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Transition surface design for blank holder in multi-point forming
ARTICLE IN PRESS
International Journal of Machine Tools & Manufacture 46 (2006) 1336–1342 www.elsevier.com/locate/ijmactool
Transition surface design for blank holder in multi-point forming Linfa Penga,, Xinmin Laia, Mingzhe Lib a
School of Mechanical and Engineering, Shanghai Jiaotong University, Shanghai 200030, People’s Republic of China b Multi-Point Forming Technique Center, Jilin University, Changchun, 130025, People’s Republic of China Received 18 April 2005; received in revised form 7 October 2005; accepted 21 October 2005 Available online 6 December 2005
Abstract During the last decades, the manufacturing mode and level of industry have been improving from mass production manufacturing to responsive and flexible manufacturing. Multi-point forming (MPF) is a flexible manufacturing technique, which is made up of a set of adjustable base elements controlled by MPF-CAD/CAM software. It uses discrete punches to express a continuous 3-D surface, by which the workpiece’s surface under manufacturing is formed. With respect to the characteristics of MPF, two methods for NURBS surface extension are presented in this paper to design the blending surface between the surfaces of the blank holder and the workpiece. Two design approaches of transition surface are proposed: one is a flexible surface extension, and the other is a bridge surface extension. These two algorithms, through which the transition surface designed can reach G2 continuity, are both effective and reliable. Applications show that high-quality products are manufactured in MPF by this transition surface design. r 2005 Elsevier Ltd. All rights reserved. Keywords: Multi-point forming; Surface extension; Transition surface NURBS
1. Introduction MPF (multi-point forming) [1–9] is an advanced flexible manufacturing technique for 3-D sheet metal forming, which is especially adaptable to form a variety of shapes in large sheet. In MPF, the conventional solid die is replaced by a set of discrete punches called ‘‘element group’’, the positions of which can be adjusted, to form the threedimensional shape of the workpiece to be manufactured. The original ideas of MPF were firstly put forward in the 1960s by Japanese researchers [1,2]. In later researches, Hardt [3,4] and his colleagues did considerable work on MPF at MIT. In the 1980s they successfully developed a device controlled by a closed-loop system to form a sheet metal with discrete punches. Then, in 1993 Li [5–9] from Jilin University of Technology developed an MPF mockup, which was the first MPF device controlled by a computer. In 1998 Li, Cai, Chen and Sui improved the
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0890-6955/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2005.10.012
above-mentioned computer-controlled MPF device and initiated a new type of MPF mock-up. In accordance with the different motion models (active or passive) of punches, multi-point forming can be classified into four different types: multi-point die, multipoint half die, multi-point press and multi-point half press. Several researches focus on the multi-point press type due to its sound flexibility and excellent controllability. By this technique the forming path and the press state of the work pieces can be changed during the manufacturing process, so that an optimum manufacture method can be achieved and a high-quality product can be attained. A multi-point press is made up of all active punches (upper and lower), which can be adjusted to a proper position during the manufacturing process to form the shapes of workpieces. Every punch is different from others in terms of movement, so that each one is like an independent small press. The whole process of formation is shown in Fig. 1. At the beginning of formation, the workpiece is pressed by all punches, and then every punch changes its height, which is controlled by the MPF-CAD/ CAM system, until the finial shape of the workpiece is formed. This manufacturing process has flexible forming
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(a)
(b)
(c)
Fig. 1. The process of Multi-point press: (a) beginning, (b) part formed shape and (c) finished shape.
characteristics and the forming path and the press force can be optimized, so as to obtain a good-quality product. Wrinkling is the most common defect in metal sheet formation by punch and die, so die designers have to spend much time on the elimination of this defect, and a common and practical method is proposed to design a blank holder while maintaining the periphery of the blank sheet. However, one of the key problems lies in whether the transition surface between surfaces of the workpiece and blank holder can be continuous and smooth. A poor transition surface may cause acute wrinkling and even cracking in workpieces, while a continuous and smooth one can avoid these defects, and thus improve the formability of the workpiece. For these reasons, several researches have been done in this field. For example, Yao et al. [10] did some work on how to optimize parameters of the process addendum surfaces and blank holder surfaces for an automotive deep draw die on UG platform. Wang et al. [11] also did the same work in an I-DEAS environment. Both of these two methods are based on the section line building. They also constructed the transition surfaces through line-sweeping methods so that the quality of the transition surface largely depends on the section lines’ quality. Therefore, the surface shapes of the blank sheet and the blank holder are neglected, and high-quality transition surfaces cannot be achieved. As both these methods have been developed under third software environments, they cannot be easily extrapolated. In this paper, two types of transition surface design methods are presented for MPF: one is flexible surface extension, and the other is bridge surface extension. Due to the effectiveness and reliability of these two algorithms, the blending surface designed by using these methods can reach G2 continuity. Although these methods are presented for the MPF transition surface design, they can also be used in other surface modeling areas.
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where V i;j represents the control points of the NURBS surfaces, and the number of control points in the u and v parametric directions is defined by n and m, respectively, while W i;j represents the weights of the corresponding control points, and Bi;k ðuÞ and Bj;l ðvÞ are defined as Bspline basis functions in the u and v parametric directions as follows: ( 1 ui pupuiþ1 ; Bi;0 ðuÞ ¼ 0 others; u ui Bi;k ðuÞ ¼ Bi;k1 ðuÞ uiþ1 ui uiþkþ1 u Biþ1;k1 ðuÞ; kX1, þ uiþkþ1 uiþ1 0 ¼ 0, 0 where u is the knot vector of the B-Spline function. 2.1. Tangent continuity between two joint surfaces Generally, there are three major types of continuity conditions between two joint surfaces: location continuity that is called G1 continuity (satisfied by common knot vectors, control vectors and orders at boundaries); tangent continuity, that is, C1 continuity; and curvature continuity which is most widely required in surface modeling, namely, G2 continuity. Two joint surfaces with G2 continuity are both continuous and smooth. In order to achieve the tangent continuity, this pair of surfaces must firstly reach location continuity, which means there must be a common boundary between these two surfaces. ~ ~ðup ;vp Þ and Q Suppose there are two joint surfaces: P ðuq ;vq Þ , which have a common boundary in parametric space (as shown in the left part of Fig. 2), and also a common boundary s ¼ up ¼ vq in practical model space (as shown in the right part of Fig. 2). If the common boundary is expounded by the uniform parameters, then the boundary equation can be defined as follows: ~ . ~ðs;0Þ Q P ð0;sÞ According to the definition of tangent continuity, the differential coefficients of this pair of surfaces along the ~ ð0; sÞ, ~vp ðs; 0Þ,Q common boundary must be equal, i.e., P uq ~ and Qs ð0; sÞ must be on the same plane as Fig. 2 shows. Thus, Eq. (2) is described as follows: ~ ðs; 0Þ þ aðsÞQ ~ ð0; sÞ ¼ 0, ~vp ðs; 0Þ þ lðsÞQ P uq s
(2)
2. Continuity condition analyses of transition surfaces The NURBS surface is often used in surface modeling [12]. All types of the free-form surface could be expressed as a uniform NURBS surface. An m n parametric-degree NURBS surface can be described in Eq. (1) as follows: Pn Pm i¼0 j¼0 Bi;k ðuÞBj;l ðvÞW i;j V i;j Pðu; vÞ ¼ Pn Pm , (1) i¼0 j¼0 Bi;k ðuÞBj;l ðvÞW i;j
Q s (s,0)
P Q
P up
Pvp (s,0)
Q vq (s,0)
Q
uq vp
vq
Fig. 2. Continuity condition of two joint surfaces.
