- Email: [email protected]
Curvature prediction in air bending of metal sheet
Journal of Materials Processing Technology 100 (2000) 257±261
Curvature prediction in air bending of metal sheet Leo J. De Vin* University of SkoÈvde, Box 408, S-541 28 SkoÈvde, Sweden Accepted 17 December 1999
Abstract The paper describes the use of bending models for air bending and focuses in particular on the importance of adequate determination of sheet curvatures, especially the curvature under the punch nose. After an introduction to air bending, the principles of models which are based on the assumption that the sheet wraps around the punch are described. This `wrap-around' assumption is a limitation and often not in accordance with industrial practice. A method to predict the sheet curvature under the punch, thus eliminating the need for the wrap-around assumption, is discussed and its results are compared with experimental results. The paper then describes how models based on this method can be used to improve adaptive control methods in air bending. # 2000 Elsevier Science S.A. All rights reserved. Keywords: Metal forming; Air bending; Curvatures; Bending models; Adaptive control
1. Introduction Decreasing batch sizes in sheet metal part manufacturing, together with higher accuracy demands and shorter lead times require a bending process that is ¯exible, effective and ef®cient. As opposed to closed die bending, air bending has the potential to meet these requirements. The possibility of controlling the bend angle by means of the punch displacement allows to bend sheets to different angles without having to change the tools. It is also possible to address variations in sheet behaviour, in particular it is possible to compensate for different spring-back angles, again resulting in a reduced need for tool changes. This makes the air bending process very well suited for use in a small batch part manufacturing environment with a high variety of components, material types and sheet thickness. Dimensional accuracy of brake-formed components (some examples are shown in Fig. 1) requires special attention. This is partly due to the higher accuracy requirements, for instance when parts serve as replacements for mass produced parts in the repair sector. Another reason for this is that in air bending, the material behaviour becomes very important. The material behaviour in¯uences the bend angle under loading conditions and after spring-back, as well as the bend allowance. Inadequate modelling of the process
* Tel.: 46-500-464657; fax: 46-500-464695. E-mail address: [email protected] (L.J. De Vin).
and the material behaviour, as well as problems with the identi®cation of material parameters, can result in incorrect blank sizes or incorrect bend angles [1]. Increasingly, adaptive control methods are applied to improve the accuracy of bend angles. Even then, the availability of an adequate process model can improve the results signi®cantly. 2. Air bending 2.1. Principle of air bending and resulting sheet geometry Fig. 2 shows the principles of a set-up for air bending. After positioning and clamping the sheet, the punch is lowered to a given (usually pre-calculated) position. During retracting of the punch (`unloading'), spring-back occurs. The relationship between the punch penetration and the bend angle under loading is non-linear, as is the relationship between unloading and spring-back angle. Fig. 3 shows the geometry of a bend in more details. In principle, four different zones can be distinguished. The zone indicated with `a' follows the shape of the punch. This zone is usually called `wrap-around zone'. Zone `b' is plastically deformed and has a varying curvature due to the different bending moments at different locations. Zone `c' has undergone elastic deformation only and hence becomes straight after unloading. Zone `d' is undeformed for monolithic materials. When bending laminates, a socalled `gull wing effect' can occur. This, however, is beyond
0924-0136/00/$ ± see front matter # 2000 Elsevier Science S.A. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 9 9 ) 0 0 4 8 9 - 6
258
L.J. De Vin / Journal of Materials Processing Technology 100 (2000) 257±261
Fig. 1. Examples of brake-formed components.
