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Mechanics work

1. Electrical and Electronics Engineering Department EEE113 Electrical Engineering Drawing Dr. M.E Smith

2. Engineering curves 2 EEET Department. Electrical Engineering Drawing Introduction Conic section Ellipse Parabola Hyperbola

3. Engineering curves 3 . EEET Department. Electrical Engineering Drawing Curves which are commonly used in engineering drawing are: Conic section Involute Cycloidal curves Evolutes Spirals Helix.

4. Engineering curves 4 EEET Department. Electrical Engineering Drawing Conic Section Cone A cone is a surface generated by the rotation of a straight line whose one end is in contact with a fixed point while the other end is in contact with a closed curve, not lying in the plane of the curve. Vertex or Apex: The point of intersection of the axis and the generator is known as vertex or apex of the cone. Generator: The revolving line which generates the surface of the cone is known as the generator. Axis: The fixed line about which the other line revolves is known as the axis of the cone.

5. Engineering curves 5 . EEET Department. Electrical Engineering Drawing Conic Sections The sections obtained by cutting off a right circular cone by section planes at different angles relative to its axis are known as conic sections. Examples of conic section Circle Ellipse Parabola hyperbola

6. Engineering curves 6 . EEET Department. Electrical Engineering Drawing Circle: When a cutting plane AA is perpendicular to the axis and cuts all the generators, the section obtained is known as circle. Ellipse: When a cutting plane BB is inclined to the axis of the cone and cuts all the generators on one side of the apex, the section obtained is known as an ellipse. Parabola: When a cutting plane CC is inclined to the axis of the cone and parallel to one of the generators, the section obtained is known parabola Hyperbola: When the cutting plane DD makes a smaller angle with the axis then that of the angle made by the generator of the cone, the section obtained is known as hyperbola

7. Engineering curves 7 . EEET Department. Electrical Engineering Drawing

8. Engineering curves 8 . EEET Department. Electrical Engineering Drawing Eccentricity, Focus and Directrix Focus (F): The fixed point is called Focus. Directrix (DD): The fixed line is called Directrix. Axis (CA): Axis is the line passing through the focus and perpendicular to the directrix. Vertex (V): Vertex is a point at which the conic cuts its axis.

9. Engineering curves 9 . EEET Department. Electrical Engineering Drawing Conic: a conic section is the locus of a point moving in a plane such that the ratio of its distance from a fixed point to a fixed straight line is always a constant Ellipse: Ellipse is the locus of a that point moves so that its distance from a fixed point called the focus bears a constant ratio, always less than 1 to its perpendicular distance from a straight line called the directrix.

10. Engineering curves 10 . EEET Department. Electrical Engineering Drawing Parabola: It is the locus of a point moving in a plane such that the ratio of its distance from a fixed point (F) to the fixed straight line (DD) is a constant and is always = 1. Hyperbola: It is the locus of a point moving in a plane in such a way that the ratio of its distance from a fixed point (F) to the fixed straight line (DD) is constant and is greater than one.

11. Engineering curves 11 . EEET Department. Electrical Engineering Drawing The eccentricity is always: < 1 for Ellipse (0.1, 0.27, 0.75, etc ) = 1 for Parabola > 1 for Hyperbola (1.5, 5.3, 7.3, etc ) When e = 1/2, the curve is an Ellipse, when e =1, it is a parabola and when e = 2, it is a hyperbola.  

12. Engineering curves 12 . EEET Department. Electrical Engineering Drawing Ellipse Uses: Elliptical shapes are used in the construction of: Arches Bridges Dams elliptical gears of textile machines and printing presses ends of cylindrical tanks top or bottom of manholes Many other artifacts.

13. Engineering curves 13 . EEET Department. Electrical Engineering Drawing Construction of Ellipse By Concentric Circle Method Draw an ellipse by concentric circle method. Take major axis 100 mm and minor axis 70 mm long. Draw the given minor axis RS and major axis PQ . Draw two concentric circles with centre O using each axis as diameter. Divide each circle into 12 equal parts. Where these radial lines cut the outer circle, draw vertical lines and where the radial lines cut the inner circle, draw horizontal lines. Draw a smooth curve through these points.