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where l(s) and a(s) could be any function; however, l(s) shall not be equal to 0. 2.2. Curvature continuity condition of two surfaces In order to achieve curvature continuity, the tangent continuity of the two joint surfaces must be obtained first. Kahmann [13] and Boehm [14] put forward an equation on curvature continuity between two surfaces. For two surfaces P and Q with tangent continuity, they must satisfy Eq. (3) to achieve curvature continuity. Pvv ðs; 0Þ þ V ðsÞQuu ð0; sÞ þ ZðsÞQus ð0; sÞ þ jðsÞQss ð0; sÞxðsÞQu ð0; sÞ þ sðsÞQs ð0; sÞ ¼ 0.
ð3Þ
Eq. (3) shows that the first two rows of the control points in the same control net along the extension direction must be coplanar to achieve curvature continuity. So the two surfaces’ differential coefficients to parameter s must be equal to 0, i.e., ZðsÞ ¼ jðsÞ ¼ sðsÞ ¼ 0. In Eq. (3), if xðsÞ ¼ 0, it represents a 2-degree parametric continuity, i.e., C2 continuity. As curvature continuity needs more flexible requirements than C2 continuity does, v(s) and xðsÞ should be set as constants (v0 and x0 , respectively). Eqs. (2) and (3) define the continuity conditions of tangent and curvature continuity for two joint NURBS surfaces. From these equations, it is obvious that in order to construct a transition surface Qðu; vÞ which joints the base surface Pðu; vÞ, and reaches curvature continuity between these two surfaces, you only need to adjust the two rows of control points adjacent to the common boundary of these two joint surfaces. In most practical engineering applications, the curvature continuity (G2) can meet the design requirements because this continuity condition can guarantee surface joint and even the smoothness along the common boundary, while the G2 continuity can be easily understood and achieved if it is presented by geometric conditions. In this paper, the study on G2 continuity of geometric condition is necessary to simplify the methods to design the transition surface. The G2 geometric condition of two joint NURBS surfaces can be re-expressed as follows: (1) These two joint NURBS surfaces must have a common boundary, i.e., reach location continuity first. (2) The three lines in the control nets of these two surfaces must be coplanar. The tangent vectors of the two surfaces on the same point along the shared common boundary must have the same direction and an equal value. (3) For NURBS surfaces that are more than cubic, the Dupin curves must be the same.
in realizing a flexible blank holder via the traditional die and punch forming technique. However, it can be easily achieved in MPF machines, which can quickly form any shape with movable discrete punches. The method of surface flexible extension is adopted in this study to design this kind of blank holder transition surface. The main idea of this method is to extend the original surface directly to obtain an extension surface. The original surface is the workpiece surface, while the extension surface is a blank holder transition surface. According to the constraints of its original surface, the transition surface and blank holder surface can be achieved directly. NURBS surface flexible extension is a method based on NURBS curve extension, through which curvature continuity between the extension surface and the original surface can be attained. Therefore, the workpiece surface, the transition surface and the flexible blank holder are continuous and smooth. In most applications, surface flexible extension could be realized by the reflection method. Dividing the Bezier curves into sub-curves, the extension curves in the Bezier surface modeling can be obtained. However, this method does not make sense in NURBS curves extension, because the weights of NURBS surface control points cannot be assured to be positive. In this paper, an NURBS surface flexible extension method is presented, which can ensure that flexible extension surfaces are achieved with positive weights of control points of extension surface if the weights of the original surface control points are positive. Hence, this pair of surfaces is continuous and smooth and the quality of the extension surface is fairly good. This method is proved to be perfect in the design of the flexible transition surface and blank holder surface in MPF. As Fig. 3 shows, the main idea of this method is the reflection strategy [15]. Firstly, normal planes at the end of the original surface are established. Every control curve in the extension direction has its own normal plane which can be calculated by the last two points on this control curve. Then, according to these normal planes, the control points on the original surface are reflected to form the extension surface. Therefore, the quality of extension surface is only determined by the shape of the original surface. If the curvature of the NURBS surface increases in the extension direction, the extension surface’s curvature must decrease Normal plane
P0=Q0
Q
Q2
P1 Q3 P2
3. Blank holder transition surfaces designed through the surface-flexible extension method
P3 Original surface
(a)
A flexible blank holder is very special, whose shapes change with different workpiece shapes. The costs are high
(b) Extension surface
Fig. 3. Extension of curve and surface: (a) curves extension, and (b) surface extension.
