General Principles of Design, Drawing & Importance of Safety Short Notes Part 2

By Sachin Singh|Updated : December 2nd, 2019

Engineering curves (conics

Introduction 

  • Curves formed by the intersection of a plane with a right circular cone. e.g. Parabola, hyperbola and ellipse.
  • Right circular cone is a cone that has a circular base and the axis is inclined at 90o to the base and passes through the centre of the base. 

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(Figure shows a right cone and the various conic curves that can be obtained from a cone by sectioning the cone at various conditions)

Conic is defined as the locus of a point moving in a plane such that the ratio of its distance from a fixed point and a fixed straight line is always constant.

  • Fixed point is called Focus
  • Fixed line is called Directrix

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(Figure. illustrates the directrices and foci of a conic curve)

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Eccentricity

Type of Conic Section

e<1

Ellipse

e=1

Parabola

e>1

Hyperbola

 

Ellipse

  • An ellipse is obtained when a section plane, inclined to the axis of the cone, cuts all the generators of the cone.

Parabola

  • A parabola is obtained when a section plane, parallel to one of the generators cuts the cone.
  • Some of the applications are in searchlight mirrors, telescopic mirrors, a beam of uniform strength in design applications, the trajectory of the weightless flight, etc.

Hyperbola

  • A Hyperbola is obtained when a section plane, parallel/inclined to the axis cuts the cone on one side of the axis.
  • A Rectangular Hyperbola is obtained when a section, parallel to the axis cuts the cone on one side of the axis.

Hyperbolic shapes finds large number of industrial applications like the shape of cooling towers, mirrors used for long distance telescopes, etc.

Spirals

  • A spiral is a curve traced by a point moving along a line in one direction, while the line is rotating
    in a plane about one of its ends or any point on it. 
  • The point which generates the curve is called the generating point or tracing point.
  • The point will move along a line called the radius vector while the line itself rotates about one of its end points.

Generally, for engineering applications, two types of spirals are encountered. They are:

Archimedian Spiral: The curve traced out by a point moving in such a way that its movement towards or away from the pole is uniform with the increase in the vectorial angle from the starting line.
Applications include teeth profile of helical gears, profile of cams, etc.

Logarithmic Spiral: The ratio of the lengths of consecutive radius vectors enclosing equal angles is always constant. 
i.e. the values of the vectorial angles are in arithmetic progression and the corresponding values of radius vectors are in geometric progression.

 

Golden ratio

  • Two quantities are in golden ratio if their ratio is same as ratio of their sum to the larger of the two quantity. It is is also called golden mean or golden section or extreme and mean ratio or divine proportion or divine section.

Mathematically,

(a+b)/b = a/b =  Φ

where Φ is Golden Ratio

Roulettes

  • Roulettes are curves generated by the rolling contact of one curve or line on another curve or line.
  • There are various types of roulettes: Cycloids, trochoids, and Involutes.
  • Depending on the position of the point and the geometry of the surface on which the wheel rolls, different curves are obtained.

Cycloid:  Cycloid is generated by a point on the circumference of a circle rolling along a straight line without slipping. 
Epicycloid:  The cycloid is called Epicycloid when the generating circle rolls along the circumference of another circle outside it. 
Hypocycloid: Hypocycloid is obtained when the generating circle rolls along the circumference of another circle but inside it.

Classification of a Cycloidal curve

 

Generating Circle

 

On the directing line

Outside the directing line

Inside the directing line

Generating  point

On the generating circle

Cycloid

Epicycloid

Hypocycloid

Outside the generating circle

Superior trochoid

Superior epitrochoid

Superior Hypotrochoid

Inside the generating circle

Inferior trochoid

Inferior epitrochoid

Inferior hypotrochoid

 

Trochoids

  • Trochoid is a curve generated by a point fixed to a circle, within or outside its circumference, as the circle rolls along a straight line. 
  • If the point is outside the rolling circle, the curve obtained is called an inferior trochoid and when outside the circle is called superior trochoid.

 

Involute 
An Involute is a curve traced by the free end of a thread unwound from a circle or a polygon in such a way that the thread is always tight and tangential to the circle or side of the polygon.

Theory of Projections

In engineering, 3-D objects and structures are represented graphically on a 2-dimensional media. The act of obtaining the image of an object is termed “projection”.  The image obtained by projection is known as a “view”.

Projection Methods

  1. Parallel Projection
  • Distance from the observer to the object is infinite projection lines are parallel – object is positioned at infinity.
  • Less realistic but easier to draw.
  1. Perspective projection 
  • Distance from the observer to the object is finite and the object is viewed from a single point – projectors are not parallel.
  • Perspective projections mimic what the human eyes see, however, they are difficult to draw.