14. Engineering curves 14 EEET Department. Electrical Engineering Drawing To inscribe an ellipse in a rectangle EFGH when EF = 120 mm and EG = 85 mm. Steps: Draw a rectangle taking major and minor axes as sizes. In the rectangle draw both axes as perpendicular bisectors of each other. For construction, select upper left part of rectangle. Divide vertical small side and horizontal long side into same number of equal parts . 5. Join all vertical points 1,2,3,4, to the upper end of minor axis. And all horizontal points i.e.1,2,3,4 to the lower end of minor axis. Extend C-1 line up to D-1 and mark that point. Similarly extend C-2, C-3, C-4 lines up to D-2, D-3, & D-4 lines. Mark all these points properly and join all along with ends A and D in smooth possible curve. Do similar construction in right side part along with lower half of the rectangle. Draw a smooth curve through these points

15. Engineering curves 15 EEET Department. Electrical Engineering Drawing STEPS ARE SIMILAR TO THE PREVIOUS CASE (RECTANGLE METHOD) ONLY IN PLACE OF RECTANGLE, HERE IS A PARALLELOGRAM. Draw ellipse by Oblong method. Draw a parallelogram of 100 mm and 70 mm long sides with included angle of .Inscribe Ellipse in it.  

16. Engineering curves 16 . EEET Department. Electrical Engineering Drawing Taking a focus 50 mm from the directrix construct (a) an ellipse, eccentricity, e =2/3. Draw the directrix AB and the axis OX perpendicular to AB. The focus is located on the axis 50 mm from O. Since the eccentricity is 2/3, the distance OF is divided into five equal parts. The vertex V is located on OF such that (VF)/(VO) = 2/3. Draw VE perpendicular to the axis such that VE = VF. Join OE and extend it. Draw a line inclined at 45 to the axis at F to cut the extension of OE at G. Draw a line perpendicular to the axis from G to locate the other vertex V1 of the ellipse. Mark points 1, 2, 3, 4, 5 and 6 between V and V1 along the axis.

17. Engineering curves 17 . EEET Department. Electrical Engineering Drawing Draw perpendiculars through these points above and below the axis. The perpendiculars intersect the line OEG at 1’, 2’, 3’, 4’, 5’ and 6’. With centre F and radius 11’, draw arcs to cut the perpendicular drawn through 1 at P1 above and P1’ below the axis. Similarly, points P2, P2’, P3, P3’, P4, P4’, P5, P5’, P6 and P6’ are obtained following the same procedure. Draw a smooth curve passing through the points V, P1, P2, P3, P4, P5, P6, V1, P6’, P5’, P4’, P3’, P2’, P1’ and V to obtain the curve ellipse.

18. Engineering curves 18 . EEET Department. Electrical Engineering Drawing

19. Engineering curves 19 . EEET Department. Electrical Engineering Drawing “Arcs of Circles” Method Draw the given minor and major axis. Mark the Foci and . With centre R and radius PO, draw arcs cutting PQ at and . Mark a number of points 1, 2, 3 on O . With centre and radius , draw arcs on both sides. With the centre and radius , draw arcs intersecting on both sides at and . Similarly, obtain other points and draw the required ellipse  

20. Engineering curves 20 . EEET Department. Electrical Engineering Drawing Draw Rhombus of 100 mm & 70 mm Long Diagonals and Inscribe an Ellipse in it. Steps: Draw rhombus of given dimensions. Mark mid points of all sides & name as A,B,C,& D Join these points to the ends of smaller diagonals. Mark points 1,2,3,4 as four centers. Taking 1 as center and 1-A as radius draw an arc AB. Taking 2 as center draw an arc CD. Similarly taking 3 & 4 as centers and 3-D radius draw arcs DA & BC.