the scope of this paper. The wrap-around zone is often absent, depending on material behaviour, bend angle and tool geometry. This makes calculation of bend allowance and punch penetration more complicated. 2.2. Simple models The calculation of the punch penetration, and also of the bend allowance, can be carried out with the use of an analytical bending model. The simplest bending model is based on the assumption that the cross-section of a bent sheet consists of a circular part and two straight legs (Fig. 4). This results in a fairly simple equation for the relationship between tool geometry, sheet thickness, punch penetration and bend angle: Z
W tan b=2 1 ÿ cos b=2 ÿ tan b=2sin b=2 2 (1) Rp Rd s
It will be obvious that due to the large number of simplifying assumptions, results obtained from this model are generally not very good. Sometimes the model is used to calculate the punch penetration for the ®rst bending step when adaptive control or a trial-and-error method is deployed to arrive at a suf®ciently accurate bend angle. The assumption made when using this method is that the model will always give a safe value for the punch penetration, i.e. due to springback, a larger penetration would be needed to arrive at the required angle. However, this is not always the case. When no wrap-around occurs, the use of the simple model can result in overbending of the sheet. This can occur when the inner radius of the sheet is signi®cantly larger than the punch radius. Therefore, this method should only be applied when a realistic prediction of the sheet radius under the punch can be made. When a realistic value for the sheet radius under the punch is used in Eq. (1), then the model can serve to calculate the
Fig. 2. Main stages in air bending.
Fig. 3. Sections in a bent sheet.
sensitivity of the bend angle to the punch penetration. This can be useful when applying trial-and-error or adaptive control, especially when only small corrections are needed to arrive at the required angle. The model is easily extended with spring-back by applying elastic beam theory. However, this makes the danger of overbending even larger. Provided again that a realistic value for the sheet radius is used and that material behaviour is addressed adequately, such models can give a rough indication of the spring-back angles to be expected. 2.3. Wrap-around models The expression `wrap-around models' is used here for models which calculate the sheet geometry on the basis of the local bending moments, and which use punch radius information as a starting point [2,3]. Wrap-around behaviour is assumed which allows to calculate the bending moment under the punch. From this, local bending moments and local curvatures can be calculated although it takes some iterations to determine the size of the wrap-around zone. These models provide a more realistic determination of the sheet geometry and hence of the required punch displacement than the simpler models do. Furthermore, the calculation of the bend allowance is superior to the use of values provided by standards. The deformed zone is found to be much larger than would be expected on the basis of the punch radius alone. This can provide useful information for design and helps to avoid assembly problems such as shown in Fig. 5. Alternatively, the admissible die opening can be determined for a given product design. Although wrap-around models offer a better description of the bending process than the simpler models, the wrap-
Fig. 4. Simpli®ed geometry.
L.J. De Vin / Journal of Materials Processing Technology 100 (2000) 257±261
259
Fig. 5. Unexpected assembly problem.
around assumption remains a limitation. In many cases, wrap-around does not occur during bending. The assumption also renders the models useless when a faceted (instead of a radiused) punch is used. 3. Curvature prediction under the punch 3.1. Need for curvature prediction The wrap-around assumption reduces the applicability of analytical models. Therefore, adequate prediction of the sheet curvature under the punch would result in a major improvement of these models. Some practical thumb-rules to predict the relationship between the die opening and the radius exists, but these are not suf®ciently accurate and pertain to 908 bends only. 3.2. A method for curvature prediction A method to address the problem is based on the minimisation of the bending energy [4]. The required bending energy is different for different radii (due to different bending moments and leverage). It is assumed that at each point of the bending process, the process will adjust itself in such a way that the bending energy is minimised. In this way, the radius decreases gradually. This can be simulated by comparing the force±displacement curves for two slightly different radii as shown in Fig. 6. Initially, the sheet will bend with the larger radius R (a suf®ciently large start value should be taken). When, at punch penetration Z*, the two
Fig. 6. Minimisation of bending energy.
Fig. 7. Radius under the punch nose as function of the die opening.
curves cross, the sheet will continue to bend with the smaller radius R0 . This is a continuous process and the radius will decrease gradually until (i) the end of the punch travel is reached, (ii) the wrap-around radius is reached, or (iii) a decrease of radius does no longer result in a reduced bending energy. In the case of the latter situation, this means that further plastic deformation takes place by reducing the local radii in the bend's legs and by deformation of fresh material that is pulled into the die opening. 3.3. Simulation results To test the validity of the method, mild steel with a sheet thickness of 2.95 mm has been bent to an angle of 908 with the use of various dies. Fig. 7 shows the inner product radius Ri as a function of the die opening W. The radii predicted with the model form an almost perfectly straight line. The equation for this line is Ri0.183Wÿ3.43, which corresponds fairly well with practical thumb rules for 908 bends. With the model, the bend radius for bend angles other than 908 can be predicted as well. Fig. 8 shows the predicted radius as a function of the punch displacement Z. The curve shows a good correspondence with the four measured
Fig. 8. Radius under the punch nose as function of the punch penetration.
