Fig. 82 Developing the plan shape of the inclined cone. 92
at my drawing. The lengths of these lines are equal to the width of the base (i.e. «, measured vertical to the plane of the paper) at the corresponding points on BD. (7) Set out these lengths on the 'plan in direction Y' as shown on the drawing. Draw a smooth curve through the points so found, giving the outline of the shape.
This may seem to be complicated, but if you have followed the steps you should have had no problems. If you haven't done so, do it now, for while it is easy to forget what you have read, that which you have done will stick in your mind. However, if you look back to Fig. 76 you will see that all we have done is the reverse of that procedure. Imagine the cone sliced off along AX. We look down the cone axis while we move from X" to X, so that this movement is a true arc. Then the distance X'-y in Fig. 76 must be half the width of the sliced-off part at this point. We then look down again, but this time normal to the base of the inclined cone (i.e. along the arrow Y in Fig. 82) to plot this as a true length on the plan view. The only difficulty with these development and intersection curves lies in deciding from which direction to view the solid in order to see the true length.
This is a very common case, where pipes or tubes of differing diameters meet. We will deal with the simple case of tubes at right angles first. We start by drawing the elevation and plan to scale, but leave space on the paper for the development, the length of which will be the circumference of the larger tube. Draw circles centred on the side tube centrelines, as you see. However, note that I have used only a semi-circle on the elevation. There is, in fact no need for a full circle in any of the constructions done so far; I have used them only to make the projections a bit clearer. Stick to the full circle on the plan in this case as well, until you get used to the drill. Divide the circles, as before, into sectors.
Now, observe: the figure 10 on the elevation lies at the top - at 1200 hrs. This means that it is on the centreline of the assembly. In the plan, therefore, it must also be on the centreline i.e. at 0900 hrs. The figure 7 is at the front - visible in the elevation - and I have marked it outside the circle on the elevation. Figure 1, however, is at the back, invisible in elevation, and is marked inside the circle. In this particular case these distinctions do not matter, but they often do, and it is important to be able to distinguish between these falsely similar positions. Now proceed as follows:
(1) From the intersections between radials and the circumference of both circles draw horizontal lines. They should reach far enough to pass the points'm' and 's', already identified in the plan view.
(2) Look at the plan view. Identify the intersection of each of these horizontal projections with the circumference of the vertical tube (a circle seen in plan) and mark it with a letter. I have used m to s.
(3) From each of these points draw a vertical line to meet the horizontal line from the corresponding point on the elevation's circle thus 6-r-e-6, or 8-r-e-8.You must take care to identify the points and lines 'back' or 'front'. In this case they lie one above the other, but this is not necessarily so, if, for example, the tubes do not have a common centreline.
(4) Mark these intersections as I have done, using letters a to g. Draw a smooth curve through these points. This is the true shape of the intersection when seen in elevation.
Now to draw the developments, needed if the joint were to be made from sheet metal. Let us do the small tube first.
(1) Draw a horizontal line ABA, equal in length to the circumference of the small tube, and divide it into the same number of equal parts as there are radials on your circles. Number these to correspond. Note that I have started at point '1' - this means that the joint will be there when it is fabricated. Adjust the numbering to bring the joint where you need it.
(2) Draw vertical lines at each of the numbered points, a little longer than the length A-m in the plan view, and add the identifying letter to correspond with each of the numbers. Note that these letters will repeat in mirror fashion; I have used a dashed letter (e.g. p') to identify letters corresponding to the numbers 8 to 12.
(3) Set your dividers to the length A-m in the plan and step this off at 1,7 and 1 again on the development. (Bs = Am in this example.)
(4) Repeat this procedure for lengths from n, o, p. q, r to the line AB, setting each in turn at its proper place on the development.
(5) Draw a smooth curve through the points so determined. This gives the profile to which the sheet must be cut to make a smooth joint to the intersection. (Don't forget to leave on a tab at one end if you are using a lap joint.)
Just a point of detail. It is prudent to arrange the joint exactly on one of the ordinates you have constructed, as if you set it elsewhere and perhaps make a slight error the stub will not point in the desired direction. I usually make my joints on one of the two centrelines, then it is easier to line up accurately.