21. Engineering curves 21 . EEET Department. Electrical Engineering Drawing PARABOLA Parabola is used for suspension bridges, reflectors for parallel beams such as head lights of automobiles, solar concentrators, etc. Path of a thrown missile and path of a jet of water issuing from vertical high are of parabolic shape.

22. Engineering curves 22 . EEET Department. Electrical Engineering Drawing To draw the parabola using focus and directrix . Steps Draw the directrix as a vertical line and the axis as a horizontal line. Mark the focus F on the axis at a distance 40 mm from the directrix. Mark the vertex V at the mid-point of AF such that VF/AV = 1 to get the parabola. Draw a vertical line from V and mark C on it such that VC = VF, then draw a line from A passing through C for convenient length. Draw a vertical line at any distance and mark 1 on the axis and 1′ on the inclined line AC. Use the length 1 – 1′ as radius, focus F as centre, draw an arc to cut the line 1 – 1′ at M1 and N1. Repeat this procedure by drawing vertical lines 2 – 2′, 3 – 3′ etc. and get M2, N2, M3, N3 ... etc. Join these points by drawing a smooth curve to complete the parabola.

23. Engineering curves 23 . EEET Department. Electrical Engineering Drawing To draw tangent and normal to the parabola Mark a point P on the parabola and join it with focus F. Then draw a line from F at 90° to the line FP to get B on the directrix. Draw a line from B passing through P which is the tangent to the parabola. Draw another line through P, perpendicular to the tangent is a normal to the curve.

24. Engineering curves 24 . EEET Department. Electrical Engineering Drawing A ball is thrown into the air and it attained a height of 100mm and return to the ground covering a horizontal distance of 150 mm. Trace the path created by the ball. Steps: Rectangle Method Draw a rectangle of the above size and divide it into two equal vertical parts Consider left part for construction. Divide height and length in equal number of parts and name them 1,2,3,4,5 & 6 Join vertical 1,2,3,4,5 & 6 to the top center of rectangle Similarly draw upward vertical lines from horizontal 1,2,3,4,5 And wherever these lines intersect previously drawn inclined lines in sequence join them with a smooth curve. Repeat the construction on right side rectangle also. Join all in sequence. This locus is Parabola.

25. Engineering curves 25 . EEET Department. Electrical Engineering Drawing 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 5 4 3 2 1

26. Engineering curves 26 . EEET Department. Electrical Engineering Drawing “Tangent” method Draw the base PQ . Draw the vertical axis AB at the mid-point A . Produce AB to O so that AB = BO . Join OP and OQ . Divide the line OP and OQ into the same number of equal parts and mark them. Join 1 with 1′, 2 with 2′, 3 with 3′, etc. Draw a curve joining tangent to lines 1 – 1′, 2 – 2′, 3 – 3′, 4 – 4′ etc.

27. Engineering curves 27 . EEET Department. Electrical Engineering Drawing Hyperbola Hyperbola is used in the design of cooling towers, hydraulic channels, electronic transmitters and receivers like radar, antenna, etc. Rectangular hyperbola is used to represent the Boyle’s Law expansion curve or the Theoretical Indicator Diagram of an Engine.

28. Engineering curves 28 . EEET Department. Electrical Engineering Drawing Example The vertex of a hyperbola is 65 mm from its focus. Draw the curve if the eccentricity is . Draw a tangent and normal at any point on the curve.     ,   To draw the hyperbola using focus and directrix Draw the directrix as a vertical line and the axis as a horizontal line. Mark the vertex V at a distance 43.3 mm from directrix and the focus F 65 mm from the vertex V. Draw a vertical line from V and mark C on it such that VC = VF , then draw a line from A passing through C for convenient length.

