Optimum design of roll forming process of slide rail using design of experiments

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1 Journal o Mechanical Science and Technology 22 (28) 1537~1543 Journal o Mechanical Science and Technology DOI 1.17/s Optimum design o roll orming process o slide rail using design o experiments Minjin Oh and Naksoo Kim * Department o Mechanical Engineering, Sogang University, Seoul, , Korea (Manuscript Received December 6, 27; Revised April 4, 28; Accepted April 26, 28) Abstract In the design o the roll orming process, design errors can be determined in advance by using an FE simulation tool such as SHAPE-RF. In the case o a product such as a slide rail having a complicated shape and requiring highprecision orming, a standard is necessary or quantitatively evaluating the quality o the ormed shape. In the analysis o the roll orming process o a slide rail, the pass having the largest deormation is designated as the target pass and the positions and shapes o the rolls are set as design variables. A minimum number o simulations was perormed by using the table o orthogonal arrays. A cost unction was obtained rom the results by using the design o experiments such as the response surace method and it was minimized or satisying the design constraints. By improving the design o the target pass, the shape o the inal product approaches that intended by the designer. Keywords: Design o experiments; Finite element method; Roll orming process; Shape dierence actor Introduction Roll orming is a process that progressively bends a lat strip o sheet metal through pairs o orming rolls, and it can be used or inexpensively manuacturing long sheet metal products with a constant cross section. Since roll orming requires manpower only or loading the strip and unloading the product, the manpower required can be reduced. I the shape o the product is simple, it takes little time to change the die and to set up a process. Since the length o the product can be controlled easily, roll orming can also be used or the batch production o small quantities o a product. Since the roll orming process was designed based on the designer s experience or developing a new product or improving the quality o existing products, the design deects were conirmed ater the production o the prototype; thereore, the compatibility o the corrected design could be veriied ater the * Corresponding author. Tel.: , Fax.: address: nskim@sogang.ac.kr KSME & Springer 28 production o the prototype. This process leads to an increase in the production cost, which reduces the competitiveness o manuacturers. In order to solve this problem, an FE simulation o the roll orming process is used prior to the production o a prototype in order to predict design deects and reduce the cost o design correction. Bhattacharayya et al. [1] perormed a semiempirical approach and by minimizing the total energy produced an expression or predicting deormation length o a channel section. Duggal et al. [2] compared the FE simulation results with Bhattacharayya s experimental results. And other numerical [3-6] and experimental [7, 8] studies have been perormed. Hong and Kim [9] developed a 3D FEM program or the roll orming process and predicted the scratch deect o the roll orming process with the rigidplastic inite element method. The analysis using the rigid-plastic inite element method has also been extended to predict the edge shape [1] and roll wear [11]. Kim et al. [12] made the prediction o buckling