Orthographic Projection

Orthographic projection is a parallel projection technique in which the plane of projection is perpendicular to the parallel line of sight.

Multi-view Projection

In an orthographic projection, the object is oriented in such a way that only two of its dimensions are shown. The dimensions obtained are the true dimensions of the object .

Orientation of views from projection planes - Multi-view drawings gives the complete description of an object. Top view is always positioned and aligned with the front view, and side view is always positioned to the side of the Front view and aligned with the front view. Depending on whether 1st angle or 3rd angle projection techniques are used, the top view and Front view will be interchanged. Also the position of the side view will be either towards the Right or left of the Front view.

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Figure. Relative positions and alignment of the views in a multi-view drawing.

Projection Methods

  1. First Angle Projection

In this the object in assumed to be positioned in the first quadrant and the object is assumed to be positioned in between the projection planes and the observer.

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Figure. Illustrating the views obtained using first angle projection technique 

  1. Third Angle Projection

In the third angle projection method, the object is assumed to be in the third quadrant. i.e. the object behind vertical plane and below the horizontal plane. In this projection technique, Placing the object in the third quadrant puts the projection planes between the viewer and the object.

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Difference between first- and third-angle projections

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Either first angle projection or  third angle projection are used for engineering drawing.  Second angle projection and fourth angle projections are not used since the drawing becomes complicated.

Symbol of projection- The symbol recommended by BIS is to draw the two sides of a frustum of a cone placed with its axis horizontal The left view is drawn.

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Axonometric Projection:

  • In this projection, only one view showing all the three dimensions of an object is drawn on a plane of projection (POP).
  • The object is so placed that its three mutually perpendicular edges will remain inclined to the plane of projection (POP).
  • Axonometric projection is a parallel projection technique to create a pictorial drawing of an object by rotating the object on an axis relative to a projection.

Isometric Projection:

  • In this projection, the three mutually perpendicular edges of an object makes equal inclination with plane of projection (POP) because of which true shape of the object is obtained.
  • In this isometric scale is used due to which its true size cannot be ascertained.

Diametric Projection:

  • In this projection two of the three mutually perpendicular edges of an object are equally inclined to the plane of projection (POP).
  • In this two different diametric scales are used.

Trimetric projection:

  • In this projection, all the three mutually perpendicular edges of an object make different inclination with plane of projection (POP).
  • In this three different trimetric scales are used.

Oblique Projection:

  • In oblique projection, the object is aligned such that one face (front face) is parallel to the projection plane.
  • In such projection, the projectors are not perpendicular to the plane of projection rather inclined to the plane of projection (POP) at 30°, 45° or 60°. Here oblique axis is called as receding axis. In oblique projection, projectors from various points on the object are drawn parallel to each other and inclined to the plane of projection.

Projection of Points

  • The position of a point in engineering drawing is defined with respect to its distance from the three principle planes i.e., with respect to the VP, HP, & PP.

Projections of lines

  • The location of a line in projection quadrants is described by specifying the distances of its endpoints from the VP, HP and PP.  A line may be:
  • Parallel to both the planes.
  • Parallel to one plane and perpendicular to the other.
  • Parallel to one plane and inclined to the other.
  • Inclined to both the planes.
  • The projection of a line can be obtained by projecting its end points on planes of projections and then connecting the points of projections. The projected length and inclination of a line can be different compared to its true length and inclination.

Traces of a line

  • The trace of a line is defined as a point at which the given line, if produced, meets or intersects a plane.
  • When a line meets HP, (or if necessary, on the extended portion-of HP), the point at which the line meets or intersects the horizontal plane, is called horizontal trace (HT)of the line and denoted by the letter H.
  • When a line meets VP (or if necessary, on the extended portion of VP), the point at which the line meets or intersects the vertical plane, is called vertical trace (VT) of the line and denoted by the letter V.
  • When the line is parallel to both HP and VP, there will be no traces on the said planes.

Projections on Auxiliary Planes 

Sometimes none of the three principal orthographic views of an object show the different edges and faces of an object in their true sizes, since these edges and faces, are not parallel to any one of the three principal planes of projection.  In order to show such edges and faces in their true sizes, it becomes necessary to set up additional planes of projection other than the three principal planes of projection in the positions which will show them in true sizes. These additional planes of projection which are set up to obtain the true sizes are called Auxiliary Planes. The views projected on these auxiliary planes are called Auxiliary Views.

Next - Projections of Solids

ESE 2019 Prelims Paper I - Revision Plan

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