Introduction

In engineering practice, it is often required to make a drawing that shows the interior details of the object. If the object is simple in its construction, the interior portion of the object can be easily interpreted by dotted lines in the orthographic projections. When the dotted lines of hidden parts are too many, the views become confusing and hard to read. In such cases, views can be drawn by cutting the object by an imaginary cutting plane so as to expose its interior or hidden details. The part of the object between the cutting plane and the observer is assumed to be removed so as to show the internal constructional features or details of the invisible surface. The exposed interior details are drawn in continuous thin lines instead of dotted lines. Such views are known as sectional views or views in section. The section surfaces are indicated by section lines, evenly spaced and inclined at 45° to the reference line.

Section Planes

These are generally perpendicular planes. These may be perpendicular or parallel to one of the principal planes and either perpendicular, parallel or inclined to the other plane. These planes are usually described by their traces.

Sections

Basically, sections are of two types:

• (i) Apparent Section

• (ii) True Section

• (i) Apparent Section:The projection of the section on the principal plane to which the section plane is perpendicular is a straight line coinciding with the trace of the section plane on it, whereas its projection on the other plane to which it is inclined is called apparent section.

• (ii) True Section:The projection of the section on a plane parallel to the section plane shows the true shape of the section. When the section plane is parallel to the HP or ground plane, the true shape of the section is seen in the sectional top view. When it is parallel to the VP, the true shape is projected in the sectional front view.

But when the section plane is inclined to one of the principal planes, the section has to be projected on an auxiliary plane parallel to the section plane to obtain its true shape.

When the section plane is perpendicular to both the principal planes, the sectional side view shows the true shape of the section.

Introduction

Drawing is a graphic language by which communication is accomplished through sketches (drawings). Drawings can be of various types. If compared with verbal or written description, drawings provide a far better idea about the shape, size and appearance of any object or situation or location, that too fairly quickly. From a manufacturing point of view, 2D and 3D drawings are very important and commonly used in engineering industry. Drawings are prepared manually or by using a computer. In 2D view (orthographic projection), one view is not enough to get all the details of the object. So it is necessary to draw the front view, top view, bottom view, right side view and left side view.

To be more useful, in particular from the manufacturing point of view, the sketch or drawing should include dimensions, manufacturing details, materials used, etc. Drawing equipment and instruments are needed to record information on drawing paper or any other suitable surface. Drawing, mainly consisting of straight lines, curves, circles and arcs, is prepared with certain instruments. The quality of the drawing mainly depends on the quality of the instruments, their adjustment, handling and care. Therefore, the equipment must be reliable and accurate, as the same will result in good quality drawing, which will further enhance a student's interest. The correct selection and use of pencils and drawing instruments should be taught throughout the course. Beginners certainly need guidance on the selection and purchase of the drawing instruments and equipment essential for drafting. The various instruments and other drafting equipment are described below.

List of Draughting Tools

The following is the list of draughting tools which every student must possess:

1. Drawing Board

2. Mini-Draughter

3. Small Instrument Box, containing the following:

• • Large Size Compass

• • Small Bow Compass

• • Large Size Divider

• • Small Bow Divider

4. Set-Squares

5. Set of Scales

6. Protractor

7. French Curves

8. Drawing Sheets

9. Drawing Pencils

10. Paper Fasteners

11. Pencil Sharpener

12. Sand Paper Pad

13. Eraser

14. Duster

Some of these have been explained in greater detail in the subsequent sections.

Drawing Board

A drawing board is usually made of well-seasoned soft wood. To prevent warping, narrow strips of wood are glued together.

Introduction

Although a pictorial view helps in understanding the shape of the object, however it suffers from the drawback that it fails to convey the actual size and inner details of the object. This is because a pictorial view is drawn by seeing the object, as a three directional task. However, for design purpose, it requires the actual details of the object. For this purpose, the pictorial view of the object is converted into orthographic views by applying the principles of orthographic projections. Conversion of a pictorial view into the orthographic view requires sound knowledge of the principles of pictorial projection and some imagination.

Direction of Sight

For converting a pictorial view of an object into orthographic views, the direction from which the object is to be viewed for its front view is generally indicated by means of an arrow. The arrow must be parallel to the sloping axis. If there is no arrow, the direction for the front view may be decided to give the most prominent view; other views are drawn by looking in the directions perpendicular to the first direction.

Orthographic Views

Orthographic views can be drawn by two methods:

• (i) First-Angle Projection Method

• (ii) Third-Angle Projection Method

First-Angle Projection Method

In the first-angle projection method, the object lies in the first quadrant, i.e., above the HP and in front of the VP. The object lies in between the observer and the plane of projection. In this method, when the views are drawn in their relative positions, the top view is placed below the front view and the left side view is placed to the right side of the front view. Thus in the first-angle projection method, either of the side views is so placed that it represents the side view of the object away from it, as shown in Fig. 17.1.