260
L.J. De Vin / Journal of Materials Processing Technology 100 (2000) 257±261
values, also shown in Fig. 8. After a punch penetration of 45 mm, the radius did not decrease any further and wraparound did not occur. 4. Adaptive control of brake-forming 4.1. Adaptive control methods Adaptive control methods can be divided in two main groups. The ®rst group consists of methods which are based on angle measurement. The other group of methods are based on analysis of the process curve (force±displacement data). Since the objective of adaptive control is to obtain accurate bend angles, it seems obvious to try adaptive control based on measurement of the bend angle. In practice, continuous measurement of the angle is not a feasible option. Although technically possible, it would increase bend processing times, impose unnecessary constraints on bending sequences, and be possible for only a very limited range of product types. Therefore, these methods are usually based on one measurement towards the end of the bending cycle, after which a new value for the punch penetration is calculated. First, a safe value for the punch penetration is calculated and the punch is lowered (for upstroke press brakes the die is raised) to this position. Next, the resulting bend angle is measured. This can be done mechanically with a probe, a faceted disc [5] or with discs of unequal size, or contactless with an overhead camera [6] or a laser sensor [7]. However, this gives information about the angle under loading conditions only. In order to be able to calculate the required punch penetration, the spring-back angle must be calculated as well. For this, often a partial unloading cycle is used, full unloading being impossible as this would release the sheet from the press brake. Partial unloading requires force measurement over a small punch travelling distance which has been reported to be not always very accurate [7]. Furthermore, it has since long been established [5] that relationships between punch retraction, spring-back angle and force reduction are nonlinear which makes adequate extrapolation even more dif®cult. Other methods are based on the analysis of in-process data. Stelson [8,9] describes the principles of this method, data logging is described in more detail in [10]. Force and displacement data are collected during bending. From this, a moment±curvature relationship can be obtained with the use of a number of mathematical operations. Prediction of the spring-back angle is less problematic since the moment± curvature relationship is determined for the whole sheet. This allows to construct the sheet geometry under loading conditions. By applying elastic beam theory, local springback can be calculated and this allows to construct the overall sheet geometry after spring-back. However, changes in Young's modulus [11] should be taken into account when calculating local spring-back [3].
4.2. The use of curvature prediction in adaptive control When applying the ®rst method of adaptive control, a safe value for the initial punch penetration must be calculated. As mentioned before, under-estimation of the bend radius can result in overbending of the sheet. Prediction of the curvature under the punch nose in combination with the use of a simple model can reduce this danger and prediction of local curvatures in combination with adequate modelling of the overall sheet geometry can eliminate it completely. When the second method of adaptive control is used, two problems occur. Firstly, the contact points between the sheet and the die change continuously, resulting in changed leverage and direction of forces. Secondly, the data acquisition zone (i.e. the part of the punch travel distance during which measurements are taken) must be determined. The ®rst problem can be solved by running an off-line simulation with provisional material data. This simulation gives adequate information about the location of the contact points and the direction of the forces. Furthermore, conversion factors can be generated which allows for fast processing of measured data. The second problem can be solved by prediction of the radius under the punch as a function of the punch penetration. When, at some stage, the radius does not decrease any more (for instance due to wrap-around), then data measured beyond this point should be excluded when constructing the moment±curvature relationship. Hence, curvature prediction can increase both the speed and the accuracy of the method [12]. 5. Conclusions Air bending potentially offers the ¯exibility required for bending sheet metal components with a high variety in small batches. However, modelling and controlling of the process present some problems. Models based on the wrap-around assumption have only a very limited applicability. A method based on minimisation of bending energy can be used to predict the sheet radius under the punch. The method gives good results when compared with experiments and makes it possible to drop the wrap-around assumption. In air bending, increasingly adaptive control is applied to address the problem of variable material properties. However, the use of adaptive control is no substitute for process knowledge. On the contrary, the use of adequate bending models in adaptive control can prevent irreversible errors such as overbending, it can avoid erroneous interpretation of data, and speed up data processing.
