laser forming (LF). Multi-point forming is a variation of die forming. In the process, a pair of reconfigurable flexible matrices of punches are used to substitute traditional dies. The shape of upper and lower die surfaces is constructed approximately by adjusting the position of punches. Li et al. [1-3] has been taking researches about the process for many years. They developed sectional MPF to realize the manufacturing of large-scale sheet metal, and found that flexible blank holder in MPF could effectively increase the forming stability and suppress wrinkling. Heo et al. [4] studied the formability of MPF for thick sheet metal. They designed and fabricated a prototype machine with capacity of kn. Abebe et al. [5] investigated how to increase quality of work-piece in MPF such as eliminating defects of dimpling and wrinkling by adjusting process parameters. Although MPF is a potential technique with flexibility of sheet metal forming, the forming force required for the process is very large. Luo et al. [6] derived another process of cyclic multi-point incremental forming (CMPIF) based on MPF. In CMPIE, the punches are cyclically controlled to move one after another. Each time every punch only moves a small distance until all punches gradually move to final shape of the work-piece. Because every time only one punch is used * This work is partially supported by Taishan Scholar of Leading Talent in Ocean Engineering - A Special Project of Numerical Control Ship Building Machines and Shenzhen Basic Research Project under the grant number JCYJ /16/$ IEEE 1444

2 to deform the sheet metal, the punching force of CMPIF is much lower than that of MPF. In hull forming the size of sheet metal is normally very large. Large capacity of press is required for manufacturing hull surface, which in turn, limits the application of MPF in ship building industry. Roll forming is an incremental sheet metal rolling process, which is mainly used to manufacture complicated curved sheet metal. Shim et al. [7, 8] prescribed a kind of RF by line array roll set (LARS). They adopted three pairs of upper and lower roll assemblies to deform the sheet metal. The middle pair is motor driven rolls, which was used to deform and move the metal. The other two pairs were idle rolls, which were used to generate the bending deformation together with the driving rolls. Cai et al. [9, 1] improved this process and prescribed continuous roll forming (CRF). Different from LARS, they reduced the number of rolling set and used only a pair of rolling tools. The sheet metal was bent along two directions by adjusting the rotation and gaps between the two forming rolls. Wang et al. [11, 1] has taken a series of deep investigation of the process. They analyzed its forming precision and explored the causes leading to shape error. They also successfully manufactured some sheet metals of spherical surface and saddle surface. RF is a potential technique to manufacture sheet metal. However curvature of the work-piece it can manufacture could not be very large along transverse direction and the process is mainly used for thin sheet metal. Its usage for hull forming is still within constrains. Laser forming, similar with line heating, deforms sheet metal by thermal stress. The key point in LF is the generation of forming trajectory. Most researchers generated forming trajectory through stress-strain analysis. Kim and Na [13, 14] utilized geometrical information of the work-piece to generate the forming trajectory, which significantly reduced the calculation time. Seong et al. [15] furthered the study and successfully applied it in the manufacturing of large-scale sheet metals with size of 7 7 1mm. But the shape error by this process is still large and cannot be ignored for actual application. It still has a long way to apply this process in actual manufacturing. To overcome limitations of current hull forming process, we developed a novel forming method of incremental forming. In incremental forming [16, 17], only a small part of sheet metal is deformed at any time. The deformation accumulates gradually among the whole work-piece. We introduced the concept of incremental forming into bending, and proposed the incremental sheet metal bending (ISMB) [18, 19] which was mainly used to form complicated curved sheet metal, especially for hull surface. In this paper, we mainly focused on the forming trajectory of ISMB. The forming trajectory was generated based on theory of minimum energy. At each step, the sheet metal would always deform to the shape that drives the sheet metal to its minimum energy. Model-less control was also used to generate the bending position based on previous steps. Some numerical simulations were programmed to validate the algorithm of forming trajectory. The corresponding experiments were carried out to examine the performance of the forming method. The experimental results showed that the forming trajectory can be applied successfully in ISMB. II. FORMING TRAJECTORY ALGORITHM A. Incremental Sheet Metal Bending Machine A schematic illustration of the ISMB process is shown in Fig. 1. The sheet metal is properly lifted regardless of its shape and size on the flexible supporting system. It is bended gradually and incrementally step by step. A curved sheet metal can be formed as a result. Fig. 1 Schematic illustration of ISMB. The machine designed and fabricated for ISMB is shown in Fig.. It was composed of machine body, flexible supporting system and bending tools. The flexible supporting system that is composed of matrix of flexible supporting pillars is fixed on the worktable. During bending process, the pillars adjust their relative heights according to the shape of the sheet metal. Worktable moves along x direction of the machine. Bending tools move along y and z directions of the machine. They are used to bend sheet metal at different positions. They can also rotate along z axis to form different angles of alignment. The machine is controlled automatically by its control terminal. z y x Fig. Machine designed for ISMB. Bending tool Sheet metal Supporting pillars Bending tools Flexible supporting system Machine body Worktable Bed 1445