Now for the hole in the large pipe. Set out the centreline and draw the rectangle, length equal to the circumference and height equal to the distance JK. Draw the horizontal centreline shown as XX. We now have to space out the vertical lines you see, which correspond to the lines bf, ce etc. on the elevation. These ought to be spaced by measuring along the circumference of the cylinder between (e.g.) o and p. This is not easy, but if you work it out you will find that the error arising from measuring the chord is very small. With a 3 inch branch and 4 inch main the error on pq is about 0.007 inches. It would be much less if we had used 16 radials instead of 12. Moreover, the error on the other spacings will be even less. This being so we can use the chords, but 'measure full' when transferring the dimensions. Proceed as follows.
(1) Set dividers to the chord op (which is the same as pq) and set off on either side of the centreline of the development. Draw the two vertical lines 3-11 and 5-9.
(2)Repeat, stepping off the chord no (or qr), set these out on the XX line of the development spacing from the points already marked. Repeat again with the chords nm/rs. You will now have five vertical lines - the two end points are, of course, «■just points.
(3) Refer to the elevation of the joint. You will see that the height of the hole at p in the plan must be the diameter of the branch. Set this out on the development. (I have marked this ag in letters and 10-4 in figures, to correspond to the elevation.)
(4) Measure from the centreline at o' to b (note that o'b = o'O and set this dimension on the appropriate vertical in the development. Note that this will also provide the heights for the vertical line 5-9.
(5) Repeat for n'c, completing lines 2-12 and 6-8 on the development.
There is no need to measure full for these vertical lines; they will be as correct as your draughtsmanship allows. A smooth curve drawn through the points gives the shape of the required opening.
Now, the more observant reader will have noticed that I have made a fair amount of extra work for myself - and for you - in all this measuring. In fact, I have done it on purpose, but never mind that. The point is that if I had set the centreline XX of the last development in line with the centreline XX on the elevation, I could have projected the heights we transferred with dividers. Similarly, if I had set the development of the branch below the plan, with the line ABA aligned to AB on the branch, I could have projected the heights here also, though the development would have 'stood on end' as it were. The reason is that circumstances can arise where direct projection is not possible - or would be very difficult -so that dimension transfer with dividers has to be resorted to.
Now, if you have forgotten what I said at the beginning and have simply tried to follow the instructions by reading, get out your paper and pencil and do it for yourself It is the only way to learn, and you will (I hope) be agreeably surprised to find how easy it is. To make it easier to follow I have emphasised one corresponding set of projection lines and the resulting lines on the developments.
By now you have had some experience, so that I have no hesitation in dealing with two elaborations at once. This time there is some justification for projecting by measurement. The development of the hole in the vertical tube could be projected easily enough, but that for the inclined branch would have to be laid out on an extension of the left-hand side of the paper; we must view the true lengths and this could only be done by looking in the direction of the arrow 'P'. Draw the outlines of the elevation and plan, leaving the end of the branch in plan blank for the time being - this must be constructed. In doing so, note that the position of such a branch may be defined from the centrelines, by the dimension DE, or at the surface of the main pipe, by dimension OF. As it is offset, the preferred way is to use the two dimensions DE and (in plan) CD. This does not affect the constructions, but is important in the design stage. Note that in plan the lengths of the side stub will not be true lengths, though the widths will be true. Further, as the stub is offset the shape of the intersection and the development will not be the same at the front as at the back.
You must take care to identify front and back points as you go along. Finally, American * readers please note that this example, like all the others, is executed in first angle projection. In this one it makes quite a difference to procedure if you wish to use third angle, as everything must be reversed. Stick to first angle.
As this example needs more construction lines than usual I am not going to draw all of them, but just 'talk you through' one or two; you can then repeat the process for the others. I do emphasise that with this example in particular you will be most ill-advised if you try to follow the procedure just by reading the page; get out the drawing-board and do each step (and the repetitions) as it arises.
