idle roll drag torque and the zero drag driven roll torque. This also implies that the displacement of the idle roll neutral point from xo/xo = 0.5 toward the entry plane due to drag is accompanied by a similar displacement of the driven roll neutral point from its zero drag location toward the exit plane. The variation of strip curvature with idle roll neutral point position is shown in Fig. 5 . It is to be noted that the vertical coordinate is the logarithm of p/ho. At constant reduction, the

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... strip is seen to curl more rapidly as the drag torque is increased. For 20 percent reduction, the wrapping condition occurs for x2/x0 approximately equal to 0.55. The curvature is also seen to depend on reduction. The idle roll frictional drag thus causes the strip to wrap on the idle roll at lower reductions and further limits applicability of single-roll drive mills. The previous results were all expressed in terms oi" the idle roll neutral point. Now, to illustrate the solution synthesis directly in terms of drag, and the nature of the effects of drag on the rolling parameters, the following example is presented: Initially neglecting the drag torque, the load factor from Fig. 3 is Ql -1.41. Using equation (19) , the idle roll neutral point location is found to be x-i/xo = 0.52S. Again referring to Fig. 3 , the load factor corresponding to this value of x2/x0 is QL = 1.39. Using this value for the load factor, equation (19) indicates that X2/X0 is essentially unchanged. Referring to Figs. 3, 4, and 5, the remaining parameters are readily determined: Ql = 1.39 (1.41) QT = 0.009S (0.00S5) log p/ho = 1.39 (1.50) p/h = 24.5 (31.5) The values in parentheses correspond to the zero drag case. The average roll pressure is seen to be reduced only slightly, while the torque factor has increased by approximately the amount of the drag factor QD as predicted by equation (20) . The drag torque has reduced the strip radius of curvature significantly. The wrapping condition, equation (14) , for this example occurs at p/ho = 20.5. Thus the effect of drag is to increase markedly the tendency of the strip to wrap on the idle roll. To illustrate the performance characteristics of single-roll drive mills with drag, variation of the roll load, torque, and strip curvature parameters with reduction and the idle roll friction drag factor, (3, was studied for a roll radius to exit strip height ratio of 20. The results of this investigation, using only the authors' model, are given in Figs. 6 through S. The load factor variation with reduction is shown in Fig. 6 for /3 values of 0, 0.025, and 0.050. The load factor decreases as the idle roll friction parameter increases for reductions through the wrapping condition. As the reduction increases, the load factor increases at a decreasing rate. The rate of increase of the load coefficient with reduction decreases as @ increases. In Fig. 7 , the variation of the driven roll torque factor for various idle roll friction magnitudes is shown as a function of reduction. The torque is seen to increase with reduction at an increasing rate. For constant reduction, the torque factor increases linearly with (3. The logarithm of the ratio of strip curvature to exit strip thickness, log p/ho, is seen to decrease at a decreasing rate with reduction, Fig. 8 . At constant reduction, as the idle roll friction parameter, /3, increases, the log p/ho decreases. The effect of idle roll drag is to significantly increase the tendency of the strip to wrap around the idle roll. The reduction at which wrapping is predicted decreases from 26.5 percent for j8 = 0 to 21.5 percent for (3 = 0.025 and IS.5 percent for (3 = 0.050. Summary The analysis of hot strip rolling on single-roll drive mills presented here recognizes the unsymmetric nature of the process and accounts for the effect of idle roll drag torque on the roll load, torque, and strip curvature. The transcendental problem introduced by the introduction of idle roll drag torque can be easily solved by an iterative procedure and, for the cases considered, rapid convergence is achieved. The theory of Holbrook and Zorowski [4] , which accounts for the unsymmetric effects, predicts slightly lower roll loads and torques and a larger radius of curvature than the theory of Sachs and Klinger [2] for the case of sticking friction. The study performed indicates that, for the range of parameters considered, the roll load decreases only slightly as the drag torque increases, while the driven roll torque is found to be approximately equal to the sum of the zero drag roll torque and the idle roll drag torque. On the other hand, strip curvature increases significantly with idle roll drag. Consequently, increased idle roll drag restricts even more severely the application of single-roll drive mills to smaller reductions to prevent the strip from wrapping about the idle roll. References 1 E. Siebel, "Zur Theorie des Walzvoganges bei ungleich angetriebenen Walzen," _4re. The ratio of roll radius to material thickness in this paper is representative of a plate-rolling situation. With .ffi//io-ratios of the order of 80 or higher, the curling effect is still a practical nuisance; but it ceases to be a limiting factor in the meaning of Fig. 6 . Numerous cold-rolling experiments with a single-roll drive performed at my laboratory indicate that there is a significant decrease of the total torque as compared with a twospindle operation. This is not withstanding the fact that the coefficient of friction was much lower than in the conditions here assumed. One question I would like to ask is whether the authors would expect a "trench" to appear in the friction mill, like the one shown in reference [3] , under cold-rolling conditions, or would Sachs and Ivlinger's scheme then be more appropriate. In the absence of a wrapping limitation, one may strive to move the boundaries of the "neutralized friction" zone as far apart as possible and thereby achieve a substantial decrease of the roll force. This may have a variety of interesting practical implications. Authors' Closure The authors are grateful for Professor Polakowski's discussion. The "trench" in the friction hill, predicted by reference [3], should not be observed according to the authors' theory under conditions of either hot or cold rolling. In the theory of reference [3], the net internal moment M and the roll pressure difference A are both taken to be zero throughout the roll gap. The net internal shear stress r is taken as k (the material yield strength hi shear) in the cross shear region, and as 2 Professor, Department of Metallurgical Engineering, Illinois Institute of Technology, Chicago, 111. Mem. ASME.