三辊卷板机上辊位置的分析建模及实验验证外文翻译.doc

2大学本科生毕业设计外文翻译Analytical and empirical modeling of top roller position for three-roller cylindrical bending of plates and its experimental verificationA.H. Gandhia, H.K. Raval bAbstract：Reported work proposes an analytical and empirical model to estimate the top roller position explicitly as a function of desired (final) radius of curvature for three-roller cylindrical bending of plates, considering the contact point shift at the bottom roller plate interfaces. Effect of initial strain and change of material properties during deformation is neglected. Top roller positions for loaded radius of curvature are plotted for a certain set of data for center distance between bottom rollers and bottom roller radius. Applying the method of least square and method of differential correction to the generated data, a unified correlation is developed for the top roller position, which in turn is verified with the experiments, on a pyramid type three-roller plate-bending machine. Uncertainty analysis of the empirical correlation is reported using the McClintocks method.Keywords: Roller bending Spring back Analytical study Empirical modelingUncertainty analysis1. IntroductionLarge and medium size tubes and tubular sections are extensively in use in many engineering applications such as the skeleton of oil and gas rigs, the construction of tunnels and commercial and industrial buildings (Hua et al., 1999). In view of the crucial importance of the bending process, it is rather surprising to find that roller-bending process in the field has been performed in a very nonsymmetrical manner. Normal practice of the roller bending still heavily depends upon the experience and skill of the operator. Working with the templates, or by trial and error, remains a common practice in the industry. The most economical and efficient way to produce the cylinders is to roll the plate through the roll in a single pass, for which the plate roller forming machine should be equipped with certain features and material-handling devices, as well as a CNC that can handle the entire production process (Kajrup and Flamholz, 2003).Many times most of the plate bending manufacturers experience Low productivity due to under utilization of their available equipment. The repeatability and accuracy required to use the one-pass production method has always been a challenging task。Fig. 1 shows the schematic diagram of three-roller bending process, which aimed at producing cylindrical shells. The plate fed by two side rollers and bends to a desired curvature by adjusting the position of center top roller in one or several passes. Distance between bottom rollers can be varied. During deformation, axes of all the three rollers are set parallel to each other. Desired curvature in this case is the function of plate thickness (t), plate width (w), material properties (E, n, K, and v),center distance between two bottom rollers (a), top-roller position(U), top-roller radius (rt) and bottom-roller radius (r1)(Raval, 2002). The capacity of the plate bender is defined by the parameters such as tightest bend radius with the maximum span and designed thickness of the plate and the amount of straight portion retained at the end portions of the plate.Fig. 1 Schematic diagram of Fig2 Deformation in fiber AB0three-roller bending process. 2. Bending analysis Bending analysis is based on some of the basic assumptions summarized below:The material is homogeneous and has a stable microstructure throughout the deformation process. Deformation occurs under isothermal conditions. Plane strain conditions prevail. The neutral axis lies in the mid-plane of the sheet. Bauschinger effect is neglected. Analysis is based on power law material model, Pre-strain is neglected. Change of material properties during deformation is neglected. Plate is with the uniform radius of curvature for supported length between bottom rollers.2.1. Geometry of bendingIn thin sheets, normal section may be considered to remain plane on bending and to converge on the center of curvature (Marciniak and Duncan, 1992). It is also considered that the principal direction of forces and strain coincide with the radial and circumferential direction so that there is no shear in the radial plane and gradient of stress and strain are zero in circumferential direction. The middle surface however may extend. Fibers away from the middle surface are deformed as shown in Fig. 2. Initially the length of the fiber AB0 is assumed as l0 in the flat sheet. Then, under the action of simultaneous bending and stretching the axial strain of the fiber AB0 is of the form (1)where a is the strain associated with the extension of middle surface, b the bending strain and is the radius of curvature of the neutral surface.2.2. Moment per unit width for bending without tensionIn the case of simple bending without applied tension and where the radius of curvature is more than several times the sheet thickness, the neutral surface approximately coincides with the middle surface. If the general stressstrain curve for the material takes the form (2)Then, for the plastic bending, applied moment per unit width can be of the form (Marciniak and Duncan, 1992)2.3 Elastic spring back in plates formed by bendingIn practice, plates are often cold formed. Due to spring back, the radius through which the plate is actually bent must be smaller than the required radius. The amount of spring back depends up on several variables as follows (Raval, 2002;Sidebottom and Gebhardt, 1979): Ratio of the radius of curvature to thickness of plates, i.e. bend ratio. Modulus of