The first thing we have to do is to obtain the shape in plan of the end of the branch. On the elevation, set out a circle - I have used a semi-circle, as it is symmetrical -on the end of the branch and divide it into segments. Draw the radial lines and insert the identification numbers and letters. Drop perpendicular projection lines down to the plan from points 1 and 7. These define the extreme points on this view. Do the same for the other points a.b.c.d.e; I have shown that from 'b' only. Mark the letters on the stub centreline XD on the plan as I have done.
Take the case of line 'b' to see how to determine the widths in plan. Measure the length b-3 in the elevation semicircle and set this dimension either side of 'b' in plan to obtain the two points 3 and 3'. Do this for all the other points, noting that the lengths b-3 and d-5 are the same as are a-2 and e-6. This done, draw a smooth curve through the points. I recommend that you now draw a dotted curve as I have done, boldly marked BACK, just to remind you which is back and front.
The next job is to determine the shape of the intersection curves on the elevation. The drill is:
(1) Draw projection lines parallel to the axis of the branch from the points 1 -2-3-4 etc. at the ends of the radials, long enough to reach into the vertical cylinder.
(2) Draw projection lines from the points 1 -2-3-4- etc. on the plan of the end of the branch to meet the circumference of the vertical cylinder. Identify the points of intersection with letters.
(3) Draw vertical projection lines from the lettered points in plan to meet the corresponding points on the parallels in the elevation.
I have drawn in two. From 3-5 in plan to's'. From the intersection at s' vertically to meet the line from 3 at't' and from 5 at 'u'. Again, from 3'-5' to 'p'; vertically to meet the parallel from 3' (the same line as from 3) at 'q', and from the radial 5' (same as from 5) at 'r'. These are at the front on the elevation, and I have ringed these points; those at the back are plain dots.
(4) Once all vertical projection lines have been drawn the outline of the intersection can be drawn. Full line at the front, and dotted line at the back.
Now for the development of the branch. The procedure is as already described. Draw the base line and mark out the divisions sideways from the centreline. Note that there is an extra one at each end - this makes it easier to draw the final curve. The heights of the ordinates are measured on the elevation from the line a-b-c-d- etc.
Fig. 85 Approximate development of a spherical surface.
to the curves q-r- and t-u-, and you will see that I have marked these four on the ■developments. Take care in identifying front and back. Note also that the curve does change curvature rather rapidly in places. This is a case where it would have been prudent to use more radials, say 16 rather than 12.
The development of the hole in the main cylinder does need more care, and the use of a trick. Set out the horizontal centreline, which will, in the elevation, be a circumferential line through 'O'. Now, if the branch has been located by the dimension DE in elevation you must ascertain the length OF on the circumference. You can do this by measurement if the drawing is carefully made; or you can calculate it, from OF = FE.Tan O + DE. (EF is the radius of the cylinder.) Then draw the baseline and from this measure up a distance FG to get the position of the top line.
Now take great care and set off the distance OK; make sure that you get it on the correct side, the drawing is in first angle projection. Then proceed as follows:
(1) Use your dividers to set out the distances Om, mp, pn from the plan view (measured on the circumference of the vertical cylinder) onto the development to the right of the point '0'.
(2) Repeat this, procedure for the points on the left of 'O'. Note that these distances are not symmetrical about O; they must be measured individually.
(3) On the elevation draw the vertical line AB at some convenient point. (This line is used to measure the verticalordinates, which might be difficult along the line tu otherwise).
(4) Measure the vertical distance from 0' to the line btq and set this on the development. Make dots where this line crosses the vertical lines marked's' and 'p'. (These are the corresponding points in the plan view.)
(5) Repeat for all the other points on the development and draw a smooth curve through them.
Note that though this development may look like a perfect ellipse this is not the case. The hole is not symmetrical about the centreline GOF.
This construction will serve for any angle of inclination or degree of offset; if 0 is 90° then you have a simple right-angled tee which is offset, and if the offset is zero then the plan view is symmetrical. Such cases are less work, that is all. However, there is just one point of geometry that needs watching. If the angle is less than about 45°, or if the offset is such that the branch lies entirely to one side of the larger cylinder's centreline, then it is almost essential to use more ordinates on the originating circle on the branch, 16 at least, and at the critical points even these may need to be divided again.