2 1538 M. Oh and N. Kim / Journal o Mechanical Science and Technology 22 (28) 1537~1543 behavior o the roll orming process. Sheikh and Palavilayil [13] assessed an FE simulation program. The eectiveness o research by using the FE simulation tool has been veriied by several studies [14-17]. In this research, the accuracy o FE simulation is veriied through a comparison o the shape between the simulation and experimental results. The shape dierence actor (SDF) is suggested as a standard or quantitatively evaluating the quality o the ormed shape. With regard to the analysis o the roll orming process o a slide rail, the pass in which the largest deormation occurs is designated as the target pass. The positions and the curvature o rolls are set or the design variables. The cost unction is obtained by using the design o experiments such as the response surace method (RSM), and it is minimized by using the Broyden-Fletcher-Goldarb-Shanno (BFGS) method. Subsequently, the design is corrected by the minimized result and it causes the shape o the inal product to approach that intended by the designer. The proposed method and its complete procedure are described in this paper. 2. Research methods 2.1 Design o experiments This section describes the scheme o the experiments how to design experiments to solve a problem, how to record data, and how to obtain the maximum inormation through the least number o experiments by analyzing the data using some statistical techniques. In other words, the design o experiments selects the parameters o a problem, selects an experimental method, decides the order o experiments, and selects an optimum analysis method. For a design that has many parameters, predictable interactions o two parameters are detected, and inormation on two or more interactions is sacriiced. As a result, a table is made or an experimental plan with a small number o experiments. The table is called as the table o orthogonal arrays. The response surace method (RSM) is mainly used to obtain an explicit unction rom experimental data. Recently, it has been used to represent a relationship between experimental parameters and responses rom numerical experimental values as the explicit unction [18]. Two methods can be used to calculate a response surace. First, an approximate unction is assumed and the equations o the coeicients o the unction are solved numerically to ind a relationship between the variables and the unction values. Second, the approximation unction is evaluated by using an optimization [19]. In this study, a second-order regression model such as that given by Eq. (1) is used to calculate a response surace. nd nd nd (1) y = β + β x + β xx i i ij i j i= 1 i= 1 j i where x i denotes the design variables; n d, the number o design variables; and β i, the unknown coeicients. Eq. (2) is used to calculate the coeicients o RSM that minimize the square summation o the residuals using least square method. T 1 T β = ( XX) XY (2) where X denotes the design matrix comprising experimental points and Y denotes the response vector. 2.2 Shape dierence actor I products that are manuactured through the roll orming process do not meet the standards because o a design error, it is necessary to correct the design deects, as shown in Fig. 1. A slide rail having a complicated shape and requiring high precision in orming and straightness is manuactured by using the roll orming process [2]. It is diicult to determine the compatibility o the design since the product has a complicated shape. A standard is necessary to quantitatively evaluate the quality o the ormed shape; one such standard is called the shape dierence actor (SDF). In order to quantitatively evaluate the precision o the shape o a Fig. 1. Flow chart or the correction o the roll orming process design.

M. Oh and N. Kim / Journal o Mechanical Science and Technology 22 (28) 1537~1543 1539 3. Simulation and experimental results 3.1 Process condition Fig. 2. Comparison o the raw plan and the simulation result.

3 M. Oh and N. Kim / Journal o Mechanical Science and Technology 22 (28) 1537~ Simulation and experimental results 3.1 Process condition Fig. 2. Comparison o the raw plan and the simulation result. A slide rail is comprised o an inner member, middle member, outer member, and bearing balls. Since this research ocuses on the slide rail, the middle member is analyzed because an inner rail and an outer rail are ormed at the middle member. The middle member is manuactured with a 25-pass line. The distance between the passes is 35 mm; oddnumbered passes are set up as driving rolls, and evennumbered passes are set up as idle rolls, and the velocities o each pass roll are set up to produce a product with a constant velocity o 4m/min. The thickness and width o the initial strip is 2 mm and 6 mm, respectively; the strip is made o SCP1, whose material properties are listed in Table 1. The inal shape o the cross section o the product manuactured in the experiment is shown in Fig. 4. Table 1. Material properties o SCP1 Fig. 3. Measurement o the dierence between the raw plan and the simulation result. Young s modulus (GPa) 21 Poisson s ratio.3 Yield Strength (MPa) 433 UTS (MPa) 46 product manuactured by the roll orming process, the cross section o a simulation or experimental result is set on the center o the cross section o a raw plan with grids drawn on it, as shown in Fig. 2. As shown in Fig. 3, SDF is decided by the summation o the dierence in the distance that is measured between the raw plan and the simulation or experimental result along the direction o thickness and it is deined as given by Eq. (3). 2 SDF = n ( di / t ) (3) i = 1 where d i denotes the dierence in distance between the result and the raw plan at the i th elements and t denotes the thickness o the initial strip. Since the shape o the cross section o the product is symmetric, the SDF is measured at the right cross section o the product. Fig. 4. Cross section obtained rom the experiment. 3.2 FE simulation sotware FE simulations are perormed by the roll orming simulation program SHAPE-RF v4.. based on the rigid-plastic inite element method. This program uses the normalized plane strain condition as the initial boundary condition or initially determining the ree surace. The velocity ield is calculated by the FEA o the 3D kinematic steady state and the inal shape is determined by an iterative method that calibrates the boundary conditions and the ree surace. Inormation such as the strain rate and pressure torque

154 M. Oh and N. Kim / Journal o Mechanical Science and Technology 22 (28) 1537~1543 Table 2. Process conditions o FE simulation. Flow stress (MPa) σ = 52(.2 + ε ) Initial thickness (mm) 2.

