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An Improved Model for Camber Generation During Rough Rolling Process ISIJ International, Vol. 55 (2015),ISIJ International, No. 9 Vol. 55 (2015), No. 9, pp. 1980–1986 An Improved Model for Camber Generation during Rough Rolling Process Youngil KANG,1) Yujin JANG,2) Yongjun CHOI,3) Dukman LEE3) and Sangchul WON4)* 1) Control and Automation Laboratory, Graduate Institute of Ferrous Technology, Pohang University of Science and Technol- ogy, 77 Cheongam-Ro, Pohang, 790-784 Korea. 2) Department of Information Communication Engineering, Dongguk University, 123, Dongdae-ro, Gyeongju, 780-714 Korea. 3) Technical Research Laboratories, POSCO, 6261, Donghaean- ro, Pohang, 790-300 Korea. 4) Department of Electronic & Electrical Engineering, Pohang University of Science and Technology, 77 Cheongam-Ro, Pohang, 790-784 Korea. (Received on February 11, 2015; accepted on May 12, 2015) A model to describe the curvature of the centerline of a strip at the delivery side during hot rolling can be used to predict the longitudinal shape of the strip after rolling, and can be useful when designing a controller to reduce camber generation. However, the existing model was obtained from studies about side-slipping, and is based on incorrect assumptions. This paper uses the definition of curvature and tan- gential angle to clarify why the assumptions of the former model are incorrect. Also, this paper proposes a new model for curvature of the centerline at the delivery side; this model is based on the assumption that the curved strip is generated only by the difference between the velocities of the sides of the strip. The accuracy of the proposed model is validated by FEM simulation. KEY WORDS: camber; side-slipping; finite element method (FEM). tangent vector are known.1,2) To produce straight strip after 1. Introduction rolling, curvatures should be zero at all points on the cen- Hot rolling mill accepts a slab produced by continuous terline.3) Therefore, an accurate model of curvature of the casting process, then converts it to strip that has predefined centerline of the strip at the delivery side is necessary. thickness. The main concern of the hot rolling process is The first study model of the curvature of the centerline to maximize productivity, by minimizing the production of at the delivery side was produced by Nakajima et al.4) In poor final product and by stabilizing the process. However, this work, the time varying locations of arbitrary points on problems due to irregular longitudinal shape of the strip the centerline were considered to obtain dynamic relation such as the lateral movement of the strip (side-slipping), between side-slipping and wedge ratio. They also derived and the longitudinal bending (camber) are the main fac- a model for delivery-side curvature, which is affected by tors that cause defects such as strip-edge folds, scrapes and the entry-side curvature and wedge ratio. Shiraishi et al.5) telescoped coils. Also, a curved strip is dangerous because introduced a modified delivery-side curvature model that it could strike and damage the side of the process line; such includes a camber change coefficient that should be tuned events can cause production stoppages. to account for various rolling conditions such as front and Both camber and side-slipping are caused by asymmetric back tension. factors during rolling such as thickness deviation, tempera- These two models were obtained by considering two ture difference, roll gap difference, mill constant variation, fundamental principles: (1) the curvature at the delivery and work roll wear of the strip along its transverse direc- side of the strip is affected by entry-side curvature scaled tion. Moreover, camber and lateral motion are coupled to by the square of the reduction ratio; (2) the velocity differ- each other. When camber generated in the roughing mill ence between Drive-Side (DS) and Work-Side (WS) at the remains until the strip is fed into the finishing mill, side- delivery-side is proportional to the wedge ratio. From these slipping increases during passage through the finishing mill. two principles, the delivery-side curvature models were rep- Therefore, camber should be regulated from an early stage resented using an equation that includes entry-side curvature of roughing mill. and wedge ratio. A mathematical model that describes generation of cam- However, the process used to obtain the delivery- ber should calculate the curvature of the centerline of the side curvature model contains an inconsistency. Previous strip at the delivery side because shape of the curve can researchers assumed that the curvature at the delivery-side be reconstructed if the curvatures of all points and initial is affected by both delivery-side velocity difference and entry-side curvature, but, they also assumed that