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in that direction [16]. Thus, self-intersection in the extension control net may occur, especially for a large extension range. Therefore, it is better to divide it into several steps during which small extensions are presented in each of them to avoid self-intersection. From Fig. 3(a), it is clear that the tangent continuity and curvature continuity are achieved at the extension point using this reflection method, through which P1, P0, and Q1 are collinear (P0 ¼ Q0 ), and the control points P2, P1, P0, Q1, and Q2 are coplanar, and therefore, meets the requirements of the tangent and curvature continuity. The principles of NURBS surface extension are the same as the NURBS curves extension shown in Fig. 3(b). According to the characterizations of the B-Spline tensor, Eqs. (4) and (5) are the necessary and sufficient curvature continuity conditions for two joint NURBS surfaces, described as follows: ðkp 1Þ ðkq 1Þ ðPH i;1 PH 0;i Þ ¼ ðQH 0;i QH 1;i Þ þ GQH 0;i , DP0 Dq0 (4a) ðkp 1Þ ðW pi;1 W pi;0 Þ DP0 ðkq 1Þ ¼ ðW Qi;1 W Qi;0 Þ þ GW Qi;0 , Dq0
ð4bÞ
ðkP 1ÞðkP 2Þ ½Dp0 PH 1;2 ð2Dp0 þ Dp1 ÞPH i;1 Dp20 ðDp0 þ Dp1 Þ ðkq 1Þðkq 2Þ þ ðDp0 þ Dp1 ÞPH i;0 ¼ oV 0 Dq20 ðDq0 þ Dq1 Þ ½Dq0 QH 2;i þ ð2Dq0 þ Dq1 ÞQH 1;i ðDq0 þ Dq1 ÞQH 0;i þ fox0 þ 2Gl0 g
kq 1 ½QH 0;i QH 1;i þ yQH 0;i , Dq0
ð5aÞ
ðkP 1ÞðkP 2Þ ½Dp0 W Pi;2 ð2Dp0 þ Dp1 ÞW Pi;1 Dp20 ðDp0 þ Dp1 Þ ðkq 1Þðkq 2Þ þ ðDp0 þ Dp1 ÞW Pi;02 ¼ oV 0 Dq20 ðDq0 þ Dq1 Þ
continuity can be achieved. Therefore, the final surface model is ‘‘watertight’’, with the extension surface sharing a common boundary, knot vectors, and surface order. Fig. 4 shows an example of flexible surface extension. The original surface has the shape of a human bone, which is used in surgery application. The extension surface is made of eight patches, which joints each other with curvature continuity. Therefore, the whole surface model is very smooth. This extension part can be taken as the flexible blank holder surface and transition surface in the design of the MPF processes. This NURBS surface extension method makes the blank holder design more flexible than before, and the pair of surfaces, original surface and extension surface, are both continuous and smooth. 4. Transition surface designed through the bridge surface extension method
½Dq0 W Q2;i þ ð2Dq0 þ Dq1 ÞW Q1;i ðDq0 þ Dq1 ÞW Q0;i kq 1 ½W Q0;i W Q1;i þ yW Q0;i þ fox0 þ 2Gl0 g Dq0 ði ¼ 0; 1; 2; . . . ; nq 1Þ,
Fig. 4. Flexible surface extension of the shape of a human bone: (a) shade model of the original surface, (b) control mesh of the extension surface, (c) mesh model of the flexible surface extension, and (d) shade model of the flexible surface extension.
ð5bÞ
where Dp1 ¼ vkp þ2 vkp þ1 ; Dq1 ¼ ukq þ2 ukq þ1 ; kp and kq are the parametric orders of the surfaces of P and Q, respectively. From Eq. (4b), the value of Gi can be calculated. Suppose G ¼ maxðGi Þ; substitution of G into Eqs. (5a) and (b) leads to the values of y and x. Then, the control points and their corresponding weights can be calculated from Eqs. (4a) and (b) according to the tangent continuity condition. Finally, the control nets must be adjusted according to Eqs. (5a) and (b), so that the curvature
There is a traditional blank holder in the MPF device, which is made of four flat rectangle surfaces jointed together. The main objective is to design a transition surface between the workpiece surface and the blank holder surface to make them continuous and smooth so that it is very effective to improve the formability and avoid defects. The bridge surface extension presented here can be used to design the transition surface between the workpiece surface and the fixed blank holder surface. The method changes the complex mathematical equations to simple visualized geometric relations of these three surfaces (original workpiece surface, transition surface and blank holder surface), so that all the continuity conditions can be easily achieved via adjustment of related off-boundary
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control points. The transition surface designed by bridge surface extension is made of four steps as follows:
localization of the blank holder, establishment of original transition surface, adjustment of off-boundaries control points.
4.1. Localization of the blank holder The relative location of blank holder and workpiece influences the form direction and punch displacement in the process design. Therefore the quality of sheet parts is also affected. Here the workpiece is supposed to be stationary. The objective of locating the blank holder is to optimize the holder location to form the workpiece. Generally, the largest projected area is considered as the best location for the blank holder because of minimum stroke displacement and reasonable punch direction. According to the punch stroke limit and size of the MPF device, the blank holder size and the height between the holder and workpiece can be determined, and then the holder surface can be achieved (as Fig. 5 shows). 4.2. Establishment of original transition surface As shown in Fig. 5, the original transition surface is located between the blank holder surface and the workpiece surface. It is a prototype surface model of the later transition surface via adjustment of some control points. The method is expounded in detail as follows: In the beginning, project the end points p1 and pn of boundary B on the workpiece surface to the blank holder surface, so that the projection points t1 and tn are obtained. Then project these two points t1 and tn on to blank holder boundary M, so that the points m1 and mn are attained (as Fig. 5 shows). The line m1mn is formed by the end points of m1 and mn, which are located at the boundary of the blank holder. Divide the line m1mn according to the number of control points at the corresponding workpiece boundary B; therefore, a series of points M(n) ðn ¼ 0; 1; 2; 3; . . . ; lÞ are gained and each point Mn corresponds to control point B(n) at the workpiece boundary B. On connecting the points Bn and Mn, a serial of lines L(n) ðn ¼ 0; 1; 2; 3; . . . ; lÞ are achieved. Divide each line L(n) according to the parametric degree needed to model the transition surface.
