Chapter Five Geometrical Constructions

1. Chapter Five Geometrical Constructions

2. Purpose This chapter provides an overview of how to: • competently use a range of drawing instruments to construct curves and shapes commonly found in engineering drawings • understand the principle and construction of cams, conic sections and templates.

3. Geometrical constructions used in engineering drawing The following basic constructions are given for reference to aid you in your engineering drawings. Page 106: • 5.7 To construct an angle equal to a given angle. • 5.8 To bisect a given angle. • 5.9 To draw a line parallel to a given line at a given distance from it. • 5.10 To draw a line parallel to a given line through a given point.

4. Geometrical constructions used in engineering drawing Page 107: • 5.11 To draw a perpendicular from the end of a line. • 5.12 To draw a perpendicular from a point in a line. • 5.13 To draw a perpendicular from a point outside a line (Method 1). • 5.14 To draw a perpendicular from a pint outside a line (Method 2).

5. Geometrical constructions used in engineering drawing Page 108: • 5.15 To divide a line into any number of equal parts, say six. • 5.16 To trisect a right angle. • 5.17 To construct a regular pentagon on a given line. • 5.18 To construct a regular hexagon on a given line.

6. Geometrical constructions used in engineering drawing Page 109: • 5.19 To construct a regular polygon, say a heptagon, on a given line. • 5.20 To draw an arc tangential to a given arc, centres on opposite sides. • 5.21 To draw an arc tangential to two arcs and enclosing one of them. • 5.22 To draw a curve tangential to three intersecting lines.

7. Geometrical constructions used in engineering drawing Page 110: • 5.23 To draw arcs tangential to a given arc, with centres on the same side. • 5.24 To draw an arc tangential to two arcs (externally). • 5.25 To draw an arc tangential to two arcs (internally). • 5.26 To draw an arc tangential to two straight lines.

8. Geometrical constructions used in engineering drawing Page 111: • 5.27 To draw an arc tangential to two straight lines at right angles. • 5.28 To draw an arc tangential to a line and another arc. • 5.29 To draw a tangent to a circle from an outside point. • 5.30 To draw a direct common tangent to two given circles.

9. Geometrical constructions used in engineering drawing Page 112: • 5.31 To draw a transverse common tangent to two given circles. • 5.32 To draw a straight line approximately equal in length to a given arc. • 5.33 To mark on a given circle an arc approximately equal to a given length. • 5.34 To draw a straight line approximately equal in length to the circumference of a given circle.

10. Geometrical constructions used in engineering drawing Page 113: • 5.35 To construct a regular hexagon given the distance across corners. • 5.36 To construct a regular hexagon given the distance across flats. • 5.37 To draw a cycloid, given the diameter of the rolling circle.

11. Geometrical constructions used in engineering drawing Page 114: • 5.38 To draw an epicycloid, given a rolling circle and a base circle. Page 115: • 5.39 To draw a hypocycloid, given a rolling circle and a base circle. • 5.40 To draw an involute to a given circle.

12. Application of the involute curve • A gear tooth is a typical engineering example of an involute curve. Figure 5.1, p.116, illustrates the gear tooth, showing the part of the tooth between the top and the base circle is of involute form. • Figure 5.2, p.116, illustrates involutes formed from various shapes.

13. The cylindrical helix • A helix is the path traced around the surface of a cylinder and, for each circumference traversed, moves a constant length (called lead) in a direction parallel to the axis. • The geometrical construction of the helix is illustrated in construction 5.41, p.118. • Practical applications of the helix are illustrated on p.117.

14. Geometrical constructions used in engineering drawing Page 118: • 5.41 To construct a cylindrical and a conical helix. • 5.42 To construct the sine and cosine curves. Page 119: • 5.43 To construct the spiral of Archimedes.

15. Cams • Cam – a machine shaped to guide another machine (follower) which presses against it and moves according to the cam surface. • Follower – moves over a specific distance called stroke or displacement via a motion governed by the cam profile.

16. Cams Types of cam • Radial or disc cams (follower moves at right angles to the cam axis). • Cylindrical and end cams (follower moves parallel to the cam axis). Cams are used as machine elements in a variety of applications including machine tools, motor cars, textile machinery and many other machines found in industry.Refer to Figure 5.5, p.120, for various configurations of cam and follower.

17. Cams • Displacement diagram – plots the followers rate of speed and its various positions during one revolution of the cam and aids in cam design; see p.121 and Figure 5.7. • Three types of motion are commonly used in cam design: • Constant velocity or straight line motion (Figure 5.6(a)). • Simple harmonic motion (Figure 5.6(b)). • Constant acceleration – deceleration or parabolic motion (Figure 5.6(c), p.121).

18. Cams Cam construction techniques • 5.44 To construct a radial disc cam for a wedge-shaped follower, p.122. • 5.45 To construct a radial disc cam profile for a roller follower, p.123. • 5.46 To construct a radial disc cam profile for a wiper follower, p.124. • 5.47 To construct a cylindrical cam for a rod follower, p.124.

19. Conic sections • Figure 5.8, p.125, shows the side view of a cone and the curves relevant to a given plane of intersection. • When the intersecting plane: • is perpendicular to the axis, section outline is a circle. • makes a greater angle to axis than the sloping surface, section outline is an ellipse. • makes the same angle to axis as sloping surface, section outline is a parabola. • makes a lesser angle to axis than the sloping surface, section outline is a hyperbola.

20. Conic sections • The true shape of conic sections can be found by projecting an auxiliary view from the edge view of the cutting plane; see construction 5.48, p.126. • The ellipse, parabola and hyperbola may also be constructed as a result of the ratio (eccentricity) between directrix and focus; Figure 5.9, p.125, illustrates these processes.

21. Conic sections • Ellipse – a closed symmetrical curve with a changing diameter which varies between a maximum (major axis) and minimum (minor axis) length; see Figure 5.10, p.127. • 5.49 To construct an ellipse (approximate methods), p.127. • 5.50 To draw an ellipse by the concentric circles method, given the major and minor axis, p.128.

22. Conic sections • 5.51 To draw an ellipse by the intersecting arcs method, given the major and minor axes, p.128. • 5.52 To draw an ellipse by the rectangular method, p.129. • 5.53 To draw an ellipse with the aid of a trammel, given the major and minor axes, p.129.

23. Conic sections Parabola – can be found in reflection of light beams, vertical curves in highways, arch profiles, cable curves and suspension bridges. • 5.54 To construct a parabola, given the directrix and the focus, p.130. • 5.55 To construct a parabola, given the axis and the base, p.130.

24. Conic sections • 5.56 To construct a parabola, given the rise and the span, p.131. • 5.57 To construct a parabolic curve between any two lines which meet, p.131. • 5.58 To locate the focus of a parabola, given two points on the curve and its vertex, p.131.

25. Conic sections Page 132: • 5.59 To construct a hyperbola, given the foci and the transverse axis (practical method). • 5.60 To construct a hyperbola, given the two foci and the transverse axis (geometrical method). • 5.61 To draw a hyperbola, given the asymptotes and a point on the curve.

26. Summary This chapter has demonstrated how to construct curves and shapes commonly found in engineering drawings and to understand the principle and construction of cams, conic sections and templates.