References [1] A.H. Streppel, L.J. De Vin, J. Brinkman, H.J.J. Kals, Suitability of sheet bending modelling techniques in CAPP applications, J. Mater. Process. Technol. 36 (1993) 139±156.
L.J. De Vin / Journal of Materials Processing Technology 100 (2000) 257±261 [2] C. Wang, O. Kinzel, T. Altan, Mathematical modeling of plane-strain bending of sheet and plate, J. Mater. Process. Technol. 39 (1993) 279±304. [3] L.J. De Vin, A.H. Streppel, U.P. Singh, H.J.J. Kals, A process model for air bending, J. Mater. Process. Technol. 57(1/2) (1996) 48±54. [4] L.J. De Vin, U.P. Singh, W. Urquhart, H.J.J. Kals, Material behaviour in air bending, AMPT '95 Conference, 1995, pp. 88±96. [5] E. Von Finckenstein, R. Rothstein, G. von Blechen, Automatische Ruckfederungskorrektur, Industrieanzeiger 24 (1988) 19±20. [6] W. Heckel, V. Franke, In-process measurement of bending angles, SheMet '95 Conference, 1995, pp. 272±281. [7] D. Lutters, A.H. Streppel, B. Kroeze, H.J.J. Kals, Adaptive press brake control in air bending, SheMet '97 Conference, 1997, pp. 471± 480.
261
[8] K.A. Stelson, Real time identi®cation of workpiece-material characteristics from measurements during brakeforming, J. Eng. Ind., Trans. ASME 105 (1993) 45±53. [9] K.A. Stelson, An adaptive pressbrake control for strain hardening materials, J. Eng. Ind, Trans. ASME 108 (1986) 127±132. [10] S. Pickering, Intelligent processing of materials, J. Mater. Process. Technol. 36 (1993) 447±465. [11] W. Lems, The change of Young's modulus after deformation at low temperature and its recovery, Ph.D. Thesis, University of Delft, 1963. [12] L.J. De Vin, U.P. Singh, Adaptive control of mechanical processes: brakeforming of metal sheet as an example, Mechatronics '98 Conference, 1998, pp. 425±431.
Curvature prediction in air bending of metal sheet Leo J. De Vin* University of SkoÈvde, Box 408, S-541 28 SkoÈvde, Sweden Accepted 17 December 1999
Abstract The paper describes the use of bending models for air bending and focuses in particular on the importance of adequate determination of sheet curvatures, especially the curvature under the punch nose. After an introduction to air bending, the principles of models which are based on the assumption that the sheet wraps around the punch are described. This `wrap-around' assumption is a limitation and often not in accordance with industrial practice. A method to predict the sheet curvature under the punch, thus eliminating the need for the wrap-around assumption, is discussed and its results are compared with experimental results. The paper then describes how models based on this method can be used to improve adaptive control methods in air bending. # 2000 Elsevier Science S.A. All rights reserved. Keywords: Metal forming; Air bending; Curvatures; Bending models; Adaptive control
1. Introduction Decreasing batch sizes in sheet metal part manufacturing, together with higher accuracy demands and shorter lead times require a bending process that is ¯exible, effective and ef®cient. As opposed to closed die bending, air bending has the potential to meet these requirements. The possibility of controlling the bend angle by means of the punch displacement allows to bend sheets to different angles without having to change the tools. It is also possible to address variations in sheet behaviour, in particular it is possible to compensate for different spring-back angles, again resulting in a reduced need for tool changes. This makes the air bending process very well suited for use in a small batch part manufacturing environment with a high variety of components, material types and sheet thickness. Dimensional accuracy of brake-formed components (some examples are shown in Fig. 1) requires special attention. This is partly due to the higher accuracy requirements, for instance when parts serve as replacements for mass produced parts in the repair sector. Another reason for this is that in air bending, the material behaviour becomes very important. The material behaviour in¯uences the bend angle under loading conditions and after spring-back, as well as the bend allowance. Inadequate modelling of the process
* Tel.: 46-500-464657; fax: 46-500-464695. E-mail address: [email protected] (L.J. De Vin).