Suppose the parametric order is m; i points are used to divide the lines L(n) with the condition i4m þ 1. These dividing points V ði; jÞ are taken as the control points of the original transition surface. If the parametric direction u is the extension direction, the knot vectors U(j) must be the same as that of boundary B to ensure that the two surfaces share the same boundary (boundary B). Suppose the transition surface is constructed by an even B-Spline basis function along the v direction; the knot vectors V(j) can be described as follows: 8 0 ipm; > > < ði mÞ (6) V ðiÞ ¼ moioN; > N m > : 1 NpioN þ m þ 1: Suppose the weights of all control points W ði; jÞ are 1.0. Now all the parameters are gained (control points, corresponding weights, knot vectors along the u and v directions and parametric degree of the surface), so that the transition surface can be presented as an NURBS surface. And then the original transition surface is constructed. 4.3. Adjustment of off-boundary control points The original transition surface constructed above can only achieve a location continuity of these three surfaces. To ensure that the joint surface has the same boundary, Eq. (7) must be satisfied: PH 0;i ¼ QH 0;i
ði ¼ 0; 1; 2; . . . ; nq 1Þ,
(7)
where nq is the number of control points at the shared boundary of these two surfaces. Only with this continuity condition can the fairness and smoothness be not achieved and the requirements of surface modeling cannot be met. Therefore, the control points of the transition surface must be adjusted to reach the tangent continuity and curvature continuity. Only with these constraints can these three surfaces, workpiece surface, transition surface and blank holder surface, be smoothly joined with each other. The tangent vectors of these two surfaces at the shared boundary can be expounded in Eq. (8) v1 ¼ PH 0;i PH 1;i ; v2 ¼ QH 1;i QH 0;i :
(8)
In order to achieve tangent continuity, the value of tangent vectors must be equal, i.e., v1 ¼ v2 . It also can be simplified as QH 1;i ¼ PH 0;i þ QH 0;i PH 1;i .
Fig. 5. Original transition surface.
(9)
From Eq. (9) above, the control points QH 1;i of the transition surface can be calculated. If there are no special requirements, weights oQ1;i of these new control points may remain the same. Update the control points QH 1;i according to Eq. (9); the two surfaces can achieve a tangent continuity condition. For some NURBS surfaces with a
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fairly larger span between the first row and the second row of control points, the control points QH 1;i of its bridge extension surface (transition surface) are far away from the shared boundary. Hence, the fairness of the transition surface cannot be achieved and poor quality transition surface occurs. In order to solve this sort of problem, points must be inserted between the first and second row of control points (PH 0;i and PH 0;i ) according to the span compared with the bridge extension span [17], so that more control points are used to control the NURBS surface (workpiece surface) and a suitable span between these two rows of control points can be achieved to improve the quality of the transition surface. Fig. 6 shows the control points of two adjacent NURBS surfaces (the original surface P and extension surface Q). The left of the dotted line is the original surface and the other side lies on the extension one. In order to achieve curvature continuity between two joint NURBS surfaces, the adjacent six control points (PH 2;i , PH 1;i , PH 0;i , QH 0;i ðPH 0;i ¼ QH 0;i Þ, QH 1;i , QH 2;i ) of the two NURBS surfaces along the same control mesh must be coplanar, which is a prerequisite for G2 continuity. According to the tangent continuity, the points PH 2;i , PH 1;i , PH 0;i , QH 0;i and QH 1;i are already coplanar, so that the main task is to adjust the point QH 2;i to that plane. Here the projection method is used. Project the point QH 2;i on to the plane, and the projection point Q0H 2;i on that plane is obtained as Fig. 6 shows. From the above analysis, Q0H 2;i can be calculated from Eq. (10) V ¼ ðPH 0;i PH 1;i Þ ðPH 2;i PH 1;i Þ; Q0H 2;i ¼ kV 0 þ QH 2;i ;
(10)
V ¼ V 0 ; ðQH 1;i Q0H 2;i Þ V 0 ¼ 0;
where k is a constant that can be calculated from Eq. (10) above. The transition surface is located between the workpiece surface and the blank holder one, so that the next step is to adjust the off-boundary control points of the transition surface and the blank holder one to make these two surfaces smoothly jointed. Suppose that the blank holder is QH
PH
PH1,i
V
0 ,i
QH
V′ 1, i
Q ′H
PH
Original surface
2 ,i
= Q H 0 ,i
2 ,i
Extension surface
2 ,i
Fig. 6. Control points of two adjacent NURBS surfaces.
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Fig. 7. Surface bridge extension: (a) control meshes of extension surface, (b) shade model of extension surface, (c) shade model of surface after twoside extension, and (d) shade model of surface after full extension.
a rectangular plane, which can also be expressed as the NURBS surface with k parametric order along the extension direction. The value of k here is equal to the parametric order of the transition surface, which will simplify the calculation. Using the control point adjustment method above, curvature continuity between the transition surface and the blank holder is achieved. Fig. 7 shows an example of the bridge extension method used in transition surface design. It is clear that the workpiece surface, transition surface and blank holder surface are jointed smoothly. 5. Application examples As Fig. 8(a) shows, the original surface has the shape of a human bone that is broken during car accidents. It is often replaced by a metal sheet in surgery after it is broken. This metal sheet is only 0.1 mm in thickness. While it is directly formed in an MPF device (400 320 mm forming region and 100 mm stoke range), wrinkling or even cracking occurs. So it is very difficult to manufacture directly. In order to avoid these deficiencies in forming this sort of product, the blank holder must be designed to improve the formability of this workpiece. Nevertheless traditional methods are unavailable to design a reasonable transition between the blank holder surface and the workpiece surface. Fig. 8(b) shows a mesh model of these three types of surfaces after surface extension using the method described in this paper. It is clear that these three types of surfaces smoothly joint with each other and a G2 continuity condition is reached between them. From Fig. 9, it can be seen that the workpiece is formed smoothly by an
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method is used to design a transition surface for a traditional rectangle plane blank holder. It is achieved by adjusting of boundary control points to obtain smoothness and continuity. These two algorithms, through which the transition surface designed can reach G2 continuity, are both effective and reliable. Though they are developed for MPF surface modeling to design the transition surface, they are common methods and can also be used in other surface modeling. Examples show that these two methods are reliable and highly efficient.
Reference
Fig. 8. Transition surface design for metal sheet part used medical engineer: (a) shade model of the object surface, and (b) mesh model of the transient surface by extension.
Fig. 9. Sketch of deformed sheet metal.
