Theoretical and experimental study on the hydroforming of bifurcation tube

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Journal of Materials Processing Technology 142 (2003) 367 373 Theoretical and experimental study on the hydroforming of bifurcation tube Quang-Cherng Hsu Department of Mechanical Engineering,

1 Journal of Materials Processing Technology 142 (2003) Theoretical and experimental study on the hydroforming of bifurcation tube Quang-Cherng Hsu Department of Mechanical Engineering, National Kaohsiung University of Applied Sciences, 415 Chien-Kung Road, 807 Kaohsiung, Taiwan, ROC Received 23 September 2001; received in revised form 2 September 2002; accepted 14 February 2003 Abstract Hydroforming is one kind of plasticity working, which can be used to increase the diameter, to change the geometry, or to expand the outer walls of cylindrical shells or tubes. The forming principle of hydroforming is by using a die to constrain the outer geometry of a tube and then pumping high pressure oil into the inner tube, so as to force part of the tube to flow into the die cavity. In order to prevent local thinning and to improve material flow capability, an axial force may be imposed on the both sides of the tube. If the axial force is not imposed correctly, a buckling defect will occur. The forming theory and plasticity of the hydrostatic bulging process are studied. The parameter effects on the hydrostatic bulging process will be analyzed by using a DEFORM-3D finite element analysis package. A new approach to measure 3D deformations and plastic strains in forming parts is also presented. A strain analysis module is also built based on the geometry of deformed grids and initial grids. The principal stretch ratios and angle changes of grid patterns are calculated, and the logarithmic strains are also computed Published by Elsevier B.V. Keywords: Tube hydroforming; Strain measurement; Finite element analysis; Perspective transformation; CCD camera 1. Introduction In this paper, the results of analytical and experimental studies on the hydroforming of bifurcation tubes are presented. Hydroforming is one kind of plasticity working or metal forming process, which can be used to increase the diameter, to change the geometry, or to expand the outer walls of a cylindrical shell or tube. This process is very important, especially for industrial products of light weight and high strength. Hydroforming has a wide range of application areas, e.g., to produce tee fittings, bicycle structure fittings, car differential gear boxes, and trumpet door lock handles, etc. The principle of hydroforming is by using the die sets to constrain the outer geometry of a tube and pumping high pressure oil into the inner tube, so as to force part of material of the tube to flow into the die cavity. If the pressure of the oil is controlled badly, a crack defect will occur. In order to prevent local thinning and to improve capability of material flow, an axial force can be imposed on the both sides of the tube. If the axial force is not imposed correctly, a buckling defect will occur. Tel.: x5338; fax: address: hsuqc@cc.kuas.edu.tw (Q.-C. Hsu). Hydrostatic extrusion [1,2] was the earliest research made on hydroforming for the plasticity forming of brittle material. From then on, several experimental and theoretical studies were conducted. Zhang et al. [3] developed a new production method for large spherical vessels by using the hydro-bulging process. Domann et al. [4 6] treated hydroforming as a flexible production method for various kinds of products. Ahmed and Hashmi [7] developed a theoretical method of estimating the process parameters for the forming of tubular components. In addition, hydroforming can be used to produce complicated shapes of sheet metal products [8,9]. As to experimental study, the measurement of plastic strains on deformed surfaces is the major consideration. Harvey [10] has developed an optical strain measurement system to acquire the principal strains of deformed grids automatically. By using the algorithms of pattern recognition and regression analysis, Lee and Hsu [11] constructed the forming limit diagrams of sheet metals through principal strain measurement of deformed grids. Due to the complicated deformation of real parts, three-dimensional strain measurement is becoming important [12,13]. The plasticity theory of the hydroforming process is studied in this paper based on two different approaches: the force equilibrium equation analysis and the finite element analysis. The parameter effect of inner pressure on the /$ see front matter 2003 Published by Elsevier B.V. doi: /s (03)

