Problem 3.9 Engineering Curves – Construct a parabola by parallelogram method with the base dimension 140 mm and height 100 mm. The base of the parabola makes an angle of 25° with the horizontal. And also draw the tangent and normal to the parabola at any suitable point.

Problem 3.9 Engineering Curves – Construct a parabola by parallelogram method with the base dimension 140 mm and height 100 mm. The base of the parabola makes an angle of 25° with the horizontal. And also draw the tangent and normal to the parabola at any suitable point.

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                                                      Procedure:

Step-1 Draw a horizontal line of some length and then draw a major axis inclined at an angle of 25° with the previously drawn horizontal line, of the length 140 mm and give the notations A & B as shown in the figure. And mark a midpoint C on it.

Step-2 Draw a vertical axis, perpendicular to the horizontal line& passing through the point C; of the length equal to the length of minor axis, which is 100 mm and give the notations C & D as shown in the figure.

Step-3 Then make a rectangle of the sides 140mm X 100mm, passing through the end points of the major and minor axes i.e., ABCD, as shown in the figure

Step-4 Divide the major axis AB into 12 equal divisions and give the notations as 1, 2, 3 etc. as shown in the figure. And also divide the two vertical sides passing through the point A & B, into 6 equal divisions and give the notations as 1’,2’,3’,4’ etc. as shown in the figure.

Step-5 Now converge all the points 1’, 2’, 3’, 4’ etc. at the point D with straight lines. And from the points 1,2,3,4 etc. draw straight vertical lines such that the lines should intersect to the lines D1’, D2’, D3’ etc. respectively, as shown in the figure. And give the notations as P1, P2, P3 etc. respectively as shown in the figure.

Step-6 Draw a smooth free hand medium dark curve passing through the points P1, P2, P3 etc. as shown in the figure on both sides of the minor axis; in sequence, so the resulting curve is the parabola.

Step-7 Now mark a point anywhere on the parabola; i.e., the point M, and from this point M draw a horizontal straight line intersecting with the line CD and give the notation as point F. Then extend the line CD in upward direction up to the length equal to DF and give the notation as the point E as shown in the figure. Then form the point E draw a straight line passing through the point M of some suitable length and at the ends give the notations T-T’. This is the tangent to the parabola. Then draw a line of some suitable length and passing through the point M, which is perpendicular to the previously draw tangent T-T’ and give the notations N-N’, this is normal on the parabola passing through the point M, as shown in the figure.

Step-8 Give the dimensions by any one method of dimensions and give the name of the components by leader lines wherever necessary.

Problem 3.10 Engineering Curves – Construct a parabola by tangent method with the base dimension 140 mm and height 100 mm.

Problem 3.10 Engineering Curves – Construct a parabola by tangent method with the base dimension 140 mm and height 100 mm.

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                                                Procedure:

Step-1 Draw a horizontal major axis of the length 140 mm and give the notations A & B as shown in the figure. And mark a midpoint C on it.

Step-2 Draw a vertical axis, perpendicular to the horizontal axis & passing through the point C; of the length equal to the length of minor axis, which is 100 mm and give the notations C & D as shown in the figure. Like in the same way extend the line CD as DE of the length equal to 100 mm, as shown in the figure.

Step-3 Then connect the point E with the points A & B by straight inclined lines as shown in the figure. And divide these two lines AE & BE in to 10 equal divisions. Now give the notations on these points as 1,2,3 etc. but in opposite manner as shown in the figure.

Step-5 Connect these points 1-1, 2-2, 3-3 etc. by straight lines as shown in the figure.

Step-6 Draw a smooth free hand medium dark curve starting from the point A and intersecting the lines 1-1, 2-2, 3-3 etc. by tangent and ending at the point B as shown in the figure. This is the required parabola.

Step-7 Give the dimensions by any one method of dimensions and give the name of the components by leader lines wherever necessary.

Problem 3.4 Engineering Curves – A string is unwound from a circle of 30 mm radius. Draw the locus or Involute of circle of the end of the string for unwinding the string completely. String is kept tight while being unwound. Draw normal and tangent to the curve at any point.

Problem 3.4 Engineering Curves – A string is unwound from a circle of  30 mm radius. Draw the locus or Involute of circle of the end of the string for unwinding the string completely. String is kept tight while being unwound. Draw normal and tangent to the curve at any point.

