UNIT 6
Engineering Curves
QUESTION BANKS
- Draw ellipse by concentric circle method. Take major axis 100 mm and minor axis 70 mm long.
Steps:
1. Draw both axes as perpendicular bisectors of each other & name their ends as shown.
2. Taking their intersecting point as a center, draw two concentric circles considering both as respective diameters.
3. Divide both circles in 12 equal parts & name as shown.
4. From all points of outer circle draw vertical lines downwards and upwards respectively. 5.From all points of inner circle draw horizontal lines to intersect those vertical lines.
6. Mark all intersecting points properly as those are the points on ellipse.
7. Join all these points along with the ends of both axes in smooth possible curve. It is required ellipse.
2. Draw ellipse by Rectangle method. Take major axis 100 mm and minor axis 70 mm long.
Steps:
1 Draw a rectangle taking major and minor axes as sides.
2. In this rectangle draw both axes as perpendicular bisectors of each other..
3. For construction, select upper left part of rectangle. Divide vertical small side and horizontal long side into same number of equal parts.( here divided in four parts)
4. Name those as shown..
5. Now 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.
6. Then extend C-1 line upto D-1 and mark that point. Similarly extend C-2, C-3, C-4 lines up to D-2, D-3, & D-4 lines.
7. 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.
8. Join all points in smooth curve. It is required ellipse.
3. Draw ellipse by Oblong method. Draw a parallelogram of 100 mm and 70 mm long sides with included angle of 750.Inscribe Ellipse in it.
(STEPS ARE SIMILAR TO THE PREVIOUS CASE (RECTANGLE METHOD) ONLY IN PLACE OF RECTANGLE, HERE IS A PARALLELOGRAM)
4. MAJOR AXIS AB & MINOR AXIS CD ARE 100 AMD 70MM LONG RESPECTIVELY .DRAW ELLIPSE BY ARCS OF CIRLES METHOD.
STEPS:
1.Draw both axes as usual. Name the ends & intersecting point
2.Taking AO distance I.e. half major axis, from C, mark F1 & F2 On AB . ( focus 1 and 2.)
3.On line F1 - O taking any distance, mark points 1,2,3, & 4
4.Taking F1 center, with distance A-1 draw an arc above AB and taking F2 center, withB-1 distance cut this arc. Name the point p1
5.Repeat this step with same centers but taking now A-2 & B-2 distances for drawing arcs. Name the point p2
6.Similarly get all other P points. With same steps positions of P can be located below AB.
7.Join all points by smooth curve to get an ellipse/
5. Draw an isosceles triangle of 100 mm long base and 110 mm long altitude.Inscribe a parabola in it by method of tangents.
Solution Steps:
1. Construct triangle as per the given dimensions.
2. Divide it’s both sides in to same no.of equal parts.
3. Name the parts in ascending and descending manner, as shown.
4. Join 1-1, 2-2,3-3 and so on.
5. Draw the curve as shown i.e.tangent to all these lines. The above all lines being tangents to the curve, it is called method of tangents.
6. Point P is 40 mm and 30 mm from horizontal and vertical axes respectively.
Solution Steps:
1) Extend horizontal line from P to right side.
2) Extend vertical line from P upward.
3) On horizontal line from P, mark some points taking any distance and name them after P-1, 2,3,4 etc.
4) Join 1-2-3-4 points to pole O. Let them cut part [P-B] also at 1,2,3,4 points.
5) From horizontal 1,2,3,4 draw vertical lines downwards and
6) From vertical 1,2,3,4 points [from P-B] draw horizontal lines.
7) Line from 1 horizontal and line from 1 vertical will meet at P1 .Similarly mark P2 , P3 , P4 points.
8) Repeat the procedure by marking four points on upward vertical line from P and joining all those to pole O. Name this points P6 , P7 , P8 etc. and join them by smooth curve. Draw Hyperbola through it.
7. Draw Involute of a circle. String length is equal to the circumference of circle.
Solution Steps:
1) Point or end P of string AP is D distance away from A.exactly Means if this string is wound round the circle, it will completely cover given circle. B will meet A after winding. D (AP) distance into 8
2) Divide number of equal parts.
3) Divide circle also into 8 number of equal parts.
4) Name after A, 1, 2, 3, 4, etc. up D line AP as well as onto 8 on circle (in anticlockwise direction).
5) To radius C-1, C-2, C-3 up to C-8 draw tangents (from 1,2,3,4,etc to circle).
6) Take distance 1 to P in compass and mark it on tangent from point 1 on circle (means one division less than distance AP).
7) Name this point P1
8) Take 2-B distance in compass and mark it on the tangent from point 2. Name it point P2.
9) Similarly take 3 to P, 4 to P, 5 to P up to 7 to P distance in compass and mark on respective tangents and locate P3, P4, P5 up to P8 (i.e. A) points and join them in smooth curve it is an INVOLUTE of a given circle.
8. DRAW LOCUS OF A POINT ON THE PERIPHERY OF A CIRCLE WHICH ROLLS ON A CURVED PATH. Take diameter of rolling Circle 50 mm And radius of directing circle i.e. curved path, 75 mm.
Solution Steps:
1) When smaller circle will roll on larger circle for one revolution it will D distance on arc and it willcover .be decided by included arc angle = (r/R) x by formula
2) Calculate 3600. with radius OC3) Construct angle and draw an arc by taking O as center OC as radius and form sector of angle .
4) Divide this sector into 8 number of equal angular parts. And from C onward name them C1, C2, C3 up to C8.
5) Divide smaller circle (Generating circle) also in 8 number of equal parts. And next to P in clockwise direction name those 1, 2, 3, up to 8. 6) With O as center, O-1 as radius draw an arc in the sector. Take O-2, O3, O-4, O-5 up to O-8 distances with center O, d
Raw all concentric arcs in sector.
Take fixed distance C-P in compass, C1 center, cut arc of 1 at P1. Repeat procedure and locate P2, P3, P4, P5 unto P8 (as in cycloid) and join them by smooth curve. This is EPI – CYCLOID.
9. Draw a spiral of one convolution. Take distance PO 40 mm.
Solution Steps
- With PO radius draw a circle and divide it in EIGHT parts. Name those 1,2,3,4, etc. up to 8
- 2 .Similarly divided line PO also in EIGHT parts and name those 1,2,3,-- as shown.
3. Take o-1 distance from op line and draw an arc up to O1 radius vector. Name the point P1
4. Similarly mark points P2 , P3 , P4 up to P8 And join those in a smooth curve. It is a SPIRAL of one convolution.
10. Draw a helix of one convolution, upon a cylinder. Given 80 mm pitch and 50 mm diameter of a cylinder. (The axial advance during one complete revolution is called The pitch of the helix)
SOLUTION:
- Draw projections of a cylinder.
2.Divide circle and axis in to same no. Of equal parts. ( 8 ) Name those as shown.
3. Mark initial position of point ‘P’ Mark various positions of P as shown in animation.
4.Join all points by smooth possible curve.
5. Make upper half dotted, as it is going behind the solid and hence will not be seen from front side.