2 368 Q.-C. Hsu / Journal of Materials Processing Technology 142 (2003) hydroforming process is analyzed. In the experimental study, the electro-chemical etching process is used to mark square grids on the outer surface of the source tube. An adjustable two-roller frame is designed as a base structure to support the source tube in the case of different diameters when an electro-chemical etching process is applied. The tee forming process is used to produce a bifurcation tube and the square grids displacements are measured by 3D digital image processing technology due to their complicated deformation pattern. Therefore, the plastic strain distributions can be generated according to source grids and deformed grids. The algorithms and accuracy necessary to reconstruct 3D patterns from a 2D view of two CCD cameras are presented and discussed. The results of theoretical and experimental studies are also compared and discussed. The research results can be used as the process design guidelines of hydroforming. 2. Theoretical and experimental methods 2.1. Theory of tube hydroforming Assume that D 0 and t 0 are outer diameter and thickness of source tube, respectively. The source tube is sealed and is pushed from two opposite sides by punches, as shown in Fig. 1. Under the conditions of diameter remaining constant, plane strain deformation, and Von-Mises yield criterion, the required minimum inner pressure P i can be derived from force equilibrium equation [7] as P i = σ y(b 2 a 2 ) (1) 3a 2 where σ y is the yielding stress, b = D 0 /2, and a = b t 0. The reaction pressure from outer surface of source tube to die is p 0 = p i + S ( a ) 2 ln + ms (a b) (2) b x where S is the shear yielding stress. The radial stress of source tube is σ r = p i + S ( a ) 2 ln + ms (a r) (3) r x 2.2. Perspective transformation and camera model Perspective transformation is used to project one material point from spatial coordinate system (world coordinate) onto planer surface (image coordinate system). Assume that (x, y, z) is a camera coordinate system, image plane is located at x y plane, the coordinate of the center of optic lens is (0, 0,λ), and (X, Y, Z) is world coordinate system, as shown in Fig. 2. According to the similarity of triangles and homogenous coordinate system [14], the prospective projection is represented as kx ky c h = Pw h = (4) kz k λ where c h and w h are the homogenous coordinates of camera model and world model, respectively. P is a matrix of perspective projection. In general, (x, y, z) is different to (X, Y, Z). Assume that pan angle θ is the angle between x and X axes, tile angle α is the angle between z and Z axes, w 0 (X 0, Y 0, Z 0 ) is the offset of the center of the gimbal from the original of the world coordinate system, and r(r 1, r 2, r 3 ) is the offset of the center of image plane with respect to the gimbal center, as shown in Fig. 3. Then Eq. (4) is generalized as c h = PCRGw h (5) where G and C are the perspective matrices, and R the rotation matrix X r Y r 2 G =, C =, Z r cos θ sin θ 0 0 R = sin θ cos α cos θ cos α sin α 0 (6) sin θ sin α cos θ sin α cos α Fig. 1. Schematic diagram of tee hydroforming process [7]. However, these parameters of w 0, r, θ, and α are not easy to measure. During image measuring, c h is known while w h is

3 Q.-C. Hsu / Journal of Materials Processing Technology 142 (2003) Fig. 2. Projection of one material point from spatial coordinate system (X, Y, Z) onto image planer surface (x, y) with focal length λ [14]. unknown, therefore, Eq. (5) should be transformed into 1 t 11 t 12 t 13 t 14 w h = T 1 t 21 t 22 t 23 t 24 c h = c h (7) t 31 t 32 t 33 t 34 t 41 t 42 t 43 t 44 where T is equal to PCRG. Because the value of z coordinate of c h is 0, the values of t 3j are not important. Therefore, Eq. (7) is changed into (t 11 t 41 x)x + (t 12 t 42 x)y + (t 13 t 43 x)z + (t 14 t 44 x) = 0, (t 11 t 41 y)x + (t 12 t 42 y)y + (t 13 t 43 y)z + (t 14 t 44 y) = 0 (8) Fig. 3. A generalized camera model where w 0 (X 0, Y 0, Z 0 ) is the offset of the gimbal center from the original and r(r 1, r 2, r 3 ) is the offset the center of image plane with respect to the gimbal center [14].