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                                           Procedure:

Step-1 Draw a circle of the given dimension that is 30 mm radius in this problem.

Step-2 Divide this circle into 12 equal divisions and give their notations.

Step-3 Draw a horizontal line from the bottom point of  the circle, which is numbered 12; of the length equal to the circumference of the circle, which is equal to πd = 188 mm.

Step-4 Divide this horizontal line into the same number of divisions as that of the circle which is 12. And give the notations p0, p1, p2, p3 etc. up to p12 as shown into the figure.

Step-5 Now from the points 1,2,3 etc. of the circle draw tangent lines of some suitable length.

Step-6 Cut the line -1 tangent to the circle with the point-1 as center and the radius equal to the distance between the points p0 &  p1 on the horizontal line and mark the cutting point as p1 as shown into the figure.

Step-7 Like in the same way mark the points up to p12.

Step-8 Draw a smooth medium dark free hand curve passing through the points p0, p1, p2, p3 etc.  up to p12  in sequence to get Involute of the circle.

Step-9 To draw a tangent and normal to the curve mark a point say M on the involute of the circle and from this point M draw a line connecting to the center of the circle, with this line MO draw a perpendicular line in clockwise direction which will intersect on circumference of the circle, through this point from the circumference of the circle draw a medium dark line passing through the point M , which is normal to the curve and draw a line which is perpendicular to the normal of the curve and passing through the point M is tangent of the curve.

Step-9 Give the dimensions by any one method of dimensions and give the name of the components by leader lines wherever necessary.

Problem 3.5 Engineering Curves – Construct an Archemedian spiral for one and half convolution. The greatest and the least radii being 50 mm and 14 mm respectively. Draw tangent and normal to the spiral at a point 40 mm from the center.

Problem 3.5 Engineering Curves – Construct an Archemedian spiral for one and half convolution. The greatest and the least radii being 50 mm and 14 mm respectively. Draw tangent and normal to the spiral at a point 40 mm from the center.

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Procedure:

Step-1 Draw a horizontal axis of the length equal to 100 mm. And mark the center point O on it.

Step-2 Draw a vertical axis bisecting and perpendicular to the horizontal axis passing through the point O.

Step-3 With O as center and radii equal to 50 mm and 14 mm respectively draw two circles.

Step-4 Divide these circles into 12 equal divisions. And give the notations 0, 1,2,3, etc. up to 18 because of one and half convolution of the curve as shown into the figure.                                  

Note: You can give notations in any direction either in clockwise or anticlockwise.


Step-5 Now divide the distance between the two circles, which is 50 mm – 14 mm = 36mm, on the horizontal axis into the same number of division as of the circle, which is 18 because of one and half convolution. So, the distance between the two consecutive divisions is 2 mm.

Step-6 With O as center and radius equal to O1 on the horizontal axis draw an arc between the respective divisional lines of the circle OO & O1 as per the figure given above. Like in the same way draw 18 arcs.

Step-7 Draw a smooth medium dark free hand curve from the end points of the previously drawn arcs in sequence to get Archimedian Curve.

Step-8 To draw normal and tangent to the curve mark a point say K to the curve at the given distance which is 40 mm form the center O by a compass. Draw a line starting from this point K to the center of the circle O. With this line KO draw a perpendicular line of the length equal to value of the formula given below:

X = Distance between the two radius vectors in mm/ Difference of these two radius vectors in radians.

Here it is selected as OO – O3, which is equal to 3.81 mm. Now from the end point of this line draw a medium dark line which passes through the point K, that is normal to the curve. And draw a perpendicular line to the normal and passing through the point K which is tangent to the curve.

Step-9 Give the dimensions by any one method of dimensions and give the name of the components by leader lines wherever necessary.

Problem 3.7 Engineering Curves – Draw an ellipse by oblong method. Size of the rectangle is 140 mm X 100 mm. Draw the tangent and normal to the ellipse at any suitable point.

Problem 3.7 Engineering Curves – Draw an ellipse by oblong method. Size of the rectangle is 140 mm X 100 mm. Draw the tangent and normal to the ellipse at any suitable point.

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                                                 Procedure:

Step-1 Draw a horizontal major axis of the length 140 mm and give the notations A & B as shown in the figure. And mark a midpoint O on it.