entry-side * Corresponding author: E-mail: [email protected] curvature is affected only by the entry-side velocity differ- DOI: http://dx.doi.org/10.2355/isijinternational.ISIJINT-2015-088 ence; i.e., the cause of the curvature at the entry side differs © 2015 ISIJ 1980 ISIJ International, Vol. 55 (2015), No. 9 from the cause at the delivery side. To achieve consistency, 1 dθθd ω ∆v an assumption that the curved shape of the strip is caused == == . ..................... (3) by the velocity difference between DS and WS is needed. ρ dx vdtvbv In addition, delivery-side velocity difference should be However, ω in Eq. (3) is not the rotation speed of the affected by entry-side curvature. strip but the time derivative of tangential angle dθ/dt which In this paper, a new model for curvature of the centerline is defined as an angle between the tangent vector and the of the strip at the delivery side is proposed. The delivery- x-axis1,2) (Fig. 1); dθ/dt depends only on the shape of the side velocity difference is represented as an equation that strip, and not on its rotation. uses the mass-flow principle to include entry-side velocity Also, the first assumption was based on the idea that the difference and wedge ratio. The relation between curvature rotated angle of the strip at the delivery side θ2 is reduced of the strip and velocity difference was found using the to rotated angle at the entry side θ1 divided by λ. From definitions of curvature, tangential angle, and tangent vector. the definition of curvature (Eq. (3)), they explained that Delivery-side curvature of the centerline is represented as an additional curvature is generated by the rotation of the equation that includes entry-side curvature and wedge ratio. 2 strip scaled by λ because dx2 increases to λx1, whereas dθ2 A three dimensional FEM simulator for rough rolling pro- decreases to dθ2/λ. cess was used to verify the accuracy of the proposed model. The basis of the first assumption was established in the The rest of this paper is organized as follows. The pro- first study4) of side-slipping which represented the time- cess that obtained the old delivery-side curvature model varying location of points on the centerline (Fig. 2). If the is reviewed in Section 2. A new model based on the new strip is regarded as a rigid body, the location of a point on assumption is proposed in Section 3. Brief explanation for its centerline can be represented as a function of time. If the FEM simulator and simulation results are given in Section longitudinal direction of the strip is defined as the x-axis, 4. Finally, a conclusion is represented in Section 5. and the transversal direction of the strip as the y-axis, the velocity of the point v(t) at time t can be represented as 2. The Former Mechanical Model for Curvature at the vt()=⋅ytω ()+ vt1 (), ......................... (4) Delivery Side Shiraishi et al.5) introduced delivery-side curvature model ut()=−xt⋅ω (), .............................(5) that includes entry-side curvature and wedge ratio as where v, u are velocity along the x, and y-axis respectively, 11∆∆v2 v1 11 ξ ∆∆h H v1 is the average speed of the strip at the entry side, and =+ =+− . 2 2 ...... (1) ω is the rotation speed of the strip. Then, time-varying ρ2 bv2 λλbv1 ρ1 b h H location of the point (x1(t), y1(t)) at the entry side can be where 1/ρ2 is delivery-side curvature, λ is reduction ratio represented as h/H, 1/ρ1 is entry-side curvature, b is width of the strip, h t xt10()=+xv1dt is delivery-side strip thickness, H is entry-side strip thick- ∫0 ........................... (6) ness, ξ is a camber-change coefficient and Δh and ΔH are delivery-side and entry-side wedge respectively. To obtain t yt10()=−ytω1 ()()xtdt the model, two assumptions were used: (1) that entry-side ∫0 velocity difference Δv1 influences delivery-side curvature t t t 2 =−fx01()vt11−−xtωω11()dt ()tdtv11dt . 1/ρ2 scaled by 1/λ in addition to the curvature due to 0 ∫∫ 0 ()∫0 delivery-side velocity difference Δv2; and (2) that Δv2 is .......................................... (7) proportional to the wedge ratio Δψ: Side-slipping is defined as the distance between centerline ∆∆v2 h ∆H of the strip and center of the roll (x=0). Consequently, the =−α . .........................(2) model of side-slipping can be obtained as v2 h H According to this model, 1/ρ1 is only determined by Δv1, whereas 1/ρ2 is affected by both the Δv2 and 1/ρ1. Fig. 2. Coordinate system of hot rolling process and cross-section Fig. 1. Definition of tangential angle. of the strip. 1981 © 2015 ISIJ ISIJ International, Vol. 55 (2015), No. 9 tt yfc =−01()vt + vt11ω ()dtdt. .................(8) ∫∫ 00 Similar to Eqs. (6) and (7), the location of the point (x2(t), y2(t)) at the delivery side can be represented as xv22=−()t τ ..............................
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