In reproducing the above diagrams the sides of the small squares are to be

taken equal to half an inch.

21. A cone, base 2.7 inches diameter, height 2:35 inches, has its axis inclined at 40°. A curve is traced on the cone which, in development, would be a circle of 1 inch radius touching the base of the cone. Draw the plan of the cone, and of the curve traced on it, touching the base of the cone at its highest point. [B.E.] 22. The radius of a sphere is 1.5 inches. Draw the approximate development

of a lune of the surface of this sphere. Angle of lune 30°.

23. A surface is described by the revolution of an ellipse about its minor axis. The major and minor axes of the ellipse are 3 inches and 2 inches long respectively. Draw the approximate de

velopment of a lune of this surface. The lune to lie between two planes containing the axis of revolution and including an angle of 30°.

24. An elbow pipe is 6 inches in diameter and the radius of its centre line is 8 inches. Draw the approximate development of a lune of the surface of this pipe. Angle of lune 2210. Scale.

A

B

с

FIG. 699.

25. A sheet metal hood is square at the top and circular at the bottom as shown in Fig. 698. Show the shape to which the flat sheet of metal must be cut to form the hood. 26. ABC (Fig. 699) is a sheet-metal pipe, the portions A and C of circular and rectangular section respectively. By developing B set out the shape to which the flat sheet of metal forming it must be cut. Omit all allowances for overlap at the seams. [B.E.]

FIG. 698.

CHAPTER XXV

HELICES AND SCREWS

311. The Helix.-The helix is the curve described by a point which moves with uniform velocity along a generating line of a right circular cylinder while the generating line revolves with uniform angular velocity about the axis of the cylinder.

The axial pitch of a helix is the distance between one turn of the helix and the next, measured parallel to the axis of the cylinder upon which it is traced. Or, the axial pitch is the distance travelled by the describing point along the cylinder while it moves once round the cylinder. The normal pitch of a helix is the distance between one turn and the next, measured along the shortest line on the surface of the cylinder. If several helices of the same pitch be traced on the surface of the same cylinder, at equal distances apart, the distance between two adjacent helices is called the divided pitch. When the term "pitch" is used without any qualification, "axial pitch" is understood. The diameter of a helix is the diameter of the cylinder upon which it is traced.

The construction for drawing the projection of a helix on a plane parallel to the axis of the cylinder follows at once from the definition of the curve and is shown in the right hand portion of Fig. 700. The axis of the cylinder is assumed to be perpendicular to the vertical plane of projection. Divide the circle which is the elevation of the cylinder into a number of equal parts, say twelve, at the points 1, 2, 3, etc.

It is evident from the definition of a helix that if the generating point moves round any fraction of the circumference of the cylinder, it will at the same time move in the direction of the axis of the cylinder a distance equal to the same fraction of the pitch. Thus, if the point move round the cylinder a distance shown in the elevation by the arc 12, that is, through 1-12th of the circumference, it will at the same time move parallel to the axis a distance equal to 1-12th of the pitch. In like manner, in moving round another 1-12th of the circumference, it will move parallel to the axis another distance equal to 1-12th of the pitch. Hence the following simple construction.

Divide the pitch ab into as many equal parts as the circle in the elevation is divided into, in this case twelve, at the points 1, 2, 3, etc.

1

Through these points draw perpendiculars to ab to meet projectors from the points on the circle as shown. The points of intersection of these two sets of lines carrying the same numbers are points on the plan of the helix and a fair curve through them is the projection required. aeb is the plan of one turn of the helix and bfd is the plan of the next turn. The plan of the second turn of the helix may be obtained in the same way as the first, or it may be determined from the first by measuring from it, along the plans of the generating lines, a constant length equal to the pitch of the helix.

On the development of the surface of the cylinder the helix becomes a straight line. In Fig. 700 the straight line. AEB is the development of one turn of the helix and the straight line CFD parallel to AEB is the development of the next turn. A straight line MN at right angles to AB and CD is the development of a helix at right angles to the one already considered. The perpendicular distance

RS between the lines AB and CD is the normal pitch of the first helix.

The inclination of the helix or the pitch angle of the helix is the angle on the development in Fig. 700, and is the complement of the angle which the tangent to the helix at any point makes with the generating line of the cylinder through that point. If d is the diameter of the cylinder and p the pitch of the helix then tan 0 = 2. If a

Р

πα

second helix on the same cylinder is perpendicular to the first and p' is its pitch and its pitch angle, then '= 90° - 0 and p'

=

Р

The helix shown in Fig. 700 is right-handed. If the full and dotted parts of the plan of the helix shown in Fig. 700 be made dotted and full respectively the helix would become left-handed.

312. Helix of Increasing Pitch.-A point which moves round

a cylinder with uniform angular velocity and at the same time moves along the cylinder with an increasing velocity describes a curve which is generally called a helix of increasing pitch. The curve is however not a helix. The development of a helix of increasing pitch is a curved line while the development of a true helix is a straight line.

A helix of increasing pitch is shown in Fig. 701. In this example the describing point is supposed to move along the cylinder with

uniform acceleration while its angular velocity about the axis of the cylinder is constant.

The development of the curve is the parabola KPM, having KN for its axis and K for its vertex. If a tangent PQ be drawn to the parabola at P, then the inclination of P'Q to KL is the pitch angle of the helix of increasing pitch at the point whose elevation is p' and the corresponding pitch is NR obtained by drawing MR parallel to PQ to meet NK at R.

313. Screw Surfaces.-A screw surface is generated by a straight line which slides with uniform velocity along a fixed straight line or axis with which it makes a constant angle, and at the same time revolves about that axis with uniform angular velocity. It is obvious that any point in the generating line describes a helix.

Fig. 702 shows the portion of a screw surface generated by a straight line which starts from the position ao in plan and a'd' in elevation and makes half a revolution about a vertical axis. The generating line is shown in thirteen positions in plan and elevation. The outer end of the generating line describes the helix whose elevation is a'b'c'.

The curve b's'c' is the elevation of the section of the screw surface by the vertical plane whose horizontal trace is LM, and ore is the plan of the section of the screw surface by the horizontal plane whose vertical trace is PQ. The constructions for one point ss' in the former section and one point rr' in the latter are shown.

It should be observed that the boundary line of the elevation of

the left-hand portion of the screw surface, apart from the curve a'b' is not straight but is a curved line (not shown) which touches the elevations of different positions of the generating line. Also if the surface be extended upwards beyond c'f' the boundary line of the righthand portion of the screw surface will not be the straight line c'f' but a curved line touching the elevations of different positions of the generating line.

Fig. 703 shows the surface generated by a quadrant of a circle which slides along a vertical axis with uniform velocity and at the same time revolves about that axis with uniform angular velocity. ao is the plan and a'd'o' the elevation of the generating figure in its initial position. The generating figure makes half a revolution. The helices

described by four points on the moving arc are shown. The generating figure is also shown in nine positions in plan and elevation. These two sets of contour lines give to the elevation a pictorial effect and represent the surface more clearly.

The remarks on the boundary line in Fig. 702 apply also to Fig. 703 except that in the latter Fig. the boundary line on the left-hand portion of the elevation has been added.

314. Screw Threads. If the edges of a screw thread are sharp they form true helices. In practice the edges are often slightly rounded, as in the Whitworth standard V-thread, and this rounding can only be shown on the complete projection of the thread by shading or contouring. When the rounding of the edges is small it is generally neglected on drawings except in the case of a cross section.

Ordinary screws, such as are found on bolts, are generally