3 Surface of sheet metal Discretized strips Maximum error Bending position Shape error Current shape Supporting position Desired shape Fig. 3 Schematic illustration of discretized curves from sheet metal. Fig. 4 Determination of bending and supporting position. It is time-consuming to generate forming trajectory directly from sheet metal surface. To simplify the calculation complexity, we discretize the sheet metal into series of separate strips. Fig. 3 is an example of the discretization of the sheet metal. B. Principle of Minimum Energy Principle of minimum energy is applied to generate the incremental punching trajectory. This principle is widely used for sheet metal forming. Luo et al. [, 1], for example, applied the principle in his research about incremental forming. In the bending procedure according to the minimum energy, the sheet metal always deforms to the shape that drives the metal to its minimum energy, and the punching location will be at where the error between the current shape and desired shape is maximum. We have discretized the sheet metal into a series of strips. Each strip is supported at its end. We could just use beam theory to analyze them. According to Euler-Bernoulli beam theory, the energy (V) of beam can be calculated as L 1 w 1 w V = u EI + q x w x t dx t x [ ( ) ( ) ( ) (, )] (1) where the first term is the kinetic energy in which u is the mass per unit length, the second term is potential energy due to internal force in which E and I are elastic modulus and second moment of area respectively, the third term is potential energy due to external force q(x), and w is the deflection of beam. Since the bending process is very slow, we could regard it as quasistatic process: L 1 w V = [ q( x) w( x) EI( ) ] dx () x To generate the forming trajectory, we must first determine the positions of bending and supporting in every step. After we get all the bending and supporting positions, we can arrange them in a proper sequence to get the metal bending sequence. Then the NC code could be generated to guide the forming process. The bending and supporting position can be obtained from principle of minimum energy. For the bending position, we could select the point that make the energy of bending minimum, where in turn is just the position of the maximum shape error between current and desired shape. For the supporting position, it is just the point where the shape error is zero. Fig. 4 schematically shows how to determine the bending and supporting position for each bending step. The shape error is defined as the difference between current and desired shape E ( x, y) = S( x, y) M ( x, y) (3) i where E i (x,y) is the shape error, S(x,y) is the desired shape and M i (x,y) is the current shape which could be obtained by theoretical calculation or measurement. As we have mentioned before, we discretize the sheet metal into a series of strips. We transform the error equation into a more simplified formula along the strip E ( x, y ) = S( x, y ) M ( x, y ) (4) i j j i j where the sheet metal discretized along y direction, and y j is the position of discretized strips. C. Effect of Springback Springback is an unavoidable phenomenon in sheet metal forming process. It is caused by elastic recovery after plastic deformation of sheet metal. In our research the sheet metal is bended plastically by the forming tool at every step. After the load is removed, the elastic bending will be recovered, leading to the reduction of bending curvature. A lot of literature have been published to investigate the springback for sheet metal forming. Wang et al. [], for example, investigated the springback phenomenon and prescribed a control method in air bending. For our research, the springback is also an important aspect that will influence the bending accuracy. In our research the springback of the sheet metal is controlled according to the discretized strips. It can be approximated by beam theory. According to Euler-Bernoulli beam theory, the moment curvature equation is i EIκ = M (5) where κ is the bending curvature and M is the bending moment. 1446

4 When the bending force is large enough, the bending moment will exceed the elastic limit bending moment. We would use ideally plastic material to describe the bending model from elastic deformation to plastic deformation. κ 1 κ = e 3 M / M where M e and κ e are elastic limit bending moment and curvature respectively. The springback is caused during unloading process, during which bending moment and curvature exhibited linear relationship ΔM M e e e (6) Δκ = (7) κ where M and κ decrement of bending moment and curvature during unloading process. After unloading, the bending moment will be zero. The springback of the beam will be characterized as 3 3 κe 1 κe * * * κ = 1 + (8) κ κ κ where κ * and κ are initial bending curvature and curvature after springback respectively. D. Model-less Control Model-less control [3] is a useful method that is applied to the system without accurate mathematical model. It is a data based approach that only needs input and output data of the system. One of the most successful application of the method is the artificial neural network technique. In this paper, we apply this method in the calculation of deformed shape of the sheet metal. The model-less control is typically described as [4, 5] yk ( + 1) = f( yk ( ),... yk ( n), uk ( ),..., uk ( m)) (9) where y(k) and u(k) are output and input of the system at iteration of k, and n and m are orders selected for output and input. Here in our research, we just use first order of input and output. The equation is simplified as yk ( + 1) = f( yk ( ), uk ( )) (1) We use y(k) to describe the shape of the sheet metal after k times bending, and u(k) to describe the variation of the shape between two bending sequences. Equation (1) could be transformed to the shape description as Sk+ 1 ( x, y) = f( Sk( x, y), Δ Sk( x, y)) (11) where S(x,y) is the metal shape, ΔS is the shape variation between two bending steps and k is the bending step. In our work, we use linear function to describe the strip shape variation ΔS as a1 ( x xb) δ, x< x Δ S = a ( x xb) δ, x x b b (1) where x b is the bending position, δ is boundary condition caused by the tool, and a 1 and a are coefficients. Above we have covered the main algorithm used for the generation of the forming trajectory. The whole process of the algorithm is summarized in Fig. 5. No Fig. 5 Flow chart of algorithm o forming trajectory. III. SIMULATION AND EXPERIMENT FOR SIMPLY CURVED SHEET METAL In this section we investigated the algorithm of forming trajectory from both simulation and experiment. The simulation was mainly applied to visualize the forming procedure. The experiment was also carried out to manufacture the work-piece. We designed a curved sheet metal to be formed by ISMB. The shape of the desired work-piece was shown in Fig. 6. It was a simply curved sheet metal. Forming this kind of shape was the most fundamental procedure for more complicated curved sheet metal. By successfully manufacturing this kind of work-piece, the forming capacity of ISMB would be guaranteed Curved sheet metal Discretized strips Bending & supporting positions Shape and error analysis Satisfy requirement - Yes Workpiece -4-4 Fig. 6 Shape of desired work-piece. To manufacture the work-piece, we should generate the forming trajectory at first. According to previous research, we discretized the metal into five curves along y direction. After discretization we only generated the bending points along these curves by the algorithms described in previous section. - Minimum energy Model-less control