Flower pattern o the slide rail s middle member. Fig. 6. FE simulation result. is obtained based on the velocity ield. The reliability o SHAPE-RF has been veriied by several previous papers [9-13].

4 154 M. Oh and N. Kim / Journal o Mechanical Science and Technology 22 (28) 1537~1543 Table 2. Process conditions o FE simulation. Flow stress (MPa) σ = 52(.2 + ε ) Initial thickness (mm) 2. Strip width (mm) 6. Friction coeicient.1 No. o PASS 25 No. o elements Section 8 Rolling direction 2.24 Fig. 7. Longitudinal strain along the rolling direction. Fig. 5. Flower pattern o the slide rail s middle member. Fig. 6. FE simulation result. is obtained based on the velocity ield. The reliability o SHAPE-RF has been veriied by several previous papers [9-13]. The process conditions o the FE simulation are listed in Table 2. Swit s low stress equation is used to express the stress-strain relation o a strip, and it is deined as given by Eq. (4). σ = K( ε + ε) n (4) where σ denotes the low stress; K, the strength coeicient; ε, the eective strain; ε, the initial eective strain; and n, the strain hardening coeicient. The low stress o the strip is obtained by using the convert unction o SHAPE-RF and it is shown in Table 2. The lower pattern o the middle member is obtained by using the FE simulation program and it is shown in Fig. 5. The inal shape o the cross section o the roll orming product is shown in Fig Veriication o FE simulation sotware The SDF obtained rom the experimental and simu- lation results is.8745 and.91677, respectively. The dierence between the results and the raw plan mostly occurs at areas where the slide rail is bent. The relative error is 4.83%. The FE simulation cannot perectly approximate the real process because obscure parameters exist at the site o the manuacturing process. For example, or any model o riction that expresses the contact between objects to be valid, it must explain the rictional behavior o two bodies under dierent loads, speed o relative sliding, temperature, surace conditions, environment, etc., as observed in practice. Consequently, many models have been proposed with varying degrees o success [21]. Although many uncertain parameters exist, as mentioned above, the FE simulation is veriied since the shape dierence error between the FE simulation and experimental results that is evaluated at the inal section increases to 4.83% as compared to the incipient shape. 4. Procedure or design correction and discussion 4.1 Designation o target pass For design variables to be applied to the design o experiments, they should be restricted because many process variables are ound in the roll orming process. In the FE simulation o the roll orming process o the slide rail, the pass where the largest deormation occurs is designated as the target pass or the design variables. The longitudinal strain along the rolling direction is shown in Fig. 7 and the largest deormation occurs at the 6.3 m spot along the rolling direction. Thereore, the 18 th pass is designated as the target pass.

M. Oh and N. Kim / Journal o Mechanical Science and Technology 22 (28) 1537~1543 1541 Table 3. Levels o the design variables (unit : mm). Design Variables Level Level 1 Level 2 A 17.7 18.7 19.7 B 12.