MPF with a blank holder and a high-quality product is obtained. 6. Conclusions The design of transition surfaces between the workpiece surface and the blank holder is vital in MPF process design. It can prevent some defects in manufacture of metal sheet parts. Two design methods of transition surface are proposed: one is a flexible surface extension, and the other is a bridge surface extension. The main idea of flexible surface extension is reflection of control points. It is suitable for flexible transition surface and blank holder design when the shape of the workpiece is complex and it is difficult to design a transition surface via a traditional fixed rectangle plane blank holder. The bridge surface extension
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International Journal of Machine Tools & Manufacture 46 (2006) 1336–1342 www.elsevier.com/locate/ijmactool
Transition surface design for blank holder in multi-point forming Linfa Penga,, Xinmin Laia, Mingzhe Lib a
School of Mechanical and Engineering, Shanghai Jiaotong University, Shanghai 200030, People’s Republic of China b Multi-Point Forming Technique Center, Jilin University, Changchun, 130025, People’s Republic of China Received 18 April 2005; received in revised form 7 October 2005; accepted 21 October 2005 Available online 6 December 2005
Abstract During the last decades, the manufacturing mode and level of industry have been improving from mass production manufacturing to responsive and flexible manufacturing. Multi-point forming (MPF) is a flexible manufacturing technique, which is made up of a set of adjustable base elements controlled by MPF-CAD/CAM software. It uses discrete punches to express a continuous 3-D surface, by which the workpiece’s surface under manufacturing is formed. With respect to the characteristics of MPF, two methods for NURBS surface extension are presented in this paper to design the blending surface between the surfaces of the blank holder and the workpiece. Two design approaches of transition surface are proposed: one is a flexible surface extension, and the other is a bridge surface extension. These two algorithms, through which the transition surface designed can reach G2 continuity, are both effective and reliable. Applications show that high-quality products are manufactured in MPF by this transition surface design. r 2005 Elsevier Ltd. All rights reserved. Keywords: Multi-point forming; Surface extension; Transition surface NURBS
1. Introduction MPF (multi-point forming) [1–9] is an advanced flexible manufacturing technique for 3-D sheet metal forming, which is especially adaptable to form a variety of shapes in large sheet. In MPF, the conventional solid die is replaced by a set of discrete punches called ‘‘element group’’, the positions of which can be adjusted, to form the threedimensional shape of the workpiece to be manufactured. The original ideas of MPF were firstly put forward in the 1960s by Japanese researchers [1,2]. In later researches, Hardt [3,4] and his colleagues did considerable work on MPF at MIT. In the 1980s they successfully developed a device controlled by a closed-loop system to form a sheet metal with discrete punches. Then, in 1993 Li [5–9] from Jilin University of Technology developed an MPF mockup, which was the first MPF device controlled by a computer. In 1998 Li, Cai, Chen and Sui improved the
Corresponding author. Tel.: +86 021 629321250; fax: +86 021 62932125103. E-mail address: [email protected] (L. Peng).
0890-6955/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2005.10.012
above-mentioned computer-controlled MPF device and initiated a new type of MPF mock-up. In accordance with the different motion models (active or passive) of punches, multi-point forming can be classified into four different types: multi-point die, multipoint half die, multi-point press and multi-point half press. Several researches focus on the multi-point press type due to its sound flexibility and excellent controllability. By this technique the forming path and the press state of the work pieces can be changed during the manufacturing process, so that an optimum manufacture method can be achieved and a high-quality product can be attained. A multi-point press is made up of all active punches (upper and lower), which can be adjusted to a proper position during the manufacturing process to form the shapes of workpieces. Every punch is different from others in terms of movement, so that each one is like an independent small press. The whole process of formation is shown in Fig. 1. At the beginning of formation, the workpiece is pressed by all punches, and then every punch changes its height, which is controlled by the MPF-CAD/ CAM system, until the finial shape of the workpiece is formed. This manufacturing process has flexible forming
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(a)
(b)
(c)
Fig. 1. The process of Multi-point press: (a) beginning, (b) part formed shape and (c) finished shape.
characteristics and the forming path and the press force can be optimized, so as to obtain a good-quality product. Wrinkling is the most common defect in metal sheet formation by punch and die, so die designers have to spend much time on the elimination of this defect, and a common and practical method is proposed to design a blank holder while maintaining the periphery of the blank sheet. However, one of the key problems lies in whether the transition surface between surfaces of the workpiece and blank holder can be continuous and smooth. A poor transition surface may cause acute wrinkling and even cracking in workpieces, while a continuous and smooth one can avoid these defects, and thus improve the formability of the workpiece. For these reasons, several researches have been done in this field. For example, Yao et al. [10] did some work on how to optimize parameters of the process addendum surfaces and blank holder surfaces for an automotive deep draw die on UG platform. Wang et al. [11] also did the same work in an I-DEAS environment. Both of these two methods are based on the section line building. They also constructed the transition surfaces through line-sweeping methods so that the quality of the transition surface largely depends on the section lines’ quality. Therefore, the surface shapes of the blank sheet and the blank holder are neglected, and high-quality transition surfaces cannot be achieved. As both these methods have been developed under third software environments, they cannot be easily extrapolated. In this paper, two types of transition surface design methods are presented for MPF: one is flexible surface extension, and the other is bridge surface extension. Due to the effectiveness and reliability of these two algorithms, the blending surface designed by using these methods can reach G2 continuity. Although these methods are presented for the MPF transition surface design, they can also be used in other surface modeling areas.
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where V i;j represents the control points of the NURBS surfaces, and the number of control points in the u and v parametric directions is defined by n and m, respectively, while W i;j represents the weights of the corresponding control points, and Bi;k ðuÞ and Bj;l ðvÞ are defined as Bspline basis functions in the u and v parametric directions as follows: ( 1 ui pupuiþ1 ; Bi;0 ðuÞ ¼ 0 others; u ui Bi;k ðuÞ ¼ Bi;k1 ðuÞ uiþ1 ui uiþkþ1 u Biþ1;k1 ðuÞ; kX1, þ uiþkþ1 uiþ1 0 ¼ 0, 0 where u is the knot vector of the B-Spline function. 2.1. Tangent continuity between two joint surfaces Generally, there are three major types of continuity conditions between two joint surfaces: location continuity that is called G1 continuity (satisfied by common knot vectors, control vectors and orders at boundaries); tangent continuity, that is, C1 continuity; and curvature continuity which is most widely required in surface modeling, namely, G2 continuity. Two joint surfaces with G2 continuity are both continuous and smooth. In order to achieve the tangent continuity, this pair of surfaces must firstly reach location continuity, which means there must be a common boundary between these two surfaces. ~ ~ðup ;vp Þ and Q Suppose there are two joint surfaces: P ðuq ;vq Þ , which have a common boundary in parametric space (as shown in the left part of Fig. 2), and also a common boundary s ¼ up ¼ vq in practical model space (as shown in the right part of Fig. 2). If the common boundary is expounded by the uniform parameters, then the boundary equation can be defined as follows: ~ . ~ðs;0Þ Q P ð0;sÞ According to the definition of tangent continuity, the differential coefficients of this pair of surfaces along the ~ ð0; sÞ, ~vp ðs; 0Þ,Q common boundary must be equal, i.e., P uq ~ and Qs ð0; sÞ must be on the same plane as Fig. 2 shows. Thus, Eq. (2) is described as follows: ~ ðs; 0Þ þ aðsÞQ ~ ð0; sÞ ¼ 0, ~vp ðs; 0Þ þ lðsÞQ P uq s
(2)
2. Continuity condition analyses of transition surfaces The NURBS surface is often used in surface modeling [12]. All types of the free-form surface could be expressed as a uniform NURBS surface. An m n parametric-degree NURBS surface can be described in Eq. (1) as follows: Pn Pm i¼0 j¼0 Bi;k ðuÞBj;l ðvÞW i;j V i;j Pðu; vÞ ¼ Pn Pm , (1) i¼0 j¼0 Bi;k ðuÞBj;l ðvÞW i;j
Q s (s,0)
P Q
P up
Pvp (s,0)
Q vq (s,0)
Q
uq vp
vq
Fig. 2. Continuity condition of two joint surfaces.
