Stability Analysis of Mathematical Model for New Hydraulic Bilateral Rolling Shear

Transcription

1 SJ nternational, Vol. 56 (06), SJ nternational, No. Vol. 56 (06), No., pp Stabilit Analsis of Mathematical Model for New Hdraulic Bilateral Rolling Shear QingXue HUANG, ) Jia L, ) HongZhou L, ) HeYong HAN ) * and LiDong MA ) ) Heav Machiner Eng. Research Center of Education Ministr, Taiuan Univ. of Sci. and Technol., Taiuan, 0004 PR China. ) Hebei Wenfeng ron and Steel Co. Ltd, Hebei, Wuan, PR China. (Received on June 7, 05; accepted on November 5, 05) n this stud, the model of new hdraulic bilateral rolling shear is researched based on kinematics theor of a hbrid spatial linkage mechanism. Combined with research method of -RRR planar parallel mechanisms kinematic, the relationship between clinder displacements and blade positions of the shearing mechanism is developed. Nonlinear sstem stabilit of the shearing mechanism in a new hdraulic bilateral rolling shear is researched based on Krasovskii method, the result shown that the sstem is stable and reliable. Moreover, based on the bilateral rolling shear of a large steel compan, the displacement of clinders is obtained through eperiments. Compared with differential between the theoretical formula and the actual value, the differential is small. This stud has certain instruction to the design and the development of similar product and it also has a higher practical value to the general industrial production. KEY WORDS: spatial linkage mechanism; new hdraulic bilateral rolling shear; mathematic model; Krasovskii method.. ntroduction A bilateral rolling shear for cutting the width of a medium steel plate in a production line while simultaneousl shearing both sides of the plate via the stepping cut method,) is designed as shown in Fig.. A mechanical bilateral rolling shear generall applies to the comple three-shaft tri-eccentric architecture and inclined-angle shear,4) (Fig. ) and their requirement for snchronous cutting control are etremel high. And the equipment maintenance is difficult. As shown in Fig., Horizontal component F acted on the steel plate produced from upper blade increase the friction of the cross-section, which lead to some built-up edges. nclined-angle shear with an additional large travel, which causing serious collapse angle in the cut end of plate as shown in Fig.. However, the arcuate blade has no sideto-side motion in the process of shearing for new hdraulic bilateral rolling shear. n stable generating phase for the new hdraulic rolling shear, the actual effective area of the steel sheet is limited to the shaded ACD, P is the maimum shear force what happened on the area in the process of rollcutting 5) as shown in Fig. 4. Due to the role of the guide rod HG in Fig. 5, the center of gravit of the blade does not swing during cutting process, which ensures good qualit of section. Friction force between the upper blade and plate is reduced so as to prolong the service life. Amount of scissors overlapping is held constant, which can guarantee that the curvature, levelness and the final cut shape of cutting end of the medium heav steel plates are appropriate. The shearing mechanism of the new hdraulic bilateral rolling shear uses servo clinders, which are positioned horizontall on both sides of the frame to drive the rods while the upper shear blade rolling cut the plate. t has a good effect on reinforcement and the minimum swinging angle of clinders. As shown in reference, 6) the larger force which can cut through the steel plate ma be produced b smaller thrust of hdraulic clinder and the thrust forces depend on the load, so the device has a simple structure, a large shearing force, Second cut First cut Fig.. Cutting steps demonstration of the rolling cut of the bilateral shear. * Corresponding author: mkj_lj@6.com DO: Fig.. The structure of hdraulic inclined-angle shear and force diagram at cutting shear. 06 SJ 88