elasticity of the material. Shape of true stress versus true strain diagram of the material for loading under tension and compression. Shape of the stressstrain diagram for unloading and reloading under tension and compression, i.e. the influence of the Bouschinger effect. Magnitude of residual stresses and their distribution in the plate before loading. Yield stress (y). Bottom roller radius, top roller radius and center distance between bottom rollers. Bending history (single pass or multiple pass bending, initial strain due to bending during previous pass).Assuming linear elastic recovery law and plane strain condition (Marciniak and Duncan, 1992; Hosford and Caddell, 1993), for unit width of the plate, relation between loaded radius of curvature (R) and desired radius of curvature (Rf) can be given by3. Analytical models of top roller position (U) for desired radius of curvature (Rf)For the desired radius of curvature (Rf), value of loaded radius of curvature (R) can be calculated using the Eq. (4). From the calculated value of loaded radius of curvature (R), top roller position (U) can be obtained using the concepts described below.3.1. Concept 1Application of load by lowering the top roller will result in the inward shift of contact point at the bottom roller plate interface (towards the axis of the central roller). Fig. 1 shows that distance between plate and bottom roller contact point reduces to a from a. Raval (2002) reported that for the larger loaded radius of curvature (R), top roller position (U) is very small, and hence, contact point shift at the bottom roller plate interface can be neglected for simplification (i.e. aa_). Fig. 3 shows the bend plate with uniform radius of curvature (R) between roller plate interfaces X and Y, in the loaded con- dition. As top roller position (U) is small for the larger loaded radius of curvature(R), in triangle OYX, segment YX can be assumed to be equal to half the center distance between bottom rollers (i.e. a/2).fig3 Bend plate in loadedcondition without considering Fig. 4 Bend plate in loaded contact point shift conditionSo, from triangle OYX in Fig. 3(a/2)2 + (R U)2 = R2Simplification of the above equation will result in the form AU2 BU + C = 0 (5)where A= (4/R), B = 8 and C = a2/R.From Eq.(5), top roller position (U) can be obtained for the loaded radius of curvature（R）,calculated from desired radius of curvature(Rf).3.2. Concept 2Concept 1 discussed above, neglects the contact point shift at the bottom rollers plate interfaces, whereas concept 2 suggests the method for the approximation of these contact point shift for the particular top roller position (U). It was assumed that the plate spring back after its exit from the exit side bottom roller and hence between the roller plate interfaces, plate is assumed to be with the uniform radius of curvature. Then, for the larger loaded radius of curvature (R), length of arc (s) between the points LH in Fig. 4 is assumed to be equal to LH(i.e. a/2). In order to obtain contact point shift at bottom roller plate interface, portion of the plate in between the bottom rollers plate interfaces is divided into total N number of small segments defining the nodal points s1, s2, . . ., sN1 at each segment intersection as shown in Fig. 4. Each small segment of the arc s, i.e. Ls1, s1s2. . . sN1 H being the arc length d(s)equal to (a/2)/N) is considered as a straight line at an angle of (/N), (2/N), . . ., , respectively with the horizontal. Incremental x and y co-ordinates at each nodal point are calculated using the relationship (Gandhi and Raval, 2006):where for total N number of segment (i.e. i=1, 2, . . .,N)Then, from the summation of x co-ordinates and y co-ordinates of all the nodal points, top roller position (U) for the particular value of loaded radius of curvature (R) can be obtained in two different ways as follows.In Fig. 4, considering the GHO (6)In Fig. 4, considering the HOLX2 + R (U JG)2 = R2 (7)This can be derived to the formU2 2(R + JG)U + X2 + (R + JG)2 R2 = 0 (8)The contact point shift between the plate and bottom rollers are obtained by (9)3.3. Concept 3Fig. 5 shows the loaded plate geometry assuming constant loaded radius of curvature (R) between the bottom roller plate interfaces with top roller position (U) and center distance between bottom rollers (a). Relationship of top roller position (U)with other operating parameters viz loaded radius of curvature (R), center distance between bottom rollers (a) and bottom roller radius (r1) considering actual contact point shift can be obtained as discussed below.Fig. 5 Geometry of three-roller bending process.From the OPQ in Fig. 5OQ2 (OQ U)2 = PQ2where OQ= R + r1, PQ=a/2 and OQ=OQ=OP+UExpanding and rearranging, this can be derived to the formReplacing R from Eq. (10) into Eq. (4) and simplifying, (11)whereG = 4U2 8r1U + a2Eq. (11) represents the top roller position (U) as a function of final radius of curvature (Rf). From Eq. (11), it can be observed that top roller position (U) is the function of Bottom roller radius (r1) Center distance between bottom rollers (a). Material property parameters (E, v K, and n). Thickness of plate (t). Final radius of curvature (Rf).Assumption of constant radius of curvature between the roller plate interfaces and plane strain condition has eliminated the effect of top roller radius (rt) and width of the plate (b).4. Development of empirical modelAs described earlier, top roller position (U) is the function of loaded radius of curvature (R), center distance between bottom rollers (a), radius of the bottom rollers (r1) and radius of the top roller (rt). Further, loaded