29. Engineering curves 29 . EEET Department. Electrical Engineering Drawing Draw a vertical line at any convenient distance and mark 1 on the axis and 1′ on the inclined line AC . Use the length 1 – 1′ as radius, focus F as centre, draw an arc to cut the line 1 – 1′. Repeat this procedure by drawing vertical lines 2 – 2′, 3 – 3′ etc. Join the points by drawing a smooth curve. To draw tangent and normal to the hyperbola Mark a point P on the curve and join focus F and P , then draw a line from F at 90° to the line FP to get B on the directrix. Draw a line from B passing through P which is the tangent to the hyperbola. Draw another line through P , perpendicular to the tangent and which is a normal to the curve.

30. Engineering curves 30 . EEET Department. Electrical Engineering Drawing

31. Engineering curves 31 . EEET Department. Electrical Engineering Drawing Example: Point P is 40 mm and 30 mm from a horizontal and vertical axes respectively. Draw a Hyperbola through it. Steps: Extend horizontal line from P to right side. Extend vertical line from P upward. On horizontal line from P, mark some points taking any distance and name them after P-1, 2,3,4 etc. Join 1-2-3-4 to point O. From horizontal 1,2,3,4 draw vertical lines downwards and from vertical 1,2,3,4 points draw horizontal lines. Line from 1 horizontal and line from 1 vertical will meet at P 1 .Similarly mark P 2 , P 3 , P 4 points. Repeat the procedure by marking four points on upward vertical line from P and joining all those to pole O. Name this points P 6 , P 7 , P 8 etc. and join them by smooth curve.

32. Engineering curves 32 EEET Department. Electrical Engineering Drawing P O 40 mm 30 mm 1 2 3 1 2 1 2 3 1 2

33. Engineering curves 33 . EEET Department. Electrical Engineering Drawing Example A sample of gas is expanded in a cylinder from 10 unit pressure to 1 unit pressure. Expansion follows the gas law PV=Constant. If initial volume is 1 unit, draw the curve of expansion. Also Name the curve. Pressure (P) x Volume (V) Constant (C) 10 x 1 10 5 x 2 10 4 x 2.5 10 2.5 x 4 10 2 x 5 10 1 x 10 10 Form a table giving few more values of P & V Now draw a Graph of Pressure against Volume. It is a PV Diagram and it is Hyperbola. Take pressure on vertical axis and Volume on horizontal axis.

34. Engineering curves 34 EEET Department. Electrical Engineering Drawing VOLUME:( M 3 ) PRESSURE ( Kg/cm 2 ) 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10

35. Engineering curves 35 EEET Department. Electrical Engineering Drawing Example Two points and are located on a plane sheet 100 mm apart. A point P on the curve moves such that the difference of its distances from and always remains 50 mm. Find the locus of the point and name the curve. Mark asymptotes and directrices. locus of the point and name the curve. Mark asymptotes and directrices.  

36. Engineering curves 36 EEET Department. Electrical Engineering Drawing Solution A curve traced out by a point moving in the same plane in such a way that the difference of the distances from two fixed points is constant, is called a hyperbola. Draw a horizontal line and mark the fixed points and in such a way that . Draw a perpendicular bisector to . Mark the points and on the horizontal line such that and . With centre O and radius equal to O draw a circle. Draw tangents at and to intersect the above circle at J, M, K and L as shown. Draw a line joining JOL and produce it and this line is one asymptote. The other asymptote is the line passing through KOM.  

37. Engineering curves 37 EEET Department. Electrical Engineering Drawing Mark any number of points 1,2,3, etc., on the axis of the hyperbola. With as centre and radius equal to 2 draw an arc to cut the arc drawn with as centre and radius equal to 2 . The point of intersection is marked as Similarly obtain other points of intersection etc. It may be noted that Draw a smooth curve passing through the points etc., which is the required hyperbola. Also draw another hyperbola on the other side of the axis as shown  

38. Engineering curves 38 EEET Department. Electrical Engineering Drawing

39. 39 . THE END THANK YOU EEET Department. Electrical Engineering Drawing