2 SJ nternational, Vol. 56 (06), No. and high control accurac. The new shearing mechanism has more advantage than the conventional in the qualit of the cross section of plate steel. At present, a prototpe of the device has been developed and applied in a production test. Hence, the reliabilit of the shearing mechanism should be further studied. Nowadas, mathematical model on spatial shear mechanism optimization of steel rolling shear is established and applied for some large ron & Steel plant. 7) Based on the establishment of composite linkage position loop equation, moment balance equation and hdraulic control sstem model is calculated. 8) Above literatures studied bilateral rolling shear onl based on the simulation test and verif, but there is no in-depth research in theor for the shearing mechanism. The motion of mechanism of new hdraulic bilateral rolling shear is comple and difficult, which bring a great deal of difficult in researching the analsis theor of mechanism dnamic. Different from the traditional geometric construction, the motion wa of different institutions of shear mechanism need to be acted as the given track. The upper blade of the hdraulic bilateral rolling shear ehibits rolling shear movement, which is comple. The displacement of the upper blade cannot be directl measured, and there is no a set of accurate calculation method about the position and posture of the upper blade. Moreover, analzing the mechanism motion is attracted attention b so man scholars in different periods. Approimate integrated approach developed in recent ears is rising, a comprehensive method used is diversit, but the comprehensive theoretical stud has not been perfect, there are still some limitations on the method. Based on kinematics theor of a hbrid spatial linkage mechanism, combined with research method of -RRR planar parallel mechanisms kinematic, the mathematical model between the displacements of four hdraulic servo clinders and the position of the upper blade is developed and the stabilit of shearing mechanism is put forward based on the Krasovskii method. Moreover, it provides a theoretical basis for the nonlinear coupling control of nonlinear multi-dof sstems and new theories and methods is developed for the kinematics and dnamic analsis of the mechanism. Fig.. Collapsed angle default b inclined-angle shear.. Building the Mathematical Model of the Shearing Mechanism of the Bilateral Rolling Shear The new hdraulic bilateral rolling shear mainl consists of fied and mobile scissors (Fig. 5). The beginning and end of the cutting stage is driven onl b a single clinder, in which the movement of the upper blade is simple. This stud aims to analze the action of two hdraulic servo clinders during the shearing process and to establish the relationship between the position of the upper blade and the displacement of the hdraulic clinders. The mobile shearing mechanism is fied with an appropriate coordinate sstem,, and z, which is assumed to be a right-handed Cartesian space with an origin point that is coincident with the position of A in the frame (shown in Fig. 6). The angle between the connecting rod AE and the ais is designed as θ. The angle between the connecting rod BF and the ais is designed as θ. The angle between Fig. 4. Sketch map of shear mechanics model. Fig. 5. New hdraulic bilateral rolling shear. Fig. 6. Agenc coordinates of the mobile mechanism SJ

3 SJ nternational, Vol. 56 (06), No. the guide rod HG and the ais is designed as θ. The counterclockwise direction is specified as the positive direction. The translational coordinate sstem is inclined at angle φ toward the direction of the absolute coordinate sstem. The coordinates of the hinge point is designed as A(a, a ), B(b, b ), C(c, c ), D(d, d ), E(e, e ), F(f, f ), G(g, g ), and H(h, h ). The hinge points of the upper blade attachment form a triangle with the following side lengths: l cd = 00 mm, l dg =845 mm, and l cg = 048 mm. The coordinate transformation matri 9 ) T is presented as cosϕ sinϕ sinϕ cosϕ... () 0 0 Where,, and φ indicate the position of the moving coordinate sstem O based on the static coordinate sstem O. The coordinates of C, D and G can be epressed as C D G T C D G =... () The coordinates of all the joints of the upper blade based on the moving coordinate sstem O are C ( 5, 5), D (75, 665), G ( 5, 5) and H(4 5, 465). Then, the transformed matri is presented as C D G cosϕ sinϕ C D G = sinϕ cosϕ () The starting angle between the moving coordinate sstem and the static coordinate sstem is φ=6.7. Then, the position coordinates of E and F are given as E = C + r cos α, E = C + r sinα... (4) F = D + rcos β, F = D + rsin β... (5) Where r = l ce = l df =70 mm, α, β are an angle of trigonometric function. The displacements of E obtained from Eq. (4) in the horizontal and vertical directions are respectivel given as E = 5cosϕ+ 5sin ϕ+ 55,... (6) E = 5sinϕ+ 5cosϕ 469 Where Δ E and Δ E are the horizontal and vertical displacements, respectivel, of # clinder. The displacements of F obtained from Eq. (5) in the horizontal and vertical directions are respectivel given as F = 75cosϕ+ 665sin ϕ 04,... (7) F = 75cosϕ+ 665sinϕ 565 Where Δ F and Δ F are the horizontal and vertical displacements, respectivel, of # clinder. The displacements of G obtained from Eq. () in the horizontal and vertical directions are respectivel given as G G = 5cosϕ+ 5sin ϕ 768,... (8) = 5sinϕ+ 5cosϕ Where ΔG and ΔG are the horizontal and vertical displacements of point G, respectivel. n addition, E, F, and G ehibit a circular motion at the centers of A, B, and H, respectivel. The coordinates of E, F, and G are respectivel given as E = r cos θ, E = r sinθ... (9) F = r cos θ + 00, F = r sinθ... (0) G = rcos θ + 4 5, G = rsinθ () Where r =l gh = 60 mm, r =l ea =l bf =70 mm, θ = 08, θ = 65, θ =9.7, and φ=6.7. Where the displacements of E, F, and G obtained from Eqs. (9), (0), and () in the horizontal and vertical directions are respectivel given as E F G = 70cos θ + 7, E = 70sinθ () = 70cos θ 97, F = 70sinθ () = 60cos θ + 4, G = 60sinθ +... (4) Where Δ E and Δ E are the horizontal and vertical displacements, respectivel, of # clinder; Δ F and Δ F are the horizontal and vertical displacements, respectivel, of # clinder; ΔG and ΔG are the horizontal and vertical displacements, respectivel, of point G. n the preceding equations, E = E = E, E = E = E. Solve the equation group as follows to obtain the φ value: θ ϕ = arcsin cos (5) 95. The upper blade is rigid, in which the movement of an point are the same during the rolling-cut process. Thus, hinged point G is used to analze the motion of the upper blade. The relationship between clinder displacement and the position of the upper blade can be epressed as = E + E = 70 ( + sin( θ + 8 ))... (6) = F + F = 70 ( + sin( θ 5 ))... (7) S S = sinϕ (8) 0. cosϕ Where S and S are the horizontal and vertical displacements of upper blade for mobile scissors, respectivel; and and are the displacements of # and # hdraulic clinders for the mobile scissors, respectivel. The relationship between the displacement of the clinders and the position of the upper blade for mobile scissors is described in Eq. (8). The bilateral rolling shear consists 06 SJ 90

4 SJ nternational, Vol. 56 (06), No. of mobile and fied scissors with snchronous kinematic velocities, # hdraulic clinder whose movement rule is the same as that of # servo-clinder, and 4# hdraulic clinder whose movement rule is same as that of # servoclinder, which can be epressed as S S = sinϕ (9) cosϕ Where S and S are the horizontal and vertical displacements of upper blade for fied scissors, respectivel; and and 4 are the displacements of # and 4# hdraulic clinders for the fied scissors, respectivel. According to the ideal cutting path, theoretical displacement of clinders are developed b using the mathematical model established in this paper. So the pose of upper blade is automaticall adjustable on line and the kinematics problem of the sstem is solved, which contribute to a deep understanding of control problems. What s more, to decrease horizontal displacement of blade arc and to increase evenness degree of blade overlapping is an important guarantee for the production of high qualit steel plate shear section, so the orbit of the upper blade need to be adjusted to meet steel plates with different thickness.. Analzing the Sstem Stabilit of the New Hdraulic Bilateral Rolling Shear Based on Eq. (8), the horizontal and vertical displacements of upper blade can be epressed as S = ( ) (0) S = ( ) Take the derivative of both sides of Eq. (0) with respect to, components S = ( ) () S = ( ) Nonlinear state equation of sstem 4) are formulated f( ) = () The equilibrium state of the sstem known as in the initial position, namel, f(0)=0. The Jacobi matri of sstem is f f( ) F ( ) = = Τ f f... () f Sufficient condition for asmptoticall stable of the sstem under equilibrium state is that F ˆ( ) = F Τ ( ) + F ( ) is negative definite matri under an value of. 5) n order to verif the conditions, detail derivation process as follows. 