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where l(s) and a(s) could be any function; however, l(s) shall not be equal to 0. 2.2. Curvature continuity condition of two surfaces In order to achieve curvature continuity, the tangent continuity of the two joint surfaces must be obtained first. Kahmann [13] and Boehm [14] put forward an equation on curvature continuity between two surfaces. For two surfaces P and Q with tangent continuity, they must satisfy Eq. (3) to achieve curvature continuity. Pvv ðs; 0Þ þ V ðsÞQuu ð0; sÞ þ ZðsÞQus ð0; sÞ þ jðsÞQss ð0; sÞxðsÞQu ð0; sÞ þ sðsÞQs ð0; sÞ ¼ 0.
ð3Þ
Eq. (3) shows that the first two rows of the control points in the same control net along the extension direction must be coplanar to achieve curvature continuity. So the two surfaces’ differential coefficients to parameter s must be equal to 0, i.e., ZðsÞ ¼ jðsÞ ¼ sðsÞ ¼ 0. In Eq. (3), if xðsÞ ¼ 0, it represents a 2-degree parametric continuity, i.e., C2 continuity. As curvature continuity needs more flexible requirements than C2 continuity does, v(s) and xðsÞ should be set as constants (v0 and x0 , respectively). Eqs. (2) and (3) define the continuity conditions of tangent and curvature continuity for two joint NURBS surfaces. From these equations, it is obvious that in order to construct a transition surface Qðu; vÞ which joints the base surface Pðu; vÞ, and reaches curvature continuity between these two surfaces, you only need to adjust the two rows of control points adjacent to the common boundary of these two joint surfaces. In most practical engineering applications, the curvature continuity (G2) can meet the design requirements because this continuity condition can guarantee surface joint and even the smoothness along the common boundary, while the G2 continuity can be easily understood and achieved if it is presented by geometric conditions. In this paper, the study on G2 continuity of geometric condition is necessary to simplify the methods to design the transition surface. The G2 geometric condition of two joint NURBS surfaces can be re-expressed as follows: (1) These two joint NURBS surfaces must have a common boundary, i.e., reach location continuity first. (2) The three lines in the control nets of these two surfaces must be coplanar. The tangent vectors of the two surfaces on the same point along the shared common boundary must have the same direction and an equal value. (3) For NURBS surfaces that are more than cubic, the Dupin curves must be the same.
in realizing a flexible blank holder via the traditional die and punch forming technique. However, it can be easily achieved in MPF machines, which can quickly form any shape with movable discrete punches. The method of surface flexible extension is adopted in this study to design this kind of blank holder transition surface. The main idea of this method is to extend the original surface directly to obtain an extension surface. The original surface is the workpiece surface, while the extension surface is a blank holder transition surface. According to the constraints of its original surface, the transition surface and blank holder surface can be achieved directly. NURBS surface flexible extension is a method based on NURBS curve extension, through which curvature continuity between the extension surface and the original surface can be attained. Therefore, the workpiece surface, the transition surface and the flexible blank holder are continuous and smooth. In most applications, surface flexible extension could be realized by the reflection method. Dividing the Bezier curves into sub-curves, the extension curves in the Bezier surface modeling can be obtained. However, this method does not make sense in NURBS curves extension, because the weights of NURBS surface control points cannot be assured to be positive. In this paper, an NURBS surface flexible extension method is presented, which can ensure that flexible extension surfaces are achieved with positive weights of control points of extension surface if the weights of the original surface control points are positive. Hence, this pair of surfaces is continuous and smooth and the quality of the extension surface is fairly good. This method is proved to be perfect in the design of the flexible transition surface and blank holder surface in MPF. As Fig. 3 shows, the main idea of this method is the reflection strategy [15]. Firstly, normal planes at the end of the original surface are established. Every control curve in the extension direction has its own normal plane which can be calculated by the last two points on this control curve. Then, according to these normal planes, the control points on the original surface are reflected to form the extension surface. Therefore, the quality of extension surface is only determined by the shape of the original surface. If the curvature of the NURBS surface increases in the extension direction, the extension surface’s curvature must decrease Normal plane
P0=Q0
Q
Q2
P1 Q3 P2
3. Blank holder transition surfaces designed through the surface-flexible extension method
P3 Original surface
(a)
A flexible blank holder is very special, whose shapes change with different workpiece shapes. The costs are high
(b) Extension surface
Fig. 3. Extension of curve and surface: (a) curves extension, and (b) surface extension.