There is, by the way, no geometrical reason for equal divisions; you can place them as you wish. We use equal divisions only because they are easier to set out.
No surface which curves in two or more planes can be developed truly from the flat, but it is possible to make a reasonable approximation. Fig. 85 shows the treatment of a hemisphere. Those of you who have seen geographical globes will have noticed that the map is cut up into slices rather like the segments of an orange; these are known as gores, and their construction is not at all difficult. In fact, it is possible to achieve a reasonable spherical shape with very little construction indeed.
Draw the plan and elevation, and divide the plan into sectors with radial lines. In this case you do need closer intervals, and 16 radials is the minimum. As before, identity each radial with a number. Project from the intersection of radials and circumference vertically upwards to meet the base AB and continue upwards to meet the outline of the sphere in elevation. Set your compass to the radii shown as R2, R3, R4 in the elevation. (R, is the radius of the sphere, D/2.) Describe arcs on the plan using these radii. Identify (i) the lengths of the arcs between adjacent radials, shown as a, b, c, d; and (ii) the lengths of the corresponding arcs on the elevation, shown as w, x, y, z.
Now, look at the plan. You will see that a sector has been shaded. This is the plan view of one of the gores. Project upwards from the points of intersection of radials and circles (shown with dots) and you get the shape OXY seen in elevation. To prepare the development first set out a base line equal in length to the circumference of the sphere. Divide this into as many equal parts as you have segments in plan. You will see that I have shown one shaded, and XY corresponds to X'Y' on the plan. Draw a parallel line above the base, distance away equal to one quarter of the circumference, and further parallels at distances w, x, y, and z apart. (These distances will be equal, and should add up to the quarter-circumference.) On these lines set out the widths a, b, c, d as shown. Then draw smooth curves through the points so found to form the gores. There will, of course, be as many of these as there are sectors between the radials. This is not a difficult construction, and can, of course, be adapted. The back of a bend, for example, can be developed simply by halving the number of gores - making a quarter sphere. However, it can be much easier than that. On the drawing I have given the necessary algebra (trigonometry?) needed to calculate the various dimensions required to construct the chain of gores. There is no need to draw it out at all. Finally, for a 'rough job' advantage can be taken of the fact that the profile of the gores approximates to the arc of a circle. It is only necessary to set out the three points X, Y, and 0 and, by trial, strike an arc which is centred on the base line and passes through 0 and Y. In any case, it is not necessary to construct a row of gores. You can mark out one on a sheet of tinplate or brass, file to shape, and use it as a template.
This is not a book on sheet metal work, but it may be advisable to mention two points. First; developments for work in thin sheet can be constructed and used as drawn. However, if the metal has any appreciable thickness - more than, say 2% of the diameter of the object - you will be well advised to allow for this. Each case must be considered individually, but in general you will not go far wrong if you make your initial drawing to dimensions which correspond to the mid thickness of the work. For example, for a cylinder 4 inch dia x 1/8 inch thick, make the drawing for 3-7/ 8 inch diameter.
In the case of a sphere the workpiece will braze up with a series of flat segments. You can ease this by doing a bit of panel ^beating on the segments to form a slight lateral curvature. Further, when making such a hemisphere, for a dome cover or similar, I use the technique applied to the cone in Fig. 78. Turn up a little disc with a spherical top, a ridge on which the ends of the gore can sit, and then file or machine all flush after brazing.
You may well come across even more elaborate cases, especially if you ever see any examination papers in so-called 'engineering' drawing. Academic examiners get high on things like 'ellipsoidal cones intersecting hypertrochoidal solids of revolution'. However, if you have followed me so far you have mastered sufficient of the art of development to cover most practical cases that arise in the ordinary workshop. If anything odd does turn up just remember that you need the true length. Draw an arrow showing the direction from which you will 'see' the true length, and then arrange your construction so that the views are seen from the direction of the arrow or arrows if more than one.
Finally, you may now appreciate why the college courses always start with this sort of construction. Once you have done a couple of developments you should have no difficulty at all in projecting one view of a pair of inside cylinders from another.
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