Cone onto a Cone: flat pattern

A cone set square onto a conical shell, such as a reducing branch on a hopper, with the hole in the main cone. A cone set square onto a conical shell, such as a reducing branch on a hopper or on a conical transition: the branch cone with its saddle cut, and the main cone with its hole.

Configurations

• Square to the main axis. The axis of the branch cone crosses the axis of the main cone at right angles.

A worked example

The square to the main configuration with the sizes SheetMorph opens with, in mm. 2 plates: the main cone with its hole, and the branch cone, with 32 divisions: 2 parts, 3 seams.

How it is developed

Triangulation and parallel-line development. This is a triangulated approximation: the curved surface is divided into small flat triangles, each laid out at its true size. The finer the division, the closer the pattern is to the surface; SheetMorph states the deviation it found.

• The branch cone is set onto the main cone. Its end is cut to the saddle, the line where it meets the wall of the main cone, and a hole of the same shape is cut in the main cone.
• The saddle is found on the main cone as it is drawn, with flat strips between its divisions, so the hole and the cut end agree exactly with each other; more divisions bring both closer to the true curve.
• A cylinder is unrolled; a cone is unfolded triangle by triangle from true lengths. The main is cut along the line opposite the branch, so the hole is in the middle of its plate.
• The 3D view shows the main without its hole. The weight is for the plates with the hole cut out.

Checked before it can be exported

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

• All three ends are square-cut circles (passed): Every end point lies in its end plane at the radius of its end (max deviation 1.14e-13).
• Saddle on the main and on the branch (passed): All 32 saddle points lie on the wall of the main cone and on the branch cone (max deviation 4.26e-14).
• Surface connectivity (passed): 128 facets form a main cone and a branch cone set onto it: every edge inside a part joins exactly two facets, every open edge one (V - E + F = 0).
• Developed edges equal true 3D lengths (passed): 384 facet edges compared between the 3D surface and the flat pattern (max deviation 2.30e-12).
• Shared edges and seam closure (passed): 126 internal edges are shared vertex-for-vertex between adjacent facets; 2 seam(s) have mating edges of equal true length (max deviation 1.14e-13).
• 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 7.22e-9).
• Cut outlines closed and non-overlapping (passed): 2 closed outline(s); no outline crosses itself or another outline.
• Hole matches the joint (passed): The hole lies within its plate and its edge has the length of the joint it is cut for (max deviation 1.95e-4).
• Symmetric about the XZ plane (passed): Every point has its mirror image across the XZ plane (max deviation 2.80e-13).

Assumptions

• The pattern is developed on the mid-thickness surfaces; inside or outside diameters are converted to them by one thickness.
• The branch is set onto the main, square to its axis and on its centre; it does not pass into it.
• No seam, lap, flange or weld allowance is added.

Limits of this module

• The branch cone must be narrower than the main cone where it meets it.
• The branch is square to the axis of the main and on its centre; angled or offset branches are not covered.
• The branch is set onto the main, not passed through it; the views show the main without its hole.
• No seam, lap, flange or weld allowances.

Conventions

Coordinate system. The main cone stands on the Z axis, with its base diameter at End A (Z = 0) and its top diameter at End B. The branch cone leaves toward +X; its axis crosses the main axis at the branch position. It is given by the diameter it would have on the main axis and the diameter of its open end, at the branch length from the main axis.

Two cones. Either cone may widen or narrow toward its open end, and either may be a cylinder when its two diameters are equal. Where two cones meet there is no closed form for the saddle: each line of the branch is followed in to the wall of the main and the crossing found by halving the step, to rounding.

Saddle and hole. The branch ends on the wall of the main, on a saddle-shaped line. Its plate is cut to that line, and a hole of the same line is marked on the plate of the main. The whole saddle must fall on the main, clear of both its ends.

Numbering. Every circle is divided into 4N equal parts. On the main, division 0 is toward +X (under the branch). On the branch, division 0 is its lower side (toward End A), N its +Y side and 2N its upper side. A and B are the ends of the main, C the open end of the branch and J the saddle. Plate S1 is the main and S2 the branch.

Material thickness. Say whether the diameters are inside, mid-thickness or outside sizes; the pattern uses the mid-thickness surfaces, and the views show the sizes as entered.

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 intersections

• Tee or Lateral: A branch set onto a run: equal or reducing tees and angled laterals, with the saddle cut and the hole in the run.
• Cross: Two branches set onto opposite sides of a run: equal, reducing or unequal, with both saddle cuts and both holes in the run.
• Manifold: A round run with a row of equal branches at an even pitch: the run unrolled with every hole marked, and the branches with their saddles.
• Cone on Pipe: A cone set onto a round pipe, square as a boot or hopper outlet, or at an angle as a conical lateral, with the hole in the pipe.
• Pipe on Cone: A round branch set square onto a conical shell, such as a nozzle on a hopper, with the hole in the cone.
• Rectangular Tee: A rectangular branch set square onto a rectangular duct, straight or with a shoe, with the hole marked on the duct.
• Rectangle on Round: A rectangular branch set square onto a round run, on its centre or offset: the run with its hole, and the branch folded at its corners.