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in that direction [16]. Thus, self-intersection in the extension control net may occur, especially for a large extension range. Therefore, it is better to divide it into several steps during which small extensions are presented in each of them to avoid self-intersection. From Fig. 3(a), it is clear that the tangent continuity and curvature continuity are achieved at the extension point using this reflection method, through which P1, P0, and Q1 are collinear (P0 ¼ Q0 ), and the control points P2, P1, P0, Q1, and Q2 are coplanar, and therefore, meets the requirements of the tangent and curvature continuity. The principles of NURBS surface extension are the same as the NURBS curves extension shown in Fig. 3(b). According to the characterizations of the B-Spline tensor, Eqs. (4) and (5) are the necessary and sufficient curvature continuity conditions for two joint NURBS surfaces, described as follows: ðkp 1Þ ðkq 1Þ ðPH i;1 PH 0;i Þ ¼ ðQH 0;i QH 1;i Þ þ GQH 0;i , DP0 Dq0 (4a) ðkp 1Þ ðW pi;1 W pi;0 Þ DP0 ðkq 1Þ ¼ ðW Qi;1 W Qi;0 Þ þ GW Qi;0 , Dq0
ð4bÞ
ðkP 1ÞðkP 2Þ ½Dp0 PH 1;2 ð2Dp0 þ Dp1 ÞPH i;1 Dp20 ðDp0 þ Dp1 Þ ðkq 1Þðkq 2Þ þ ðDp0 þ Dp1 ÞPH i;0 ¼ oV 0 Dq20 ðDq0 þ Dq1 Þ ½Dq0 QH 2;i þ ð2Dq0 þ Dq1 ÞQH 1;i ðDq0 þ Dq1 ÞQH 0;i þ fox0 þ 2Gl0 g
kq 1 ½QH 0;i QH 1;i þ yQH 0;i , Dq0
ð5aÞ
ðkP 1ÞðkP 2Þ ½Dp0 W Pi;2 ð2Dp0 þ Dp1 ÞW Pi;1 Dp20 ðDp0 þ Dp1 Þ ðkq 1Þðkq 2Þ þ ðDp0 þ Dp1 ÞW Pi;02 ¼ oV 0 Dq20 ðDq0 þ Dq1 Þ
continuity can be achieved. Therefore, the final surface model is ‘‘watertight’’, with the extension surface sharing a common boundary, knot vectors, and surface order. Fig. 4 shows an example of flexible surface extension. The original surface has the shape of a human bone, which is used in surgery application. The extension surface is made of eight patches, which joints each other with curvature continuity. Therefore, the whole surface model is very smooth. This extension part can be taken as the flexible blank holder surface and transition surface in the design of the MPF processes. This NURBS surface extension method makes the blank holder design more flexible than before, and the pair of surfaces, original surface and extension surface, are both continuous and smooth. 4. Transition surface designed through the bridge surface extension method
½Dq0 W Q2;i þ ð2Dq0 þ Dq1 ÞW Q1;i ðDq0 þ Dq1 ÞW Q0;i kq 1 ½W Q0;i W Q1;i þ yW Q0;i þ fox0 þ 2Gl0 g Dq0 ði ¼ 0; 1; 2; . . . ; nq 1Þ,
Fig. 4. Flexible surface extension of the shape of a human bone: (a) shade model of the original surface, (b) control mesh of the extension surface, (c) mesh model of the flexible surface extension, and (d) shade model of the flexible surface extension.
ð5bÞ
where Dp1 ¼ vkp þ2 vkp þ1 ; Dq1 ¼ ukq þ2 ukq þ1 ; kp and kq are the parametric orders of the surfaces of P and Q, respectively. From Eq. (4b), the value of Gi can be calculated. Suppose G ¼ maxðGi Þ; substitution of G into Eqs. (5a) and (b) leads to the values of y and x. Then, the control points and their corresponding weights can be calculated from Eqs. (4a) and (b) according to the tangent continuity condition. Finally, the control nets must be adjusted according to Eqs. (5a) and (b), so that the curvature
There is a traditional blank holder in the MPF device, which is made of four flat rectangle surfaces jointed together. The main objective is to design a transition surface between the workpiece surface and the blank holder surface to make them continuous and smooth so that it is very effective to improve the formability and avoid defects. The bridge surface extension presented here can be used to design the transition surface between the workpiece surface and the fixed blank holder surface. The method changes the complex mathematical equations to simple visualized geometric relations of these three surfaces (original workpiece surface, transition surface and blank holder surface), so that all the continuity conditions can be easily achieved via adjustment of related off-boundary
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control points. The transition surface designed by bridge surface extension is made of four steps as follows:
localization of the blank holder, establishment of original transition surface, adjustment of off-boundaries control points.
4.1. Localization of the blank holder The relative location of blank holder and workpiece influences the form direction and punch displacement in the process design. Therefore the quality of sheet parts is also affected. Here the workpiece is supposed to be stationary. The objective of locating the blank holder is to optimize the holder location to form the workpiece. Generally, the largest projected area is considered as the best location for the blank holder because of minimum stroke displacement and reasonable punch direction. According to the punch stroke limit and size of the MPF device, the blank holder size and the height between the holder and workpiece can be determined, and then the holder surface can be achieved (as Fig. 5 shows). 4.2. Establishment of original transition surface As shown in Fig. 5, the original transition surface is located between the blank holder surface and the workpiece surface. It is a prototype surface model of the later transition surface via adjustment of some control points. The method is expounded in detail as follows: In the beginning, project the end points p1 and pn of boundary B on the workpiece surface to the blank holder surface, so that the projection points t1 and tn are obtained. Then project these two points t1 and tn on to blank holder boundary M, so that the points m1 and mn are attained (as Fig. 5 shows). The line m1mn is formed by the end points of m1 and mn, which are located at the boundary of the blank holder. Divide the line m1mn according to the number of control points at the corresponding workpiece boundary B; therefore, a series of points M(n) ðn ¼ 0; 1; 2; 3; . . . ; lÞ are gained and each point Mn corresponds to control point B(n) at the workpiece boundary B. On connecting the points Bn and Mn, a serial of lines L(n) ðn ¼ 0; 1; 2; 3; . . . ; lÞ are achieved. Divide each line L(n) according to the parametric degree needed to model the transition surface.