5 M. Oh and N. Kim / Journal o Mechanical Science and Technology 22 (28) 1537~ Table 3. Levels o the design variables (unit : mm). Design Variables Level Level 1 Level 2 A B C Table o orthogonal arrays The strip is bent by the let and right rolls at the 18 th pass. Since the slide rail has a symmetric shape, the design variables are limited to the right roll. Design variable A is the x-coordinate o the lat part o the right roll and B is the y-coordinate o the same part. C is the curvature o the right roll. The design variables and levels are listed in Table 3 and a table o orthogonal arrays L 9 (3 4 ) is used. Table 4 shows the table o orthogonal arrays or the SDF obtained rom the FE simulation results. Table 4. Table o orthogonal arrays or the SDF. No. A B C SDF o the simulation Optimization o the cost unction Based on the table o orthogonal arrays, the cost unction obtained by RSM is given by Eq. (5) as: Φ= x x x +.533x x x-.825 xx xx (5) Fig. 8. Comparison o Φ c and Φ a where x denotes the design variable A; 1 x 2, the design variable B; and x 3, the design variable C. In order to examine the adequacy o the cost unction, Fig. 8 shows the comparison o the values between the cost unction in which the conditions o Table 4 are applied and the SDF obtained rom the FE simulation results. In order to investigate how the numerical dierences in the compared values exist, it is veriied through Eq. (6) that the error is less than 1%. Thereore, the cost unction can represent the SDF between the inal shape o the product and the raw plan when the 18 th pass is corrected. c a Φ - Φ Error(%) = 1 c Φ (6) where Φ c denotes the SDF computed rom each simulation and Φ a denotes the value o the cost unction when the same variables are inputted. In order to minimize the cost unction, the BFGS method, which directly updates a Hessian matrix, is used. Initial design variables and the constraints are given as ollows: x1 = 17., x2 = 12., x3 = 5.5 (7) 17.7 x x (8) 5. x The result o minimization is given as ollows: x = 19.7, x = , x = Φ=.6159 Based on this result, the 18 th pass is corrected and the FE simulation is perormed. There is a dierence o 3.87 % between the minimum value o the cost unction and the SDF o the FE simulation result o

1542 M. Oh and N. Kim / Journal o Mechanical Science and Technology 22 (28) 1537~1543 Fig. 9. Comparison o the SDF between the original design and the optimum design. Fig. 1. Comparison o the raw plan and the optimized simulation result.

91677; the comparison o the results is shown in Fig. 9. The cross section o the optimized simulation result and the raw plan are compared, as shown in Fig. 1.

6 1542 M. Oh and N. Kim / Journal o Mechanical Science and Technology 22 (28) 1537~1543 Fig. 9. Comparison o the SDF between the original design and the optimum design. Fig. 1. Comparison o the raw plan and the optimized simulation result Although the result indicates a wide gap in the minimum o the cost unction, the SDF o the optimized result decreases by 5.34 % as compared to the original result o.91677; the comparison o the results is shown in Fig. 9. The cross section o the optimized simulation result and the raw plan are compared, as shown in Fig. 1. A signiicant dierence is observed between Φ c and Φ a since the cost unction obtained rom the restricted design variables does not consider all conditions o the target pass such as the design o the top and bottom rolls. Further, roll orming has many design variables such as roll velocities, riction condition, and angle o roll. I more process variables are contained in the design variables, then the error between the FE simulation result and the cost unction will be smaller than that in the above result. 5. Conclusions In order to improve the eiciency o the roll orming process, it is very important to immediately correct a design that has some deects. There is a product called a slide rail that has a complex shape and whose design is diicult to modiy. In this paper, the roll orming design was corrected by the design o experiments. The SDF was also introduced to determine the compatibility o the roll design. The conclusions drawn rom this study are listed below. The correction o the design o the target pass, which is designated through the measurement o the longitudinal strain along the rolling direction o the entire process, aects the inal shape o the roll orming product. The SDF, which represents the dierence between the cross section o the product that is aected by the change o the design variables and the raw plan, is suggested as a standard. Further, the cost unction that can evaluate the SDF is derived by using the design o experiments such as the RSM. The optimum design is determined through the minimization o the cost unction. The minimum value o the cost unction is applied to the design o the target pass and it decreases the SDF by 5.34 %. Consequently, the cross-sectional shape o the slide rail obtained by the simulation approaches the shape intended by the designer. Nomenclature x i : Design variable d i : Dierence between the simulation or experimental result and the raw plan t : Thickness o initial strip Φ : Cost unction σ : Flow stress ε : Eective strain ε : Initial eective strain c Φ : Computed shape dierence actor a Φ : Analytical shape dierence actor Reerences [1] D. Bhattacharayya, P. D. Smith, C. H. Yee and I. F. Collins, The prediction o Deormation length in cold roll orming, J. Mech. Working Technol., 9 (1984) [2] N. Duggal, M. A. Ahmetoglu, G. L. Kinzel and T. Altan, Computer aided simulation o cold roll orming-a computer program or simple section proiles, J. Mater. Proc. Technol., 59 (1996) [3] M. Brunet and S. Ronel, Finite element analysis o roll-orming o thin sheet metal, J. Mater. Proc. Technol., 45 (1994)