and the material behaviour, as well as problems with the identi®cation of material parameters, can result in incorrect blank sizes or incorrect bend angles [1]. Increasingly, adaptive control methods are applied to improve the accuracy of bend angles. Even then, the availability of an adequate process model can improve the results signi®cantly. 2. Air bending 2.1. Principle of air bending and resulting sheet geometry Fig. 2 shows the principles of a set-up for air bending. After positioning and clamping the sheet, the punch is lowered to a given (usually pre-calculated) position. During retracting of the punch (`unloading'), spring-back occurs. The relationship between the punch penetration and the bend angle under loading is non-linear, as is the relationship between unloading and spring-back angle. Fig. 3 shows the geometry of a bend in more details. In principle, four different zones can be distinguished. The zone indicated with `a' follows the shape of the punch. This zone is usually called `wrap-around zone'. Zone `b' is plastically deformed and has a varying curvature due to the different bending moments at different locations. Zone `c' has undergone elastic deformation only and hence becomes straight after unloading. Zone `d' is undeformed for monolithic materials. When bending laminates, a socalled `gull wing effect' can occur. This, however, is beyond
0924-0136/00/$ ± see front matter # 2000 Elsevier Science S.A. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 9 9 ) 0 0 4 8 9 - 6
258
L.J. De Vin / Journal of Materials Processing Technology 100 (2000) 257±261
Fig. 1. Examples of brake-formed components.
the scope of this paper. The wrap-around zone is often absent, depending on material behaviour, bend angle and tool geometry. This makes calculation of bend allowance and punch penetration more complicated. 2.2. Simple models The calculation of the punch penetration, and also of the bend allowance, can be carried out with the use of an analytical bending model. The simplest bending model is based on the assumption that the cross-section of a bent sheet consists of a circular part and two straight legs (Fig. 4). This results in a fairly simple equation for the relationship between tool geometry, sheet thickness, punch penetration and bend angle: Z
W tan b=2 1 ÿ cos b=2 ÿ tan b=2sin b=2 2 (1) Rp Rd s
It will be obvious that due to the large number of simplifying assumptions, results obtained from this model are generally not very good. Sometimes the model is used to calculate the punch penetration for the ®rst bending step when adaptive control or a trial-and-error method is deployed to arrive at a suf®ciently accurate bend angle. The assumption made when using this method is that the model will always give a safe value for the punch penetration, i.e. due to springback, a larger penetration would be needed to arrive at the required angle. However, this is not always the case. When no wrap-around occurs, the use of the simple model can result in overbending of the sheet. This can occur when the inner radius of the sheet is signi®cantly larger than the punch radius. Therefore, this method should only be applied when a realistic prediction of the sheet radius under the punch can be made. When a realistic value for the sheet radius under the punch is used in Eq. (1), then the model can serve to calculate the
Fig. 2. Main stages in air bending.
Fig. 3. Sections in a bent sheet.
sensitivity of the bend angle to the punch penetration. This can be useful when applying trial-and-error or adaptive control, especially when only small corrections are needed to arrive at the required angle. The model is easily extended with spring-back by applying elastic beam theory. However, this makes the danger of overbending even larger. Provided again that a realistic value for the sheet radius is used and that material behaviour is addressed adequately, such models can give a rough indication of the spring-back angles to be expected. 2.3. Wrap-around models The expression `wrap-around models' is used here for models which calculate the sheet geometry on the basis of the local bending moments, and which use punch radius information as a starting point [2,3]. Wrap-around behaviour is assumed which allows to calculate the bending moment under the punch. From this, local bending moments and local curvatures can be calculated although it takes some iterations to determine the size of the wrap-around zone. These models provide a more realistic determination of the sheet geometry and hence of the required punch displacement than the simpler models do. Furthermore, the calculation of the bend allowance is superior to the use of values provided by standards. The deformed zone is found to be much larger than would be expected on the basis of the punch radius alone. This can provide useful information for design and helps to avoid assembly problems such as shown in Fig. 5. Alternatively, the admissible die opening can be determined for a given product design. Although wrap-around models offer a better description of the bending process than the simpler models, the wrap-
Fig. 4. Simpli®ed geometry.
