Pyramid Hopper: flat pattern

A four-sided pyramid between two rectangles, such as a hopper or a plenum, with or without flanges turned outward. A four-sided pyramid frustum between two rectangles, such as a hopper or a plenum, with or without a flange turned outward at each end.

Configurations

• Without flanges. Four flat trapezoidal sides between the two rectangles.
• With flanges. Each side carries a flange turned outward at End A, End B or both, with mitred corners.

A worked example

The without flanges configuration with the sizes SheetMorph opens with, in mm. 4 flat sides: 4 parts, 4 seams.

How it is developed

True-shape development of flat sides. This is an exact development: the surface unfolds without stretching, so the pattern closes exactly.

• Each side is a flat trapezoid whose height is the slant height of that side, so the four plates are cut to their true shape.
• Each side is a separate plate. The four corner edges of a duct are joints between plates, drawn as plain edges without allowance.

Checked before it can be exported

Every pattern SheetMorph draws is checked against the 3D part it comes from. For this example:

• Every station is a rectangle (passed): 2 stations: opposite sides and the two diagonals of each compared (max deviation 0).
• Plates flat, rolled or triangulated (passed): 4 flat, 0 folded or rolled from flat pieces (both develop exactly), 0 twisted and split along a diagonal.
• Surface connectivity (passed): 8 facets form one closed duct: every edge inside a part joins exactly two facets, every open edge one (V - E + F = 0).
• Developed edges equal true 3D lengths (passed): 24 facet edges compared between the 3D surface and the flat pattern (max deviation 2.84e-13).
• Shared edges and seam closure (passed): 4 internal edges are shared vertex-for-vertex between adjacent facets; 0 seam(s) have mating edges of equal true length (max deviation 0).
• Developed facets consistently oriented (passed): All developed facets are counter-clockwise (outside face up); none is mirrored or degenerate.
• Developed area equals surface area (passed): Facet area in 3D and in the flat pattern agree (difference 5.82e-10).
• Cut outlines closed and non-overlapping (passed): 4 closed outline(s); no outline crosses itself or another outline.

Assumptions

• The pattern is developed on the mid-thickness surface; inside or outside sizes are converted to it by one thickness across each opening.
• Folds are sharp lines; no bend allowance is applied.
• No seam, lap or weld allowance is added.

Limits of this module

• Flanges are flat and turned square to the axis, not square to the sloping side; bolt holes are not drawn.
• Both ends are level; an inclined end is made with the rectangle-to-rectangle transition.
• No seam, lap, lock-seam, flange or weld allowance is added unless the module says so; corner joints are drawn as plain edges.
• Bend allowance and springback are not modelled; folds are sharp lines on the mid-thickness surface.
• No stiffening, tie rods, access doors or nesting onto stock sheets.

Conventions

Coordinate system. End A is at Z = 0, centred on the Z axis, with its width along X and its depth along Y. A duct that turns or is offset does so in the XZ plane, toward +X.

Corners and sides. Corners are numbered 0 (-X, -Y), 1 (+X, -Y), 2 (+X, +Y), 3 (-X, +Y), seen from +Z at End A. Rj_c is corner c of station j. Each of the four sides is its own plate; the plates are joined along the four corner edges.

Plates. Plates are drawn outside face up and stacked in the order of their numbers. The blank sizes listed are the bounding rectangle of each plate.

Material thickness. Say whether the sizes are inside, mid-thickness or outside sizes. The pattern is developed on the mid-thickness surface, one thickness larger or smaller across each opening; the views show the sizes as entered.

Hopper. End A is at Z = 0 and End B at Z = height, its centre moved by the offsets. A flange is turned outward, square to the Z axis, by the width entered; its corners are cut on the mitre so that the flanges of two sides meet.

Change the sizes to your own in SheetMorph, check the drawing, add seam and flange allowances, and export DXF, PDF, SVG, CSV or G-code. Check every pattern against your own standards before cutting.

Other transitions

• Rectangle to Rectangle: Hoppers and duct transitions between two rectangular ends, centred, offset or inclined.
• Rectangle to Round: Square-to-round transitions with flat sides and conical corners, offset or inclined.
• Round to Oval: A round end to a flat-oval end: flat triangles under the straight sides, curved parts round the ends.
• Rectangle to Oval: A rectangular end to a flat-oval end: flat panels on the long sides and parts of cones at the corners.
• Rectangle to Ellipse: A rectangular end to an elliptical end, or an ellipse to a rectangle: concentric, offset, eccentric or with angled faces.
• Polygon: A regular polygon, such as a hexagon or an octagon, to a circle or to another polygon.
• Twisted Rectangle: A transition between two rectangles, one of them turned about the axis; its twisted sides are split into flat triangles.