24. Fig. 734 gives the plan and part of the elevation of a portion of a handrail of rectangular cross-section (as prepared for moulding); complete the elevation of the rail. Set out the "face mould" for this rail. Draw also the "falling mould" for the central vertical section CDDC of the rail.

[B.E.]

25. Stair rail (Fig. 735). To be made in two parts, cut from planks and connected at C by a joint perpendicular to FG. A plan and part elevation are given. Complete the elevation of the centre line ABCDE and draw a second elevation of this line on X'Y'. Show the plan and two elevations of the joint at C. Draw the "face mould" for the wreath ABC, and set out the "bevels" for its ends.

Plot the "falling moulds" for the central section and the inner and outer surfaces of the wreath ABC. Complete the elevation of the wreath ABC. [B.E.]

26. In Fig. 736 are given the flange and the maximum thickness section of one blade of a three-bladed propeller; the dotted curve is the developed face of the blade. The face of the blade is a true right-handed screw surface whose pitch is one and a quarter times the diameter of the propeller. Draw the figure double size and then construct a plan and two elevations together with sections of the blade as shown in Fig. 721, p. 378.

27. Same as the preceding exercise except that the blade is to have a set back which, measured at the tip, is 0.06 of the diameter of the propeller; also the screw is to be left-handed instead of right-handed.

28. The flange and the maximum thickness section of one of the four detachable blades of a screw propeller are shown in Fig. 737. The dotted curve is the developed face of the blade. The diameter of the propeller is 19 ft. 6 in. at the tip and 5 ft. 2 in. at the root. The pitch of the face towards the leading edge increases from 19 ft. at the root to 21 ft. at the tip, and the pitch of the face towards the following edge increases from 19 ft. at the root to 23 feet at the tip.

AXIS OF SHAFT

FIG. 736.

FACE

Draw the plan and two elevations of this blade to a scale of one inch to a foot, and determine the developed sections of the blade by cylindrical surfaces of 3, 4, 5, 6, 7 and 8 ft. radii.

AXIS OF SHAFT

FIG. 737.

12

1212

12

15

CHAPTER XXVI

INTERSECTION OF SURFACES

320. General Method. The general method of finding the intersection of two surfaces is as follows. Let A and B denote two given surfaces whose intersection with one another is required. Cut the surfaces A and B by a third surface C, the latter surface being so chosen and employed that the projections of its intersections with A and B are lines, such as straight lines or circles, which can easily be drawn. Let A' denote the line of intersection of C and A, and let B' denote the line of intersection of C and B. Let the lines A' and B' meet at a point P. Then the point P lying on the intersection of C and A, lies on A. Also since the point P lies on the intersection of C and B, it lies on B. The point P therefore lies on A and B, and therefore it must be a point on the intersection of A and B. By moving the surface C into different positions, or by cutting A and B by other surfaces similar to C, any number of points on the intersection of A and B may be determined.

On the intersection of two surfaces there are generally certain important points which should be first determined. For example if the projection of the intersection meets a boundary line of the projection of one of the surfaces, the meeting point will be an important one to determine and in the case of a curved surface this will be a tangent point.

In moving the cutting surface in one direction from a position which gives points on the required intersection to a position which gives no such points, there will be an intermediate position which gives the last of the points obtained by moving the surface in that direction and these points are usually important.

As a general rule after the important points have been determined only a few others are necessary to fix the required line of intersection.

It is a good plan to number or letter the cutting surfaces in order, and the points determined by them, each point having the same number or letter as the cutting surface on which it is situated.

321. Intersection of Two Cylinders.-The general method of finding the intersection of the surfaces of two cylinders is to cut the surfaces by planes parallel to their axes or to their generating lines. These planes will intersect the surfaces of the cylinders in straight

lines, the intersection of which with one another will determine points on the intersection required.

EXAMPLE 1. The horizontal trace of a vertical cylinder is an ellipse, 3-2 inches x 24 inches, the major axis being parallel to XY. A horizontal cylinder, 16 inches in diameter has its axis parallel to the vertical plane and 0.3 inch in front of the axis of the vertical cylinder. It is required to show the elevation of the intersection of the cylinders.