In the same way as discussed above, three more views may be obtained by placing the plane of projection on the front, top and left hand side of the object. The three views then obtained are called rear or back view, bottom view and right side view, respectively. The layout of all the six views on the drawing sheet is shown in Fig. 17.2.

Introduction

Three views of an object, viz. the front view, top view and side view, are sometimes not sufficient to provide complete information regarding true shape and size of an object. Additional views are therefore projected on other planes (auxiliary planes) and are known as auxiliary views or auxiliary projections.

Types of Auxiliary Planes and Views

Auxiliary planes are of two types:

• (a) Auxiliary Vertical Plane (AVP)

• (b) Auxiliary Inclined Plane (AIP)

Auxiliary vertical plane (AVP):Auxiliary vertical plane (AVP) is perpendicular to the HP and inclined to the VP. The projection on an AVP is called an auxiliary front view. See Fig. 11.1.

Auxiliary inclined plane (AIP):Auxiliary inclined plane (AIP) is perpendicular to the VP and inclined to the HP. The projection on an AIP is called an auxiliary top view. See Fig. 11.2.

For showing the orthographic projections of an object, the auxiliary plane should always be rotated about the plane to which it is perpendicular.

Projections of Points

(a) Projection of a point on an auxiliary vertical plane (AVP):A point A is situated above HP and in front of the VP. AVP is a plane perpendicular to the HP and inclined to the VP. The HT of this plane is inclined to x-y and VT perpendicular to x-y line. The HP and the AVP meet at right angles in the line x1 y1. From the Figs. 11.3 (a) and 11.3 (b), the following points may be observed:

• (i) The distance of the auxiliary front view from x1y1 is equal to the distance of the front view from x-y, which in turn is the distance of the point A from the HP.

• (ii) The line x1 y1 is inclined to x-y at an angle ϕ, which is the angle of inclination of the AVP with the VP.

To draw the orthographic projections, see Fig. 11.3 (c).

• (i) Draw the reference line x-y and mark the front view a’ and the top view a.

• (ii) Draw a new reference line x1 y1, making an angle ϕ with x-y.

Introduction

In this chapter, an attempt is being made to introduce the readers to the projection of points. Following the treatment here it will be easy for them to understand the projection of lines, planes and solids in the subsequent chapters. For the projection of points, the quadrant system is considered and a point lying in space is assumed in any one of the four quadrants that are obtained by the intersection of two principal planes. A point can lie with reference to both the reference planes, i.e., HP and VP. Its projections are obtained by extending projectors perpendicular to the planes.

In order to obtain the projection of a point lying in three-dimensional space on a two-dimensional plane (drawing sheet), the principal plane HP is rotated clockwise through 90° and made co-planner with the VP. This process coverts the three-dimensional quadrant system into two-dimensional front and top views, i.e., the front view VP is obtained above x-y line where this x-y line represents the elevation or front view of HP; similarly the top view of HP is obtained below the x-y line, here the x-y line represents the top view of VP.

Projection of a Point Lying in the First Quadrant

The pictorial view Fig. 8.1(a) shows a point A lying in the first quadrant, i.e., above the HP and in front of the VP. When the point is viewed in the direction of l, the view from front a’ is obtained as the intersection point between the ray of sight through A and the VP. When the point is viewed in the direction m, the top view (a) of point ‘A’ is obtained as the ray of sight intersect with HP at a. Similarly front view (a’) of ‘A’ is obtained when the ray of sight coming from l direction intersects VP at a'. Hold VP and rotate HP 90° in the clockwise direction; these projections are seen in the Fig. 8.1(b). The front view a’ is above x-y and top view below it.

The line joining a’ and a (which is called as projector) intersects x-y at right angle (90°) at a point a.

Introduction

Planes or surfaces are objects that have two dimensions, i.e., length and breadth; they have negligible thickness. Plane surfaces may be considered of infinite sizes. However, for convenience, segments of planes are only considered in the solutions. Planes are represented in space by either of the following:

• • Three non-collinear points, Fig. 10.1(a)

• • A line and a point, Fig. 10.1(b)

• • Two intersecting lines, Fig. 10.1(c)

• • Two parallel lines, Fig. 10.1(d)

• • A plane, Fig. 10.1(e)

Types of Planes

Planes are mainly of two types:

• • Principal Planes

• • Secondary Planes

Principal planes:The planes on which the projections are obtained are called the principal planes. Examples of principal planes are horizontal and vertical planes.