L.J. De Vin / Journal of Materials Processing Technology 100 (2000) 257±261
259
Fig. 5. Unexpected assembly problem.
around assumption remains a limitation. In many cases, wrap-around does not occur during bending. The assumption also renders the models useless when a faceted (instead of a radiused) punch is used. 3. Curvature prediction under the punch 3.1. Need for curvature prediction The wrap-around assumption reduces the applicability of analytical models. Therefore, adequate prediction of the sheet curvature under the punch would result in a major improvement of these models. Some practical thumb-rules to predict the relationship between the die opening and the radius exists, but these are not suf®ciently accurate and pertain to 908 bends only. 3.2. A method for curvature prediction A method to address the problem is based on the minimisation of the bending energy [4]. The required bending energy is different for different radii (due to different bending moments and leverage). It is assumed that at each point of the bending process, the process will adjust itself in such a way that the bending energy is minimised. In this way, the radius decreases gradually. This can be simulated by comparing the force±displacement curves for two slightly different radii as shown in Fig. 6. Initially, the sheet will bend with the larger radius R (a suf®ciently large start value should be taken). When, at punch penetration Z*, the two
Fig. 6. Minimisation of bending energy.
Fig. 7. Radius under the punch nose as function of the die opening.
curves cross, the sheet will continue to bend with the smaller radius R0 . This is a continuous process and the radius will decrease gradually until (i) the end of the punch travel is reached, (ii) the wrap-around radius is reached, or (iii) a decrease of radius does no longer result in a reduced bending energy. In the case of the latter situation, this means that further plastic deformation takes place by reducing the local radii in the bend's legs and by deformation of fresh material that is pulled into the die opening. 3.3. Simulation results To test the validity of the method, mild steel with a sheet thickness of 2.95 mm has been bent to an angle of 908 with the use of various dies. Fig. 7 shows the inner product radius Ri as a function of the die opening W. The radii predicted with the model form an almost perfectly straight line. The equation for this line is Ri0.183Wÿ3.43, which corresponds fairly well with practical thumb rules for 908 bends. With the model, the bend radius for bend angles other than 908 can be predicted as well. Fig. 8 shows the predicted radius as a function of the punch displacement Z. The curve shows a good correspondence with the four measured
Fig. 8. Radius under the punch nose as function of the punch penetration.
260
L.J. De Vin / Journal of Materials Processing Technology 100 (2000) 257±261
values, also shown in Fig. 8. After a punch penetration of 45 mm, the radius did not decrease any further and wraparound did not occur. 4. Adaptive control of brake-forming 4.1. Adaptive control methods Adaptive control methods can be divided in two main groups. The ®rst group consists of methods which are based on angle measurement. The other group of methods are based on analysis of the process curve (force±displacement data). Since the objective of adaptive control is to obtain accurate bend angles, it seems obvious to try adaptive control based on measurement of the bend angle. In practice, continuous measurement of the angle is not a feasible option. Although technically possible, it would increase bend processing times, impose unnecessary constraints on bending sequences, and be possible for only a very limited range of product types. Therefore, these methods are usually based on one measurement towards the end of the bending cycle, after which a new value for the punch penetration is calculated. First, a safe value for the punch penetration is calculated and the punch is lowered (for upstroke press brakes the die is raised) to this position. Next, the resulting bend angle is measured. This can be done mechanically with a probe, a faceted disc [5] or with discs of unequal size, or contactless with an overhead camera [6] or a laser sensor [7]. However, this gives information about the angle under loading conditions only. In order to be able to calculate the required punch penetration, the spring-back angle must be calculated as well. For this, often a partial unloading cycle is used, full unloading being impossible as this would release the sheet from the press brake. Partial unloading requires force measurement over a small punch travelling distance which has been reported to be not always very accurate [7]. Furthermore, it has since long been established [5] that relationships between punch retraction, spring-back angle and force reduction are nonlinear which makes adequate extrapolation even more dif®cult. Other methods are based on the analysis of in-process data. Stelson [8,9] describes the principles of this method, data logging is described in more detail in [10]. Force and displacement data are collected during bending. From this, a moment±curvature relationship can be obtained with the use of a number of mathematical operations. Prediction of the spring-back angle is less problematic since the moment± curvature relationship is determined for the whole sheet. This allows to construct the sheet geometry under loading conditions. By applying elastic beam theory, local springback can be calculated and this allows to construct the overall sheet geometry after spring-back. However, changes in Young's modulus [11] should be taken into account when calculating local spring-back [3].