There will be two identical curves in this case and in Fig. 738 only enough of each of the cylinders is shown for determining one of these curves.

Cutting planes parallel to the axes of both cylinders will in this case be vertical planes parallel to the vertical plane of projection. HT is the horizontal trace of one of

"a"

-Y

T

these planes. This plane will cut the vertical cylinder in two vertical straight lines, one of which will have the point a for its plan, and a perpendicular to XY from a for its elevation. This same plane will cut the horizontal cylinder in two horizontal lines of which ab will be the plan. To find the elevations of these lines, take an elevation of the horizontal cylinder on a plane perpendicular to its axis, this elevation will be a circle, but only half of it need be drawn, as shown, The cutting plane now being considered will have a trace on this new vertical plane which will coincide with its horizontal trace. The point b' where this trace cuts the circle, or semicircle, just mentioned will be the end elevation of AB one of the lines in which the horizontal cylinder is cut by the cutting plane HT, and the length bb,' will be the vertical distance of these lines above and below the horizontal plane containing the axis of that cylinder. Hence the required elevations a'b', a'b' will be at distances equal to bb,' above and below the elevation of the axis of the horizontal cylinder.

Habi

3

FIG. 738.

- - 1.6 ·

!

The intersections of a'b' and a'b' with the vertical line through a determine points on the intersection required.

In like manner, by taking other planes, other points can be found. The most important points on the curves of intersection in this case are those determined by the following cutting planes. (1) The two planes which touch the horizontal cylinder, these determine the points l' and 5' on the elevation. (2) The plane which contains the axis of the horizontal cylinder, this determines the points marked 3'. (3) The plane which contains the axis of the vertical cylinder, this determines the points marked 4'.

The foregoing example may also be worked by using horizontal cutting planes, and the student would do well to work the problem in

this way.

EXAMPLE 2.-A vertical tube, external diameter 2.75 inches, internal diameter 1.5 inches, has a horizontal cylindrical hole bored through it 1.25 inches in diameter. The axis of the hole is 0.25 inch from the axis of the tube. It is required to draw an elevation on a vertical plane inclined at 35° to the axis of the hole.

First determine the intersection of the boring cylinder with the outside of the tube exactly as in the preceding example. The fact of the horizontal cylinder being inclined to

the vertical plane will make no difference in the working, as the heights of all the lines in which the cutting planes intersect the horizontal cylinder will remain the same whatever be its inclination to the vertical plane.

Next in like manner the intersection of the boring cylinder with the interior surface of the tube is determined. It will be found that in this example this interior intersection consists of one line only.

The cutting planes which give the most important points on the curves of intersection in this example are the same as those in example 1, with the addition of the plane which touches the inner surface of the tube.

The result of this example is shown in Fig. 739, but the construction lines have been omitted.

FIG. 789.

This example, like the preceding one, may also be worked by using horizontal cutting planes.

EXAMPLE 3 (Fig. 740).—ab a'b' is the axis of a cylinder whose horizontal trace is a circle 2.25 inches in diameter. cd c'd' is the axis of a cylinder whose horizontal trace is an ellipse (major axis 3 inches, minor axis 2 inches), whose minor axis is parallel to XY. It is required to show, in plan and elevation, the intersection of the surfaces.

In this example the planes which will intersect the surfaces of both cylinders in straight lines will be inclined to both planes of projection.

To find the directions of the traces of the cutting planes take a point pp' in ab a'b'. Through this point draw pq p'q' parallel to cd c'd. The line aq which joins the horizontal traces of ab a'b' and pq p'q' will be the horizontal trace of a plane which contains the axis of one cylinder and is parallel to the axis of the other, and this plane, and planes parallel to it will, if they cut the cylinders at all, cut them in straight lines parallel to their respective axes. In this particular example the horizontal traces of these planes are parallel to XY.