Evolution of Generic Mathematical Models and Algorithms for the Surface Development and Manufacture of Complex Ducts

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Abstract
Complex ducts bound by developable surfaces can be obtained by cutting out the required developed shape on plane sheets and then wedge-bending or folding along certain lines called generators or rulings. The problem then reduces to the theoretical determination of the 2-D shape to be cut so that on folding along the rulings, the required 3-D surface is accurately obtained. The first step in the surface unfolding process is the determination of the parametric β-θ relationship between two adjacent cross-sections of the duct. The cross-sections of the duct can be planar or nonplanar and composed of conic or spline curve segments, placed anywhere in space. In this paper the requisite β-θ relationships of the most generic form have been derived and can be directly applied to any complex duct. Based on these relationships, an efficient and compact algorithm for surface-unfolding has also been derived. The application of this algorithm to numerous prototype cases has been shown. The theory has been verified by physical modelling of various ducts occurring in the field of hydroturbines, and now forms the basis of an Integrated CAD-CAM System.

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