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INTERPENETRATION AND DEVELOPMENTS OF SURFACES AND SOLIDS-SECTIONS OF SPECIAL SOLIDS-HELICES AND SCREW THREADS.

THERE are a number of problems of frequent occurrence in practical draughtsmanship which are best solved by the application of methods usually regarded as a part of solid geometry. Among such problems may be mentioned the drawing of an ordinary steam dome upon a cylindrical boiler, or of the semispherical ends of egg-ended boilers, and the finding of the true shape of the plates for such parts; the drawing of the contact lines of the cylindrical branches of cocks and valves with the main casing (see Fig. 166), and the drawing of the correct outline of such parts as at the junction of the crank-web and crankshaft, or the meeting of other flat and curved surfaces, as in connecting-rod ends and other similar parts.

These problems may generally be regarded as special cases of the interpenetration and development of surfaces and solids, as, for example, the steam and boiler may be treated as a case of the interpenetration of two cylinders, and the cock with its inlets and outlets as the interpenetration of a cylinder with a

cone.

It will be understood that in the case of the steam dome and boiler, and of many similar examples, it is necessary to develop the true shape of surfaces in order that the plates may be so cut when flat, so that they shall join up correctly when bent to their required form.

PROBLEM LV. (Fig. 89).-To draw the projections of the interpenetration of a horizontal and vertical cylinder and the development of their contact surfaces.

Let A be the plan, B the end elevation, and C the side elevation of the cylinders.

The line of interpenetration is evidently shown in plan by the circle 1, 7, and in the end elevation B by the arc d' l'. In order to find its side elevation on C, we imagine the vertical cylinder to have a number of stripes drawn upon it, and we then find the real length of each stripe from the top base of the cylinder to the point where it enters the horizontal cylinder. This is done as follows:-Divide the plan of the vertical cylinder, the circle on A, into a convenient number of equal parts, say 12,

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and mark as shown, 1, 2, 12. Draw the elevation of the stripes in each of the elevations B and C, and mark the stripes 1', 2', . . . 12' on B, and 1", 2", . . . 12" on C. Care must be taken not to confuse the marking of the stripes in the two elevations, notice that the outside stripes 1' and 7', on the end elevation B, are the centre stripes on the side elevation C, while the centre stripes 4' and 10', on the end elevation, are the outside stripes on the side elevation. The stripes are correctly obtained on the end elevation B, by projecting from the plan, 1011 12" 1"

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Fig. 89.

and, on the side elevation O, by drawing and dividing the semicircle on the line m n as shown.

The real length of the stripes are shown in the end elevation B, therefore mark off on each stripe in the elevation C its real length as obtained from the end elevation, thus 1"p=1' d' or

7'l', and 4′′n and 10′′ m = 4′ h', and so on for each stripe. The curve mpn drawn through the end of each stripe as thus found is the side elevation of the interpenetration. It will be noticed that it also represents the back half of the interpenetration.

To develop the surface of the vertical cylinder draw the oblong RSTV, the length of which RV equals the circumference of the cylinder base, and the height of which R S equals the length of the longest stripe on the cylinder, l' d' or 7. Divide the oblong into the same number of equal parts as stripes on the vertical cylinder, and draw lines through each point. Mark the lines 1, 2, . . . 12, 1, as shown. These lines are the development of the stripes, the two end lines coinciding to form the stripe 1 when the oblong is bent to form the cylinder. Mark off the real length of each stripe as found from either of the elevations B or C, down the lines representing the stripes from the line RV as 4 x = 4′h', and draw the curve SWT as shown through the points thus found. The complete figure, RSW TV, is then the development of the vertical cylinder, supposing it simply to rest upon the horizontal cylinder, and is the shape to which a piece of paper must be cut, so that when it is bent to bring the edges RS and TV together, it shall exactly fit the horizontal cylinder.

We will now suppose the vertical cylinder to penetrate the horizontal one for a short distance and find the true shape of the hole of penetration in the surface of the horizontal cylinder. To do this we must stripe that part of the horizontal cylinder containing the hole, and then develop it with the stripes, the length of which between the extremities of the hole will enable us to find the true shape of the hole. Divide the arc d'l' in the end elevation B into eight equal parts at the points d', e,' f', ... . . . l'. Consider these points as the elevation of the stripes, and draw their plan across the hole in the plan A as shown by the dotted lines d, e, f,.... Develop a part of the surface of the cylinder containing the hole, as in the Figure D, where the line d" l′′ is equal to the real length of the arc d' l', and is found as shown on p. 73. Divide the line d" 7" into eight equal parts, and draw the lines marked e", ƒ", g′′, . . . l" through each as shown dotted, these lines are the development of the horizontal stripes crossing the hole. The real lengths of each stripe between the extremities of the hole are shown by the length of the dotted lines crossing the hole in the plan A, thus g" b" or g′′ c" gb and gc, or measuring from the front of the cylinder a" b′′ = a b and a" c" = a c. The closed curve drawn through the points thus found is the true shape of the hole.

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The conditions of this problem are similar to the practical

example of a steain boiler and dome, for the horizontal cylinder represents the boiler, and the vertical cylinder the dome. Then the development, RSW TV, is the shape for the plates of the dome before bending, neglecting the flange, and the development D shows the shape which the hole should be cut in the plates, so that when bent it shall be a circle.

EXAMPLES.

EX. 1.-Draw the curves of interpenetration of two cylinders each 3" diameter and 4" high. Axis of one horizontal, of the other vertical; both axes parallel to the V P.

EX. 2.-A horizontal cylinder, 4" diameter, 6′′ long, is interpenetrated by an oblong block 2" wide, 2′′ thick, the sides of which are vertical. Draw the correct lines of interpenetration in the side elevation, and draw a development of the surface of the oblong block and of the hole in the cylinder. Height of block immaterial.

EX. 3.—A cylindrical boiler is 6' in diameter and has a cylindrical steam dome 2′ 6′′ in diameter and 2' high. Draw three views of the arrangement and show the development of the plates of the steam dome and of the hole in the boiler shell. Scale 1" 1'. (See Problem lv.)

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EX. 4.-A right circular cylinder of 21" diameter penetrates another of 3" diameter, the axes being at right angles and passing " from each other. Draw the projection of the curves of intersection on a plane parallel to the axes of the cylinders. (Vict. B. Sc. Hon., 1889.)

EX. 5.-Develop the surface of the cylinders in Ex. 4, the larger cylinder development to show the holes for the small cylinder. Cut out the figures and make a model of the cylinders in the given position.

Interpenetration of Cone and Cylinder.-PROBLEM LVI. (Fig. 90).—To show the curve of intersection of a vertical cone and a horizontal cylinder in plan and elevation and to develop the surface of the cone.

The cone is shown in the figure having its vertex marked v and v' in the plan and elevation respectively. The cylinder has the axis ab, and its diameter is such that it does not cut completely through the cone-that is, the diameter of the cylinder is less than the diameter of the section of the cone a' b', which contains the axis of the cylinder. In order to save drawing only one-half of the cylinder is shown.

To obtain points in the intersection we take a number of

horizontal cuts through both solids, and obtain from a plan of the section two points in the plan of the intersection. For example, if a horizontal section be taken at the line c'd', its plan will be the circle gch, equal in diameter to c' d' shown partly drawn upon the plan of the cone, and the oblong shown by the

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lines gf, fe, eh, which represents the section of the cylinder. The oblong and the circle cut at the two points g and h, which are evidently two points in the plan of the required intersection. The elevation of the points coincide at the point marked

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