7 M. Oh and N. Kim / Journal o Mechanical Science and Technology 22 (28) 1537~ [4] M. Brunet, B. Lay and P. Pol, Computer aided design o roll-orming o channel sections, J. Mater. Proc. Technol., 6 (1996) [5] C. Liu, Y. Zhou and W. Lu, Numerical simulation o roll-orming by B-spline inite strip method, J. Mater. Proc. Technol., 6 (1996) [6] M. Farzin, M. S. Tehrani and E. Shameli, Determination o buckling limit o strain in cold roll orming by the inite element analysis, J. Mater. Proc. Technol., (22) [7] S. M. Panton, J. L. Duncan, S. D. Zhu, Longitudinal and shear strain development in cold roll orming, J. Mater. Proc. Technol., 6 (1996) [8] M. Lindgren, Experimental investigations o the roll load and roll torque when high strength steel is roll ormed, J. Mater. Proc. Technol., 191 (27) [9] S. Hong and N. Kim, Study on scratch deect o roll orming process, Trans. o the KSME (A). 25 (21) [1] S. Lee and N. Kim, Prediction and design o edge shape o initial strip or thick tube roll orming using inite element method, Trans. o the KSME (A). 26 (22) [11] B. Kang and N. Kim, A study on roll wear in the roll orming process, Trans. o the KSME (A). 27 (23) [12] Y. Kim, J. Kim, Y. Jeoung and N. Kim, Buckling analysis o roll orming process using inite element method, Trans. o the KSME (A). 27 (23) [13] M. A. Sheikh and R. R. Palavilayil, An assessment o inite element sotware or application to the rollorming process, J. Mater. Proc. Technol. 18 (26) [14] A. Alsamhan, P. Hartely and I. Pillinger, The computer simulation o cold-roll-orming using FE methods and applied real time re-meshing techniques, J. Mater. Proc. Technol., 142 (23) [15] A. K. Datta, G. Das, P. K. De, P. Ramachandrarao and M. Mukhopadhyaya, Finite element modeling o rolling process and optimization o process parameter, Material Science and Engineering (A). 426 (26) [16] M. Lindgren, Cold roll orming o a U-channel made o high strength steel, J. Mater. Proc. Technol., 186 (27) [17] R. S. Senanayake, I. M. Cole and S. Thiruvarudchelvan, The application o computational and experimental techniques to metal deormation in cold roll orming, J. Mater. Proc. Technol., 45 (1994) [18] E. Kim, I. Lee and B. Kim, Optimum design o washing machine lange using design o experiment, Trans. o the KSME (A). 31 (5) (27) [19] D. Lee, S. Baek, K. Lee, S Cho and W. Joo, Multiobjective optimization in discrete design space using the design o experiment and the mathematical programming, Trans. o the KSME (A). 26 (1) (22) [2] S. Jeong, S. Lee, G. Kim, J. Kim and S. Shin, Rigid-plastic inite element method or development mold o dishwasher s under rail roll orming, Trans. o the KSMPE Annual Meeting. (26) [21] S. Kalpakjian and S. R. Schmid, Manuacturing processes or engineering materials, second ed. Addison-Wesley Publishing Company, New York, USA, (1992).

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