9.( 6 ) F ˆ( + ) = ( + ) 65.( 8 + )... (4) 9( + ) 65.( 8 + ) 97.( + ) Based on Slvester criterion, 6) then According to, 7) = 9.( 6 + ) < 0... (5) = F ˆ( ) < 0... (6) = ( k B + APL F)... (7) mt Substituting Eq. (7) into Eq. (5) is as below. + = + ( k B + AP F)... (8) m t n order to enjo the constraint condition (5) ( ) < t B B + 4kmt AP L F m.. (9) 4mt mt Where, B=00 N s/m, K=0 04 N/mm, m t =950 kg, A =60 mm, F=5. 06 N/mm. So we can obtain as below (0) < < When t = 0.5 s, =.6 mm, f( ) min = and f ( ) is a monotone increasing function when 0< <88.48 mm. Speed of clinder meets the demands: 50 mm/s < <50 mm/s, obviousl, Eq. (9) is correct, namel Δ <0. a = + < 0, b = +... () Substituting Eq. () into Eq. (4) is as below. 9. 6a 9a 658. b F ˆ( ) = a. b. b... () According to Eq. (6), the equation = F ˆ( ) < 0 is right under the hpothesis. So, ˆ( ).. (. ) F = 9 6a 97 b 9a 658b = ( 9a 0. b)( b 9a) Where, a<0 and b<0, according to Eq. (6), 0.b <9a< 58.5b Taking the integral of both sides of equation with respect to is as below. 0 bd < ad < 6. 6 bd... () Substituting Eq. () into Eq. () is as below.. 0 b< < 6. 6b... (4) Taking the integral of both sides of Eq. (4) with respect to is as below.. 0 < < (5) L 9 06 SJ

5 SJ nternational, Vol. 56 (06), No. During the process of shearing plate, Eq. (5) meet the condition that = F ˆ( ) < 0. So, F ˆ( ) is negative definite and the equilibrium state e =0 for the sstem is asmptoticall stable. n addition, when, we can get [ ] f Τ ( ) f( ) = = (.. ) + ( ) So the sstem is global asmptotic snchronization at a state of equilibrium e =0. The same procedure is easil adapted to obtain behavior models for the Eqs. (8) and (9). The proving procedures are skipped. The results suggest that the scheme meet the conditions of stabilit. 4. Eperiment Research n this stud, the relationship between clinder displacements and blade positions of the shearing mechanism is developed, which can make arcuate upper blade do movement of rolling shear. f mathematical model is not built as this article, then the shearing mechanism would be out of control, which leads to the problems that steel plate can not be cut off. Some defect such as rip can be found in the part of the plate as shown in Fig. 7. n this paper, the displacement of clinder is given in ever moment and closed-loop hdraulic servo sstem is applied to adjust the actual displacement of clinder in real time. Then the trajector of Fig. 8. Fig. 7. Fracture surface. Picture of the actual hdraulic bilateral shear. upper blade can keep the track of rolling shear trajector, which can achieve an ideal motion and satisf the shearing qualit of steel plate. 4.. Eperimental Procedures The new hdraulic bilateral rolling shear which is designed b the Taiuan Universit of Science and Technolog and produced b the Taiuan Heav Machiner Group, as shown in Fig. 8. A given numerical of the theoretical calculation need to be verified through the eperiment, displacement is detected b the displacement sensor, which are both installed on the servo clinder. The displacement of the upper blade cannot be directl measured, and thus, the displacement data of the clinder are gathered. 4.. Eperimental Results and Discussion The error between the actual and given displacements is analzed b collecting the actual displacements of the four hdraulic servo clinders. Comparing the actual and given displacements in the field. Firstl, # and # clinders move at a specific speed; then, when it arrives at a certain point, # and 4# right clinders begin to move and enter the rollingcut stage to shear the plates. When # and # clinders reach maimum displacement, i.e. (88.7 mm), it returns while # and 4# clinders continue to etend (4.8 mm is reached) until shearing is complete. Finall, the rods of the four clinders return to their original positions. As shown in Fig. 9(a), the error of # clinder between the actual and given displacements is 0.55% to 5.%, that of # clinder is 0.08% to.% (Fig. 9(b)), that of # clinder is 0.% to.