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Fig. 78 Forming the point at the vertex of a cone.

quite impossible to fold even thin sheet metal to an exact point. Now try one for yourself - make it in stiff card or even paper if you have no use for a tin funnel.

The sliced-off cone

The proper name for this is a frustum (Fig. 79), this being Latin for 'a part' or 'cut off. (Latin because the method was worked out over 2000 years ago.) At first sight it looks formidable, but remember what I said; the lines are drawn one after the other, in a system. To make the example universal I have sliced off both the top and the bottom at different angles, along BD and EF. You will see that the 'fan' is now a very odd shape indeed, but in fact, all we need are the true lengths of the lines radiating from the apex of the cone. I have marked one of these as D'F', and shown it on the fan.

In this case I suggest that you make a proper elevation drawing of the 'frustrated' cone and its plan view, but you can manage with a sketch if you are careful. Just the outline at present; never mind all the construction lines. Then proceed as follows, remembering that the object is to determine the true lengths between the slices all round the cone.

(1) Divide the plan view into equal parts, I have shown 12. More would be better, but it is seldom necessary to go beyond 16, easy with a 45deg. set-square. But work as accurately as you can in setting out.

(2) Number the radials 1 to 12 as I have shown. This makes it easier to follow.(3)

Project vertical lines from the points on the circle to meet the base of the cone. I have ^shown these points as 1', 2', 3' etc. Note: You will notice immediately that there is no need for a complete plan view; a semi-circle would suffice, as the points 8' to 12' lie on top of those from 6' to 2'. This is true for all developments which are, like this cone, symmetrical.

(4) Draw lines from the points on the base (1', 2' etc.) to the apex of the cone. Mark where they cross the planes of section, BD and EF.

(5) Project horizontally from these intersection points to meet the slant side of the cone. (I have marked these as p,q,r,s,t.) I have not marked these on the plane EF, do that for yourself.

(6) Draw the fan as if the cone were intact, using the method already described in Fig. 77. Set out radial lines as shown, by dividing the projected angle 0 by the number of divisions you have used on the plan -12 in this case. Number each of these radial lines to correspond with the numbers on the plan view.

(7) Working from the base (BC) upwards, set off the distance CD from the elevation onto the end radials of the fan.

(8)Similarly set off the distance Bp from the elevation on the fan at the radial corresponding to points 6 and 8.

(9) Set off, in turn, Bq, Br, Bs and Bt on the fan to correspond with radials 5/9, 4/10, 3/ 11 and 1/12.

(10) Join the points on the fan radials with a smooth curve.

(11) Repeat this procedure for the upper slice, EF, this time measuring lengths from the apex of the cone.

Again, you will need a jointing tab (unless you are going to make a butt-brazed joint) but this is a matter of workshop practice, not drawing.

Try this fellow with a sheet of thin tinplate or card and see how you get on. It is really very easy once you take your courage in both hands and have a go. However, there Is one point of detail worth mentioning. I usually do these developments full size on either thin tinplate or on tinsmith's template paper - a fairly stiff paper-like material which takes pencil and yet will withstand hard usage in the shop. After a trial to see that all is well I

Fig. 80 The inclined circular cone. This is, in effect, the same as Fig. 79.

then use this as a template to mark out the actual workpiece. However, if the development is very important or complicated, then I draw the diagrams either twice or five times full size, take dimensions off and reduce them to the actual workpiece. You must use very sharp pencils and take care over the setting out when an actual workpiece is involved though sketches may be quite adequate if you are just practising draughtsmanship.

The inclined cone (Fig. 80)

This looks like a new shape but it is, in fact, identical to that in Fig. 79, and can be treated as such. You can do the construction either with the centreline of the cone vertical, or in the design position, Fig. 80.The former is more convenient if all you need is the development, but the latter makes it easier to carry out the next stage. Before coming to that, however, I show in Fig. 81 an application of this conic section which you might think worth making; a funnel I made many years ago for feeding a car from a jerrican. I find it very useful

with modern cars.

Base of the inclined cone (Fig. 82)

We may need to know the shape of the cutting plane across BD - the base of the cone as drawn. This may look very formidable but, again, the lines are drawn one by one, and much of what you see on Fig. 82 you have done already. In fact, you proceed exactly as you would to prepare the development, but do it with the cone on the slant. This finds the points p,q,r,s,t, but notice that I have added points at p',q',r',s' and t' on the true centreline of the cone. Now proceed as follows.

(1) Draw a line parallel to BD some convenient distance below.

(2) From the numbered points on BD drop perpendiculars to this line; these I have marked with double arrows. (The lines with single arrows are drawn as for the construction of a development.)

(3) Set compasses to a radius equal to t-t' and, on the plan at X' scribe an arc which crosses the radial lines 2 and 12. (Note that these radials correspond to the lines used in finding the point)

(4) Set compasses to radius s-s' and draw an arc to cross the radials 3 and 11 .The same comment applies.

(5) Repeat this process for radii equal to r-r', q-q' and p-p' in turn, crossing radials 4-10, 5-9 and 6-8.

(6) Draw lines joining these intersections between arcs and radials, and set identification numbers against them - look

PLAN IN DIRECTION X B