The triangulation method offers the most flexible system - i.e. it's just a series of simple circle to circle transformers. TapB can transform a circles around a circular radius reducing or increasing the diameter at a constant rate in the form of segments. You can choose the start and end diameters, bend angle and the number of segments and like with all Ty Harness Sheet Metal Applications the patterns are drawn automatically and in this case the patterns for all the segments are drawn. Fig XX shows a 90 degree bend tapering from 200 down to 100 around a quarter circle with a radius of 200. You can store the coordinates of any segment for further investigation with the built-in functions like pattern area, bounding area and volume of the segment. Fig XX shows segment number 3's minimum bounding area which will be useful for ordering enough plate.
|
| Figure XX - 200 - 100 90Deg tapered bend |
|
| Figure XX - 6 segments describing a 60 deg bend |
|
| Figure XX - 18 segments describing 270 deg |
|
| Figure xx a,b a) The first 3D DXF export from TapB and b) The first build from TapB |
Exporting the 3D DXF is great way to mock up designs for showing to customers. They really get a feel for what the final product will look like before any hard work is started. Sometimes it's difficult to explain segmentation and faking the transformation with CAD lofting tools can be misleading but TapB 3D exports the same segmentation as the patterns will produce. Unlike the earlier SegB software where you had to assemble your own segments for a DXF and VRML rendering, TapB can create all the segments assembled. VRML is also a good way to explain the tapered bend to clients,customers and students. You don't need any expensive software just a browser with a VRML plugin like Cortona and then you can view the tapered bend from all angles or in 3D rotation.
VRML model of 200-100 6 segment tapered bend
The above tapered bends are all examples of bends around a circular radius and this makes for an easy way to divide up the segments into equal divisions by simply dividing the bend angle. The length of the bend is simply radius times bend angle (radians).
There are numerous instances when this technique is not very pleasing to the eye. Ventilator and cowls are one example where you need to keep the back edge "tangential" or inline with a circular pipe. Figure XX was copied from Rabl[xx] plate 22 where a 45 deg oblique slice of the circular pipe is then joined to 4 segments. The elevation contains 2 true circular radii the describe the bend and the segments are created with the 4 equal divisions. But note when a circular pipe is cut obliquely the connection is an ellipse. Rabl goes on to develop the patterns which are ellipse-to-ellipse being triangulated out with the final exist closely approximates a circle. This is certainly a great technique if you especially require an ellipse on the output side of the funnel.
|
| Figure XXa - Rabl Vent Cowl & Air Scoop - b) Dickason's ventilator head |
Dickason[2] tackles the same problem, but this time, with six circle-to-circle transformers which is more work but no less elegant in form. From figure xx the smaller inner arc is a pure quarter circle divided into 6 equal divisions and the outer curve is taken as elliptical. Both techniques produce nice results but they are very orientated to solution of a 90 degree ventilator funnel.
One important aspect of sheet metal design is how to specify a design or what measurements do you need to take on site or from drawings to then take to the workshop and make the job. I feel the key dimensions are the input and output diameters, the bend angle and the semi major and minor axes i.e. the ellipse. I feel the ellipse dimensions are achievable with a folding rule or 2 flat bars bolted together like a folding rule.
My first thought is why use the quadrant/ellipse method when we could easily use 3 ellipses to define the bend with a computer. Figure xx shows the equal angle division of the segments is poor an not very pleasing to the eye.
|
| Figure XX - Poor segmentation |
Figure XX shows one method of drawing an ellipse where the red arc is the semi major axis and the blue arc is the semi minor axis. To find the point on the ellipse then draw horizontal from the blue arc and vertically down from the red arc. To draw the 3 ellipses graphically then it's a lot of work and but there are shortcut methods out there which would be acceptable to sheet metal workers.
|
| Figure XX - A tapered bend using 3 ellipses to describe the tapering. |
We need a method to equally divide the ellipse arc length which means the angle must change. The workshop technique would be to set the dividers and then step off around the arc centre line which are shown as grey circles in figure xx. You'll see the error an be able to adjust the dividers and try again until you get it spot on. The mathematical approach would be to solve the arc length using elliptic intrgrals.
There are many textbooks that cover elliptic integrals [4] and tables [3] of the following equation 1 can be used to calculate the arc length. I guess all students are probably ok with incomplete elliptic integrals of the second kind to find the arc length around an ellipse.
$ E(k,phi) = int_0^phi sqrt(1- k^2 * sin^2(theta)) d theta $ where $ 0 <= phi <= pi/2 $ and $ 0 < k < 1 $ Equation 1.
It's pretty tricky but there is more information here: ellipticE.htm
Ultimately we can use the computer to find the area under the curve with a numerical integration routine like the trapezoidal rule or Simpson's Rule which is how TapB will progress.
Now it's a question of dividing the arc length equally into the number of segments.
|
| Figure XX - A tapered bend using equal arc length around the ellipse |
|
| Figure XX - 12 segment circle-to-circle transfomers. |
The Rabl style Ventilator
Some people prefer the look of the Rabl ventilator with its elliptical flared exit. So I've made the software create ellipse-to-ellipse transformers. The circular pipe is obliquley cut producing the starting ellipse and you can input the exit ellipse dimensions.
|
| Figure XX a,b - Ellipse-to ellipse transformers showing the oblique starting pipe connection |
|
| Figure XX a,b,c - A Rabl style vent with 4 elliptical segments connecting to circular pipe mitred at 45 degrees. |
|
| Figure XX - Going beyond the quadrant |
[Home]
References
(1) Rabl, Ship and Aircraft Fairing and Development For Draftsmen and Loftsmen and Sheet Metal Workers, Cornell Maritime Press, , MayLand, USA: (First Ed. 1941 Fourth Printing 1994), .
(2) Dickason, A., The Geometry of Sheet Metal, Longman Scientifc & Technical, , London: (1991), 256-257.
(3) Jahnke, E., Emde, F., Tables of Functions with Formulae and Curves, Dover, , New York: (1943), . p72
(4) Stephenson, G., Mathematical Methods for Science Students, Longman, 2nd Ed., London: (1973), 188 - 194.
ASCII to MathML used in this page: ASCIItoMathML homepage