4 370 Q.-C. Hsu / Journal of Materials Processing Technology 142 (2003) There are 12 parameters t ij (i = 1, 2, 4, j = 1 4) needed to be solved, this is called camera calibration. It needs at least six known points including world and image systems [15]. However, if t ij, x and y are given, the values of X, Y and Z cannot be solved from Eq. (6) due to the lack of enough conditions. Assume that the upper labels 1 and 2 represent the view 1 and view 2, respectively. Therefore, Eq. (8) can be represented by [A][X] = [B] (9) where t11 1 t1 41 x1 t12 1 t1 42 x1 t13 1 t1 43 x1 t21 1 [A] = t1 41 y1 t22 1 t1 42 y1 t23 1 t1 43 y1 t11 2 t2 41 x2 t12 2 t2 42 x2 t13 2, t2 43 x2 t 2 21 t2 41 y2 t 2 22 t2 42 y2 t 2 23 t2 43 y2 [X] T = [ X Y Z], [B] T = [ t44 1 x1 t14 1 t1 44 y1 t24 1 t2 44 x2 t14 2 t2 44 y2 t24 2 ] (10) Therefore, the values of X, Y and Z can be solved by [X] = [[A] T [A]] 1 [A] T [B] (11) 2.3. Calculation of three-dimensional plastic strains Assume that the deformation is linear and no shear strain occurs outside the plane. Choosing axes of X and Y coinciding with two different sides of triangle grid (half of square grid), as shown in Fig. 4. Therefore, the strains can be calculated based on the stretch ratio and angle change of these two sides [16]. The relation of Green deformation tensor C and stretch ratio ds/ds (the ratio of deformed length to original length) is represented as [ ] ds 2 = N CN (12) ds where N is an unit vector along ds direction and N is its transport. Because N is equal to [ 1 0 0], therefore, Eq. (12) is derived as [ ] ds 2 = C 11 (13) ds Assume that the length of the original square grid is D, the following relations can be obtained: C 11 = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 + (z 2 z 1 ) 2 D 2, C 22 = (x 3 x 1 ) 2 + (y 3 y 1 ) 2 + (z 3 z 1 ) 2 D 2 (14) Due to the relation of Lagrangian strain tensor E and Green deformation tensor C, i.e. C = 1 + 2E, the following equation can be obtained: E 11 = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 + (z 2 z 1 ) 2 D 2 2D 2, E 22 = (x 3 x 1 ) 2 + (y 3 y 1 ) 2 + (z 3 z 1 ) 2 D 2 2D 2, [(x 2 x 1 )(x 3 x 1 ) + (y 2 y 1 )(y 3 y 1 ) + (z 2 z 1 )(z 3 z 1 )] E 12 = E 21 = 2D 2 (15) The principal strains can be represented as E 1,2 = E (E11 ) 11 + E 22 E 2 22 ± + E (16) Due to the relation of stretch ratio and principal strain, the natural strains can be represented as ( ) ds ε 1,2 = ln = ln 1 + 2E 1,2 (17) ds 2.4. Finite element analysis of hydroforming process DEFORM-3D [16] system is applied to analyze the tube hydroforming process. In DEFORM-3D, rigid die geometry can be defined by a set of polygons. Flow stress of the workpiece can be defined as a function of strain, strain rate and temperature. Coulomb and constant shear factor frictions are supported. Mesh generation and automatic remeshing are possible. It offers a complete environment to perform the analysis and simulation of material forming processes. 3. Results and discussion Fig. 4. Deformation of half of square grid with original length D. ds and ds represent the original length and the deformed length, respectively. The proposed strain measurement system includes one Pentium 133 PC, two sets of CCD camera (one is 16 mm standard lens and the other is with 10 times zoom lens),

Q.-C. Hsu / Journal of Materials Processing Technology 142 (2003) 367 373 371 Table 2 Testing of strain calculation Number x y z Strains Uniaxial tension 1 0 0 0 E 1 = 0.105, E 2 = 0 2 1.