radius of curvature (R) can be calculated from the desired final radius of curvature (Rf) considering the spring back. To develop the empirical model, data set were generated from the geometry for the required top roller position (U) in order to obtain the particular value of loaded radius of curvature (R), with a set of values of center distance between bottom rollers and bottom roller radius. Effect of top roller radius (rt) on top roller position (U) was neglected with the assumption of no contact point shift at the top roller plate interface (i.e. uniform radius of the supported plate length). Fig. 6 shows the plot of U versus R for the data set for three different bottom roller radiuses (r1) i.e. 95, 90 and 81.5mm. These data sets were generated with top roller radius (rt) as 105mm, for range of loaded radius of curvature (R) from 1400 to 3800mm; center distance between bottom rollers (a) from 375 to 470mm and bottom roller radius (r1) from 81.5 to 105mm. From these data, correlation for top roller position (U) was derived which is described as follows.Fig. 6 U vs. R for different bottom roller radius (r1) and center distances between bottom rollers (a), rt = 105mm.From the study of the U versus R plots for the particular machine (with top roller radius (rt) equal to 105mm and bottom roller radius (r1) equal to 81.5 mm), a functional relationship of the form given by Eq. (12) can be assumed.U = cRm (12)Constants (c) and (m) were evaluated using method of least square. For the different center distance between bottom rollers (a) i.e. 375, 390, 405, 425, 440, 455 and 470mm, values of constants (c) and (m) were found to be different. Hence, variation of constant (c) and (m) were plotted against center distance (a) as shown in Figs. 7 and 8. The top roller position (U) is derived with new constant (c1) and (m1).，U = (c1)R(m1) (13)Fig. 7 Constant c for different center distance between bottom rollers (a), rt = 105mm.where constants c1 and m1 were obtained as a function of center distance between bottom rollers (a). Similarly, from the U versus R plots for the other machines with top roller radius equal to 105mmand bottom roller radius equal to 90, 95, 100 and 105mm, the empirical equation for top roller position (U) was derived in the form given by Eq. (13).Where, constants (c1) and (m1) were obtained as a function of center distance between bottom rollers (a), for the different machines and are presented in Table 1.c1 = PaQ, m1 = (0.00002)a SSo, unified empirical equation considering all different machines can be obtained as belowU = P(a)Q(R)0.00002(a)S (14)where P, Q and S are constants, which depend on bottom roller radius (r1). From the P versus r1, Q versus r1 and S versus r1 plots, constants P, Q and S were obtained by applying the generalized method of least square and method of differ- ential corrections (Devis, 1962) to the generated dataset, as a function of bottom roller radius (r1) given by Eqs. (15)(17)Fig. 8 Constant m for different center distance between bottom rollers (a), rt = 105mm.sP = 0.198r10.2582 (15)Q = 2.1199r10.0057 (16)S = 1.133r10.0373 (17)Replacing P, Q and S from Eqs. (15)(17) into Eq. (14)U = (0.198r10.2582)a2.1199r10.0057R(0.00002)a(1.133r10.0373) (18)Assuming the unit width of the plate and plane strain condition, from Eqs. (3) and (4)Replacing R from Eq. (19) into Eq. (18) (20)Eq. (20) is the empirical equation for top roller position (U) considering contact point shift, where U is the function of Bottom roller radius (r1). Center distance between bottom rollers (a). Material property parameters (E, V K, and n). Thickness of plate (t). Final radius of curvature (Rf).Eq. (20) is the generalized equation of top roller position (U) as its derivation is based on the trend equations and it is applicable to any range of the parameters under consideration. Further from Table 1, for the range of bottom roller radius (r1) from 81.5 to 105mm, range of variation of P, Q and S was observed to be 0.06360.0593, 2.06732.0651 and 0.96310.952, respectively. By averaging the P, Q and S, top roller position (U) can be derived to the form given by Eq. (21), which neglects the effect of bottom roller radius (r1) and is applicable to the machine with the range of the bottom roller radius from 81.5 to 105mm. （21）5 Uncertainty analysisThe uncertainty analysis is carried out in accordance with the McClintocks method with the following assumed uncertainties in the various parameters: Uncertainty in strain hardening exponent (n) =10%. Uncertainty in strength coefficient (K, N/mm2) =15%. Uncertainty in thickness of plate =0.29mm (5mmt 8 mm), 0.32mm (8mmt10mm), 0.35mm (10mmt12mm) and 0.39mm (12mmt15mm). Uncertainty in center distance between bottom rollers (a) =1mm. Uncertainty in loaded radius (R, mm)=1%.6 ConclusionDeveloped analytical and empirical models were verified with the experiments on three-roller cylindrical bending. Following important conclusions were derived out of the reported work:(1) Analytical model based on concept 3, simplifies the calculation procedure for the machine-setting parameters as it expresses the top roller position as an explicit function of desired radius of curvature.(2) Agreement of empirical results with that of the experiments and analytical results based on concept 3 proves the correctness of the procedure.(3) For the small to medium scale fabricators, where the volume of production does not permit the acquisition of automated close loop control systems, developed models can be proved to be simple tool for the first hand estimation of machine setting parameters for required product dimension