% (Fig. 9(c)), and that of 4# clinder is 0.4% to 5.8% (Fig. 9(d)). There is a gradual pressure for the upper blade in the process of cutting plate, so the error should be eist. The error is related to the thickness, the width and different materials of plate and so on. Therefore, this percentage can not be directl used as a standard to measure the precision of cutting mechanism. And shear qualit of the plate is most relevant to the shear forces generated from the upper blade. According to the eperience of the real eperiment, the maimum shear force is.5 times then the theoretical value, the roll cutting angle range is. ) According to reference, 5) cutting angle can be obtained, the maimum displacement error of hdraulic clinder derived is.4 mm. After analzing test data collected in this paper, the maimum displacement error of hdraulic clinder is not more than mm. So the precision of the cutting mechanism can be improved b controlling the displacement of the hdraulic clinder, then the qualit of steel plate can be improved. Both theoretical and field data indicate that the hdraulic servo control sstem can satisf the shearing track characteristics of the hdraulic rolling shear. The shearing mechanism does the movement as ideal trajector and a strong shear force induced from clinders, which will achieve the prediction accurac. According to the ideal qualit of the cross section, the required trajector can be derived. As the mathematical model of the mechanism built in this paper, displacement of clinders can be adjusted in real time, the upper blade can do the ideal rolling shear movement, which achieves the required cutting position accurac and the qualit of the cross section of plate steel as shown in Fig SJ 9

6 SJ nternational, Vol. 56 (06), No. Fig. 9. The actual and given displacements of clinders. Acknowledgements This project is supported b National Natural Science Foundation of China (Grant No , 5755, ), Provincial Fund for Young Scholars (0009-), the initial funding of doctor research of Taiuan Universit of Science and Technolog (0047). REFERENCES 5. Conclusions Fig. 0. The cross section of plate steel. () n order to establish the mathematic model of new hdraulic bilateral rolling shear, the relationship of position and attitude of shearing mechanism is developed. Then the entire motion of upper blade can be epressed in the base coordinate sstem. () n order to describe the various motions of upper blade, DOF of the mechanism is configured: the level movement toward the ais direction, the vertical movement toward the ais direction, and the rotation toward the z ais direction, and the general movement requirements are satisfied. () n this stud, the model of the mechanism is established based on the kinematics and inverse kinematics analsis. Moreover, b controlling the displacement of the hdraulic clinder to adjust the position and attitude of the upper blade in real time, the stabilit of sstem and the qualit of plant surface can be improved. ) Z. B. Chu, Q. X. Huang and L. F. Ma: J. Sichuan Univ.: Eng. Sci. Ed., 4 (0), 47. ) Y. G. Bondar, Y. N. Belobrov and A. A. Kalashnikov: Metallugist, 49 (005), 456. ) Q. X. Huang: Design of Rolling Mills, Metallurg ndustr Press, Beijing, (007),, 6. 4) M. Murakawa and Y. Lu: J. Mater. Process. Technol., 66 (997),. 5) R. Q. Zhang: The Research on mproving Thick Plates Shear Qualit for Double Side Rolling Trimmer, Yanshan Universit, HeBei, (008), 4. 6) H. Y. Han, Q. X. Huang, L. F. Ma and J. Wang: J. Sichuan Univ.: Eng. Sci. Ed., 4 (0), 9. 7) L. F. Ma, Q. X. Huang, Y. Li, J. M. Wang and X. G. Wang: J. Sichuan Univ.: Eng. Sci. Ed., 40 (008), 70. 8) L. F. Ma, Q. X. Huang, H. Y. Han, Z. B. Chu and Z. G. Li: J. Beijing. Univ. Technol., 8 (0), 60. 9) L. F. Jiang, T. Lan, Y. C. Zhang and G. Q. Ni: Proc. of SPE-The nt. Societ for Optical Engineering, OT 009, China, (009), 75. 0) S. W. Lee, K. S. Kwon and. C. Park: EEE T. Circuit.-, 54 (007), 680. ) X. G. Ding: Research on Robotic Control, Zhejiang Universit Press, Hangzhou, (006), 9. ) Y. Wei: Liner Algebra Appl., 466 (05), 0. ) E. Defez, L. Jódar and A. Law: Compt. Math. Appl., 48 (004), ) E. Fridman: Sst. Control. Lett., 4 (00), 09. 5) K. M. Xie: Modern Control Theor, Tsinghua Universit Press, Beijing, (0), 8. 6) N. Nakashima: Algebr. Represent Th., 7 (04), 6. 7) C. X. Wang: Hdraulic Control Sstem, China Machine Press, Beijing, (0), SJ