DEVELOPMENT OF SURFACES
CLASS NOTES FOR ENGINEERING
Cut one surface of the solid. Open
the entire solid on a sheet. It is development
of surfaces. It applies to those objects made
out of a thin sheet. Use true lengths in the
development. Many objects need development
for manufacture. It saves time, labor and
wastage of material. There are four methods of
development of surfaces. In engineering, sheet
metal work need development. It includes
(ix) Transition pieces
METHODS OF DEVELOPMENT OF SURFACES
(a) Parallel line development of surfaces
Used in the development of prisms and cylindrical objects. In this, the lines drawn represent parallel surfaces as in prisms, cubes, cylinders. Omit the bases of the objects from the development.
Fig. Development of a truncated cylinder
Draw the front and top views of the cylinder. Divide the top view into 12 equal parts. Project the points to the front view. Take the length of line 1-1 equal to πD (circumference of the cylinder). Divide it in 12 equal parts equal to chord length ab, using a bow divider. Make the generators. Draw horizontal lines through points a’, b’ to meet the generators at points A, B etc.. Draw a smooth curve through these points. the portion 1ABCD….A1-1 is the final development.
(b) Radial line development of surfaces
Fig. Development of a Square Pyramid (Radial Method)
Used in the development of conical and pyramids solids. In this, apex becomes is the center. Its slant edge as the radius for its development. Find the true lengths of the slant edges for the development of a pyramid. These slant edges are not parallel to the reference planes.
The development of the curved surface of a cone is a sector of a circle. Its radius is equal to the slant height and length equal to the circumference of the base circle. It is difficult to measure the length of the arc. Use any one of the two methods to find the arc length.
(i) Calculate the angle subtended by the arc at center as given below:
θ =3600 x (Radius of base circle/ Slant height of cone)
(ii) Divide the arc using a bow divider into same number of equal divisions as the base circle. It is slightly approximate.
DEVELOPMENT OF OBLIQUE SOLIDS
Used to develop the lateral surfaces of an oblique solid like a pyramid. In this, it is absolutely essential to determine the true lengths of slant edges of the pyramid. Final development uses true lengths of slant edges and base edges.
(c) Triangulation Development Method
Used in the development of transition portions of solids. In this, divide the surface into a suitable number of triangles. Place these triangles outside side by side after finding the true length of each side of the triangle. Transition pieces connect pipes of two different shapes of cross section. The two different transition pieces may have equal or unequal areas.
(d) Approximate method of development
Used in the development of double or multi curved surfaces like a spherical solid. Use an approximate method for its development. Divide the surfaces of such a solid into number of narrow conical or cylindrical segments. Do their approximate development. There are two different methods of development of a sphere. These are
(i) Zone Method
Fig. Sphere Development by Zone Method
In this, cut the sphere into number of horizontal zones. Consider each of these a frustum of a cone. Its apex is at the intersections of the extended chords. Now develop each zone by the radial line method. In Fig. top half of a sphere is divided into four zones A, B, C & D by lines ST, RU, QV and PW. For the development of zone C, Join QR and UV. Produce these lines to meet at point O (center of development). From O, draw two arcs of radii OU and OV. Now develop by radial method as already explained.
(ii) Lune Method
Fig. Development of a Sphere by Lune Method
In this, cut the sphere into a number of equal meridian sections named as lunes. Divide the top view of the sphere into 12 equal sections. There are 12 lunes. For the complete development of the sphere, develop each lune completely. The length of each lune is equal to half the circumference of the sphere. The width of each lune is equal to the arc length.
DO THE DEVELOPMENT OF SURFACES
- Cubical solid
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