Step-2 Draw a vertical axis, perpendicular to the horizontal axis & passing through the point O; of the length equal to the length of minor axis, which is 100 mm and give the notations C & D as shown in the figure.

Step-3 With the center C or D and length equal to the half of the length of major axis; which is 70 mm; cut the major axis on two sides of the minor axis and give the notations F1& F2 respectively as shown in the figure. These are the focal points of the ellipse.

Step-4 Then make a rectangle of the sides 140 mm X 100 mm, passing through the end points of the major and minor axes i.e., ABCD, as shown in the figure

Step-5 Divide the major axis AB into 10 equal divisions and give the notations as 1’, 2’, 3’ etc. as shown in the figure. And also divide the upper half of the two vertical lines passing through the point A & B, into 5 equal divisions and give the notations as 1,2,3,4 etc. as shown in the figure.

Step-6 Now set all the points 1,2,3,4 etc. at the point C with straight lines. And from the point D draw straight lines such that the lines should start form the point D-should pass through the points 1’, 2’, 3’ etc. and intersect to the lines C1, C2, C3 etc. respectively, as shown in the figure. And give the notations as P1, P2, P3 etc. respectively as shown in the figure.

Step-7 Draw a smooth free hand medium dark curve passing through the points P1, P2, P3 etc. as shown in the figure on both sides of the major axis; in sequence, so the resulting curve is the ellipse.

Step-8 Now mark a point anywhere on the ellipse; i.e., the point M, and connect this point with the focal points F1& F2 with straight lines. Then bisect the angle F1MF2 and draw a line of suitable length and give the notations N – N’ as shown in the figure. This is normal passing through the point M on the ellipse. Then draw a line which is tangent to the previously draw normal and give the notations T-T’, this is tangent passing through the point M on ellipse.

Step-9 Give the dimensions by any one method of dimensions and give the name of the components by leader lines wherever necessary.

Problem 3.8 Engineering Curves – Construct a parabola by rectangle method with the base dimension 140 mm and height 100 mm. And also draw the tangent and normal to the parabola at any suitable point.

Problem 3.8 Engineering Curves – Construct a parabola by rectangle method with the base dimension 140 mm and height 100 mm. And also draw the tangent and normal to the parabola at any suitable point.

Cick on it to zoom

                                                Procedure:

Step-1 Draw a horizontal major axis of the length 140 mm and give the notations A & B as shown in the figure. And mark a midpoint C on it.

Step-2 Draw a vertical axis, perpendicular to the horizontal axis & passing through the point C; of the length equal to the length of minor axis, which is 100 mm and give the notations C & D as shown in the figure.

Step-3 Then make a rectangle of the sides 140mm X 100mm, passing through the end points of the major and minor axes i.e., ABCD, as shown in the figure

Step-4 Divide the major axis AB into 12 equal divisions and give the notations as 1, 2, 3 etc. as shown in the figure. And also divide the two vertical sides passing through the point A & B, into 6 equal divisions and give the notations as 1’,2’,3’,4’ etc. as shown in the figure.

Step-5 Now converge all the points 1’, 2’, 3’, 4’ etc. at the point D with straight lines. And from the points 1,2,3,4 etc. draw straight vertical lines such that the lines should intersect to the lines D1’, D2’, D3’ etc. respectively, as shown in the figure. And give the notations as P1, P2, P3 etc. respectively as shown in the figure.

Step-6 Draw a smooth free hand medium dark curve passing through the points P1, P2, P3 etc. as shown in the figure on both sides of the minor axis; in sequence, so the resulting curve is the parabola.

Step-7 Now mark a point anywhere on the parabola; i.e., the point M, and from this point M draw a horizontal straight line intersecting with the line CD and give the notation as point F. Then extend the line CD in upward direction up to the length equal to DF and give the notation as the point E as shown in the figure. Then form the point E draw a straight line passing through the point M of some suitable length and at the ends give the notations T-T’. This is the tangent to the parabola then draw a line of some suitable length and passing through the point M, which is perpendicular to the previously draw tangent T-T’ and give the notations N-N’, this is normal on the parabola passing through the point M, as shown in the figure.

Step-8 Give the dimensions by any one method of dimensions and give the name of the components by leader lines wherever necessary.