SUMMARYA long established approximation, normally invoked in the development of classical rolling theory, is that the work roll-strip interface assumes a circular profile. Although valid for a wide range of practical rolling conditions, the approximation is definitely suspect for instance in the last and penultimate stand of a thin sheet tandem mill, and algorithms based on classical theory may fail to converge. This paper formulates the analysis of non-circular arc rolling conditions, and describes a new iterative procedure for computing solutions. In deriving the algorithm it is convenient to employ the formalism of functional analysis. The algorithm is of the 'hill climbing' variety, involving the calculation of suitable gradients.One notable feature is that the gradients have to be carefully defined with respect to the correct 'space', in order that physically inadmissible discontinuities are not introduced during the hill-climbing.Computed roll gap solutions sometimes display elastic deformation regions embedded within the normal plastic deformation zone. It is thought that this is the first time this possibility has been allowed for in the formulation, or solutions obtained exhibiting this feature.
Impulses in gradient functions are shown to arise when the systems contain impulses in their impulse response. Methods are described for obtaining search directions which are continuous functions and which can be used in gradient minimization algorithms.
In this paper we investigate the closure of certain filters under different definitions of diagonal intersection. The space of partitions over which filters concern us is Qκ(λ), the set of partitions of λ into fewer than κ pieces, invented by Henle and Zwicker [4] in the spirit of Pκ(λ). Various notions of normality for filters over Qκ(λ) have been introduced in [4] and [7]. Our objective is to find a notion of normality in terms of a tractable diagonal intersection which also in some sense reflects the construction of a partition. Extending the parallels between Pκ(λ) and Qκ(λ) we define two diagonal intersections, Δ1 and Δ2, under which the club filter is closed.
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