5 Step 1 Step Step Maximum error Shape error Current shape Desired shape Bending position Step n Step n Step i Fig. 7 Forming procedure simulated by MATLAB. (Most of the middle steps were ignored here) The algorithm was simulated step after step as in Fig. 7. At step 1, the bank sheet was just a flat metal. The maximum error was at position x =. We selected this point as the initial bending point. After first bending, we could obtain the shape of the strip after bending, which was shown in the graph of step. Using (4) we calculated the shape error after first bending which was also shown in the graph of step. The maximum error occurred this step was at position of x = -8, which was the second bending point along this curve. We repeated this procedure step after step, simulated the strip shape after every bending and calculated the shape error to find the next bending position until the maximum error satisfied the requirement. We got a series of bending points from the procedure above, arranged them in sequence as the final bending trajectory along this strip. The ultimate purpose of the algorithm was to generate bending sequence for the sheet metal, not the individual strips. For the combination of bending points along different strips, we should consider the shape of the work-piece. As in our experiment the wok-piece is simply curved sheet metal. We could just expand the bending points of individual strips into the whole sheet metal as in Fig. 8. Apart from the simulation by MATLAB, we also carried out the experiment to manufacture the work-piece by ISMB machine. We selected a blank metal of steel Q35 with 5mm in thickness. The bending points generated by the forming algorithm were transformed into the G-codes that could be implemented by the control center of the machine. The whole manufacturing process finished within just twenty minutes, which showed that the process was highly efficient compared with line heating. The final work-piece was shown in Fig. 9. It was obvious that the surface was smooth enough and the work-piece was of high quality. Defects such as dimpling and wrinkling which often occurred in MPF didn t appear in our work-piece. The maximum depth of the work-piece was about 5mm, much less than we desired. This was mainly caused by two reasons. (i) The springback was not accurately controlled in our algorithm. (ii) The strain hardening was not considered in our algorithm. In our future work, we will mainly focus on the control of the springback, improving the forming accuracy of the work-piece y (mm) Fig. 8 Bending points for the simply curved sheet metal Fig. 9 Experimental result of the work-piece. 1448

6 IV. CONCLUSIONS Hull forming was a fundamental process in ship building industry. The shape of the hull surface was designed to satisfy fluid dynamics requirement and reduce the resistance of water flow as much as possible. This paper presented a new incremental bending method for hull forming based on minimum energy principle. The algorithm of forming trajectory for ISMB was originated from traditional incremental forming process. The sheet metal was bended gradually step by step to achieve the final work-piece. The forming trajectory was generated by the error between the desired and current shape. The theory of minimum energy and model-less control were applied to calculate the bending positions and predict the shape after bending. According to the minimum energy principle, the punch location was at where the error between the current shape and desired shape is maximum. The algorithm can be successfully used to generate the forming trajectory of ISMB. To examine the validity of the algorithm, we designed an experiment to manufacture simply curved sheet metal. The MATLAB program was developed to simulate the bending process. Through the simulation we predicted the forming trajectory in advance. Experiment was also carried out to manufacture the work-piece. The final work-piece was of high quality and smooth enough. The result showed the effectiveness of the forming trajectory. It was also figured out that the algorithm prescribed in our paper can be successfully applied for ISMB. Comparing to the existing methods, the new method has a number of advantages, including simple, fast and highly energy efficient. The ISMB would be applicable in manufacturing hull surface. REFERENCES [1] H. Peng, M. Li, C. Liu, and J. Cao, "Study of multi-point forming for polycarbonate sheet," The International Journal of Advanced Manufacturing Technology, vol. 67, no. 9-1, pp , August 13. [] C. U. Liu, M. Z. Li, and W. Z. 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