Block diagram of the measurement system. two Matrox Meteor image adapters, and a light source. The software tools Visual BASIC and Active MIL image library are used to develop the proposed system.

5 Q.-C. Hsu / Journal of Materials Processing Technology 142 (2003) Table 2 Testing of strain calculation Number x y z Strains Uniaxial tension E 1 = 0.105, E 2 = E 12 = E 21 = ε 1 = 0.095, ε 2 = 0 Pure shear E 1 = 0, E 2 = 0 2 Cos 22.5 Sin E 12 = E 21 = Sin 22.5 Cos ε 1 = 0.267, ε 2 = Fig. 5. Block diagram of the measurement system. two Matrox Meteor image adapters, and a light source. The software tools Visual BASIC and Active MIL image library are used to develop the proposed system. The block diagram of the proposed measurement system is depicted in Fig. 5. The experimental procedures of strain measurement of tube hydroforming parts are as follows. (1) The square grids of 5.8 mm are fabricated by using electro-chemical etching processes on the surface of the source tube. (2) Perform tube hydroforming. (3) Perform camera calibration by adjusting the positions of the two cameras, the light source and the focal length of each camera so as to make the two images of the calibration part clear. (4) Substitute the calibration part with the hydroforming part. Without changing any parame- ters of the camera, adjust the position of the hydroforming part so as to make its images at both views clear. (5) Measure these grid points coordinates and calculate their plastic strain. During camera calibration, six points whose world and image coordinates values are known. In the forward test, Table 1 Calibration and testing results of the proposed system World coordinate system View 1 View 2 Number X Y Z x 1 y 1 x 2 y Forward test Reverse test Fig. 6. (a) Strain measurement from two views. (b) View 1 of tee hydroforming part. (c) View 2 of tee hydroforming part.

372 Q.-C. Hsu / Journal of Materials Processing Technology 142 (2003) 367 373 Fig. 7. Strain distributions of tee hydroforming part.

In the reverse test, the world coordinate values are calculated based on calibration parameters and image coordinate values.

6 372 Q.-C. Hsu / Journal of Materials Processing Technology 142 (2003) Fig. 7. Strain distributions of tee hydroforming part. the image coordinate values are calculated based on calibration parameters and world coordinate values. In the reverse test, the world coordinate values are calculated based on calibration parameters and image coordinate values. The testing results (as listed in Table 1) are reasonable, yet it still needs to improve. At the test of strain calculation (uniaxial tension and pure shear of square grid 1 mm), the results are fully correct as compared with theoretical values [17], as listed in Table 2. Fig. 6 shows the strain measurement of the tube hydroforming part from two views. The strain measurement of five grids that are located at the center of specimen is shown in Fig. 7. Fig. 8 depicts the arrangement of DEFORM-3D simulation of tee hydroforming. The material of the source tube is middle carbon steel, 90 mm in outer diameter, 4 mm in thickness, and 210 MPa in yielding stress. From Eq. (1), the optimal inner pressure is 24.8 MPa for tee hydroforming of this tube. Fig. 9 shows the analysis results without and with inner pressure. It shows a better deformation pattern when applying inner pressure at the tee hydroforming process. Fig. 9. Deform-3D analysis results of tee forming (top view). (a) Without inner pressure. (b) With 24.8 MPa inner pressure. 4. Conclusions Fig. 8. Schematic diagram of tee hydroforming for DEFORM-3D analysis. The plasticity theory of hydroforming process is studied in this paper based on two different approaches: the force equilibrium equation analysis and the finite element analysis. The tee forming process is used to produce a bifurcation tube and the square grids displacements are measured by 3D digital image processing technology due to their complicated deformation pattern. Therefore, plastic strain distributions can be generated according to source grids and deformed grids. The algorithms and accuracy to reconstruct 3D patterns from 2D views of one CCD camera are presented and discussed. The testing results of 3D deformation measurement are reasonable, yet it still needs to improve. The tests of strain calculation are fully correct as compared with theoretical values. From DEFORM-3D simulation of tee hydroforming,