Suppose the parametric order is m; i points are used to divide the lines L(n) with the condition i4m þ 1. These dividing points V ði; jÞ are taken as the control points of the original transition surface. If the parametric direction u is the extension direction, the knot vectors U(j) must be the same as that of boundary B to ensure that the two surfaces share the same boundary (boundary B). Suppose the transition surface is constructed by an even B-Spline basis function along the v direction; the knot vectors V(j) can be described as follows: 8 0 ipm; > > < ði mÞ (6) V ðiÞ ¼ moioN; > N m > : 1 NpioN þ m þ 1: Suppose the weights of all control points W ði; jÞ are 1.0. Now all the parameters are gained (control points, corresponding weights, knot vectors along the u and v directions and parametric degree of the surface), so that the transition surface can be presented as an NURBS surface. And then the original transition surface is constructed. 4.3. Adjustment of off-boundary control points The original transition surface constructed above can only achieve a location continuity of these three surfaces. To ensure that the joint surface has the same boundary, Eq. (7) must be satisfied: PH 0;i ¼ QH 0;i
ði ¼ 0; 1; 2; . . . ; nq 1Þ,
(7)
where nq is the number of control points at the shared boundary of these two surfaces. Only with this continuity condition can the fairness and smoothness be not achieved and the requirements of surface modeling cannot be met. Therefore, the control points of the transition surface must be adjusted to reach the tangent continuity and curvature continuity. Only with these constraints can these three surfaces, workpiece surface, transition surface and blank holder surface, be smoothly joined with each other. The tangent vectors of these two surfaces at the shared boundary can be expounded in Eq. (8) v1 ¼ PH 0;i PH 1;i ; v2 ¼ QH 1;i QH 0;i :
(8)
In order to achieve tangent continuity, the value of tangent vectors must be equal, i.e., v1 ¼ v2 . It also can be simplified as QH 1;i ¼ PH 0;i þ QH 0;i PH 1;i .
Fig. 5. Original transition surface.
(9)
From Eq. (9) above, the control points QH 1;i of the transition surface can be calculated. If there are no special requirements, weights oQ1;i of these new control points may remain the same. Update the control points QH 1;i according to Eq. (9); the two surfaces can achieve a tangent continuity condition. For some NURBS surfaces with a
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fairly larger span between the first row and the second row of control points, the control points QH 1;i of its bridge extension surface (transition surface) are far away from the shared boundary. Hence, the fairness of the transition surface cannot be achieved and poor quality transition surface occurs. In order to solve this sort of problem, points must be inserted between the first and second row of control points (PH 0;i and PH 0;i ) according to the span compared with the bridge extension span [17], so that more control points are used to control the NURBS surface (workpiece surface) and a suitable span between these two rows of control points can be achieved to improve the quality of the transition surface. Fig. 6 shows the control points of two adjacent NURBS surfaces (the original surface P and extension surface Q). The left of the dotted line is the original surface and the other side lies on the extension one. In order to achieve curvature continuity between two joint NURBS surfaces, the adjacent six control points (PH 2;i , PH 1;i , PH 0;i , QH 0;i ðPH 0;i ¼ QH 0;i Þ, QH 1;i , QH 2;i ) of the two NURBS surfaces along the same control mesh must be coplanar, which is a prerequisite for G2 continuity. According to the tangent continuity, the points PH 2;i , PH 1;i , PH 0;i , QH 0;i and QH 1;i are already coplanar, so that the main task is to adjust the point QH 2;i to that plane. Here the projection method is used. Project the point QH 2;i on to the plane, and the projection point Q0H 2;i on that plane is obtained as Fig. 6 shows. From the above analysis, Q0H 2;i can be calculated from Eq. (10) V ¼ ðPH 0;i PH 1;i Þ ðPH 2;i PH 1;i Þ; Q0H 2;i ¼ kV 0 þ QH 2;i ;
(10)
V ¼ V 0 ; ðQH 1;i Q0H 2;i Þ V 0 ¼ 0;
where k is a constant that can be calculated from Eq. (10) above. The transition surface is located between the workpiece surface and the blank holder one, so that the next step is to adjust the off-boundary control points of the transition surface and the blank holder one to make these two surfaces smoothly jointed. Suppose that the blank holder is QH
PH
PH1,i
V
0 ,i
QH
V′ 1, i
Q ′H
PH
Original surface
2 ,i
= Q H 0 ,i
2 ,i
Extension surface
2 ,i
Fig. 6. Control points of two adjacent NURBS surfaces.
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Fig. 7. Surface bridge extension: (a) control meshes of extension surface, (b) shade model of extension surface, (c) shade model of surface after twoside extension, and (d) shade model of surface after full extension.
a rectangular plane, which can also be expressed as the NURBS surface with k parametric order along the extension direction. The value of k here is equal to the parametric order of the transition surface, which will simplify the calculation. Using the control point adjustment method above, curvature continuity between the transition surface and the blank holder is achieved. Fig. 7 shows an example of the bridge extension method used in transition surface design. It is clear that the workpiece surface, transition surface and blank holder surface are jointed smoothly. 5. Application examples As Fig. 8(a) shows, the original surface has the shape of a human bone that is broken during car accidents. It is often replaced by a metal sheet in surgery after it is broken. This metal sheet is only 0.1 mm in thickness. While it is directly formed in an MPF device (400 320 mm forming region and 100 mm stoke range), wrinkling or even cracking occurs. So it is very difficult to manufacture directly. In order to avoid these deficiencies in forming this sort of product, the blank holder must be designed to improve the formability of this workpiece. Nevertheless traditional methods are unavailable to design a reasonable transition between the blank holder surface and the workpiece surface. Fig. 8(b) shows a mesh model of these three types of surfaces after surface extension using the method described in this paper. It is clear that these three types of surfaces smoothly joint with each other and a G2 continuity condition is reached between them. From Fig. 9, it can be seen that the workpiece is formed smoothly by an
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method is used to design a transition surface for a traditional rectangle plane blank holder. It is achieved by adjusting of boundary control points to obtain smoothness and continuity. These two algorithms, through which the transition surface designed can reach G2 continuity, are both effective and reliable. Though they are developed for MPF surface modeling to design the transition surface, they are common methods and can also be used in other surface modeling. Examples show that these two methods are reliable and highly efficient.
Reference
Fig. 8. Transition surface design for metal sheet part used medical engineer: (a) shade model of the object surface, and (b) mesh model of the transient surface by extension.
Fig. 9. Sketch of deformed sheet metal.
MPF with a blank holder and a high-quality product is obtained. 6. Conclusions The design of transition surfaces between the workpiece surface and the blank holder is vital in MPF process design. It can prevent some defects in manufacture of metal sheet parts. Two design methods of transition surface are proposed: one is a flexible surface extension, and the other is a bridge surface extension. The main idea of flexible surface extension is reflection of control points. It is suitable for flexible transition surface and blank holder design when the shape of the workpiece is complex and it is difficult to design a transition surface via a traditional fixed rectangle plane blank holder. The bridge surface extension
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