4.2. The use of curvature prediction in adaptive control When applying the ®rst method of adaptive control, a safe value for the initial punch penetration must be calculated. As mentioned before, under-estimation of the bend radius can result in overbending of the sheet. Prediction of the curvature under the punch nose in combination with the use of a simple model can reduce this danger and prediction of local curvatures in combination with adequate modelling of the overall sheet geometry can eliminate it completely. When the second method of adaptive control is used, two problems occur. Firstly, the contact points between the sheet and the die change continuously, resulting in changed leverage and direction of forces. Secondly, the data acquisition zone (i.e. the part of the punch travel distance during which measurements are taken) must be determined. The ®rst problem can be solved by running an off-line simulation with provisional material data. This simulation gives adequate information about the location of the contact points and the direction of the forces. Furthermore, conversion factors can be generated which allows for fast processing of measured data. The second problem can be solved by prediction of the radius under the punch as a function of the punch penetration. When, at some stage, the radius does not decrease any more (for instance due to wrap-around), then data measured beyond this point should be excluded when constructing the moment±curvature relationship. Hence, curvature prediction can increase both the speed and the accuracy of the method [12]. 5. Conclusions Air bending potentially offers the ¯exibility required for bending sheet metal components with a high variety in small batches. However, modelling and controlling of the process present some problems. Models based on the wrap-around assumption have only a very limited applicability. A method based on minimisation of bending energy can be used to predict the sheet radius under the punch. The method gives good results when compared with experiments and makes it possible to drop the wrap-around assumption. In air bending, increasingly adaptive control is applied to address the problem of variable material properties. However, the use of adaptive control is no substitute for process knowledge. On the contrary, the use of adequate bending models in adaptive control can prevent irreversible errors such as overbending, it can avoid erroneous interpretation of data, and speed up data processing.
References [1] A.H. Streppel, L.J. De Vin, J. Brinkman, H.J.J. Kals, Suitability of sheet bending modelling techniques in CAPP applications, J. Mater. Process. Technol. 36 (1993) 139±156.
L.J. De Vin / Journal of Materials Processing Technology 100 (2000) 257±261 [2] C. Wang, O. Kinzel, T. Altan, Mathematical modeling of plane-strain bending of sheet and plate, J. Mater. Process. Technol. 39 (1993) 279±304. [3] L.J. De Vin, A.H. Streppel, U.P. Singh, H.J.J. Kals, A process model for air bending, J. Mater. Process. Technol. 57(1/2) (1996) 48±54. [4] L.J. De Vin, U.P. Singh, W. Urquhart, H.J.J. Kals, Material behaviour in air bending, AMPT '95 Conference, 1995, pp. 88±96. [5] E. Von Finckenstein, R. Rothstein, G. von Blechen, Automatische Ruckfederungskorrektur, Industrieanzeiger 24 (1988) 19±20. [6] W. Heckel, V. Franke, In-process measurement of bending angles, SheMet '95 Conference, 1995, pp. 272±281. [7] D. Lutters, A.H. Streppel, B. Kroeze, H.J.J. Kals, Adaptive press brake control in air bending, SheMet '97 Conference, 1997, pp. 471± 480.
261
[8] K.A. Stelson, Real time identi®cation of workpiece-material characteristics from measurements during brakeforming, J. Eng. Ind., Trans. ASME 105 (1993) 45±53. [9] K.A. Stelson, An adaptive pressbrake control for strain hardening materials, J. Eng. Ind, Trans. ASME 108 (1986) 127±132. [10] S. Pickering, Intelligent processing of materials, J. Mater. Process. Technol. 36 (1993) 447±465. [11] W. Lems, The change of Young's modulus after deformation at low temperature and its recovery, Ph.D. Thesis, University of Delft, 1963. [12] L.J. De Vin, U.P. Singh, Adaptive control of mechanical processes: brakeforming of metal sheet as an example, Mechatronics '98 Conference, 1998, pp. 425±431.