7 Q.-C. Hsu / Journal of Materials Processing Technology 142 (2003) it shows a better deformation pattern with inner pressure than without inner pressure. The research results can be used as the process design guidelines of hydroforming. Acknowledgements This research work was partly supported by National Science Council, ROC, under grant no. NSC E The author would like to thank Ta-Chen Stainless Pipe Co., Taiwan, for their kindly support of tee hydroforming experiments. References [1] P.W. Bridgman, Studies in the Large Plastic Flow and Fracture with Special Emphasis on the Effect of Hydrostatic Pressure, McGraw-Hill, New York, [2] N. Inoue, M. Nishihara, Hydrostatic Extrusion: Theory and Application, Elsevier Applied Science Publication, London, [3] S.H. Zhang, Z.R. Wang, T. Wang, Experimental research into the hydro-bulging process for spherical vessels with a new structure, J. Mater. Process. Technol. 21 (1990) [4] F. Domann, A. Boehm, The shaping of hollow shaft-shaped workpiece by liquid-bulge-forming, in: Proceedings of Fifth International Conference on Technology of Plasticity, Beijing, China, September 1993, pp [5] F. Domann, C. Hartl, Liquid-bulge-forming as a flexible production method, J. Mater. Process. Technol. 45 (1994) [6] F. Domann, Ch. Hartl, Tube hydroforming-research and practical application, J. Mater. Process. Technol. 71 (1997) [7] M. Ahmed, M.S.J. Hashmi, Estimation of machine parameters for hydraulic bulge forming of tubular components, J. Mater. Process. Technol. 64 (1997) [8] T. Nakagawa, K. Nakamura, H. Amino, Various applications of hydraulic counter pressure deep drawing, J. Mater. Process. Technol. 71 (1997) [9] M. Kleiner, A. Gartzke, R. Kolleck, Experimental and finite element analysis of capabilities and limits of a combined pneumatic and mechanical deep-drawing process, Ann. CIRP 46 (1) (1997) [10] D.N. Harvey, Optimizing patterns and computational algorithms for automated, optical strain measurement in sheet metal, in: Proceeding of 13th Biennial Congress on Efficiency in Sheet Metal Forming, Melbourne, Australia, International Deep Drawing Research Group, 1984, pp [11] R.S. Lee, Q.C. Hsu, Image processing system for circular-grid analysis in sheet metal forming, Exp. Mech. 34 (2) (1994) [12] J.H. Vogel, D. Lee, An automated two-view method for determining strain distributions on deformed surface, J. Mater. Shaping Technol. 64 (1989) [13] N. Tan, L. Melin, C. Magnusson, Application of an image processing technique in strain measurement in sheet metal forming, J. Mater. Process. Technol. 33 (1992) [14] K.S. Fu, R.C. Gonzalez, C.S.G. Lee, Robotics: Control, Sensing, Vision, and Intelligence, McGraw-Hill, New York, [15] D.F. Rogers, J.A. Adams, Mathematical Elements for Computer Graphics, McGraw-Hill, New York, [16] Anonymous, DEFORM-3D System User s Manual, Version 3.1, Scientific Forming Technologies Co., USA, [17] L.E. Malvern, Introduction to the Mechanics of a Continuous Medium, Prentice-Hall, New Jersey, 1969.

Comparison of different analysis models to measure plastic strains on sheet metal forming parts by digital image processing

International Journal of Machine Tools & Manufacture 43 (2003) 515 521 Comparison of different analysis models to measure plastic strains on sheet metal forming parts by digital image processing Q.-C.