Secondary planes:Secondary planes are of two types:

• (i) Perpendicular planes

• (ii) Oblique planes

(i) Perpendicular planes: These planes can be divided into the following sub-types:

1. Perpendicular to both the principal planes

2. Perpendicular to one of the principal planes and parallel to the other plane

3. Perpendicular to one of the principal planes and inclined to the other plane

1. A plane perpendicular to both the principal planes.A square plane ABCD is perpendicular to both the principal planes. Its horizontal trace (HT) and vertical trace (VT) are in a straight line perpendicular to the reference line x-y, as shown in Fig. 10.2. The elevation, b'c', and plan, ab, of the square are both straight lines coinciding with VT and HT, respectively, i.e., VT and elevation, HT and plan overlapping.

2. Perpendicular to one of the principal planes and parallel to the other plane.

• (a) A plane perpendicular to HP and parallel to the VP. A square lamina ABCD is perpendicular to the HP and parallel to the VP. Its HT, is parallel to x-y and it has no VT. The front view a'b'c'd’ shows the true shape and size of the square object. The top view ab is a line, parallel to x-y, coinciding with HT, as shown in Fig. 10.3.

• (b) Plane, perpendicular to VP and parallel to the HP. A square ABCD is perpendicular to the VP and parallel to the HP. Its VT is parallel to x-y and it has no HT. The top view abcd shows the true shape and size of the square object. The front view d'c’ is a line, parallel to x-y, coinciding with VT, as shown in Fig. 10.4.

Introduction

There is a huge number of objects of different shapes which have dimensions in the three directions, i.e., length, breadth and thickness. However, these three-dimensional objects are difficult to describe on a two-dimensional sheet. Therefore, in the language of graphics, the shape of an object is described by its projection. A projection is the image of an object formed by rays of sight coming from the same direction as the object. The image is taken on a picture plane, and it appears to an observer stationed at a point from or towards which the projection is made. Therefore, in order to draw the projections of an object the following elements are considered:

• (i) An object or body (point, line, plan, solid)

• (ii) The projection plane (HP, VE, PP)

• (iii) The point of sight (Observer)

• (iv) The lines of sight or rays of sight

Methods of Projections

In an engineering drawing, four methods of projection are generally used, which are as follows:

• (i) Orthographic Projections

• (ii) Axonometric Projections

• (iii) Oblique Projections

• (iv) Perspective Projections

The first type, i.e., orthographic projection, is one of the most prevalent methods, in which an object can be represented by two or more than two views, on mutually perpendicular planes of projections or picture planes. However, a maximum of six views (front, top, rear, bottom and both left and right side) are used to represent an object. Each view represents two dimensions of an object. For the complete description of a three-dimensional object, at least two views are required. Whereas, in the other three methods, the objects are represented by a pictorial view only, a three-dimensional object is represented on a projection plane by single view only.

Perspective projection

Perspective projection represents objects as perceived by the human eye(s) (refer to Fig. 7.1). It is a pictorial drawing by the intersection of observer's visual rays (lines of sight) converging on a plane (picture plane). The observer's eye-station point or point of sight-is located at a finite distance from the picture plane (refer to Fig. 7.1). Depending on the position of the picture plane, the size of the projection may vary.

Introduction

In engineering practice, it is usual to draw two or more than two orthographic projections to show the true shape and size of an object. Such drawings can be correctly interpreted only by those persons who have a sound knowledge of the principles of orthographic projections. To make the drawing more understandable, several forms of one plane projection drawings are used to supplement the orthographic drawings. These are called pictorial drawings, which can be easily understood by everybody without any formal training. Pictorial drawings are mainly used to show complicated shapes such as aircraft, ships, buildings, etc.

Classification of Pictorial Drawings

Pictorial drawings are classified into three categories:

• (a) Axonometric Projection

• (b) Oblique Projection

• (c) Perspective Projection

Axonometric Projection:It is a three-dimensional projection of an object, as shown in Fig. 16.1 (a). It is a form of orthographic projection, since the projectors are perpendicular to the plane of projection and parallel to each other.

Oblique Projection:It is a three-dimensional projection of an object on a projection plane, as shown in Fig. 16.1 (b). In this, the projectors are parallel to each other, but are oblique to the picture plane.

Perspective Projection:It is the most realistic projection. In this, the projectors converge towards the viewer's eye, making different angles to the picture plane, as shown in Fig. 16.1 (c).

Axonometric Projection

It is a form of orthographic projection and is obtained by projecting an object placed in an oblique position to the plane of projection. Axonometric projection is classified as:

• • Isometric Projection

• • Dimetric Projection

• •Trimetric Projection

Out of these, only isometric projection is dealt with here.

Isometric Projection

It is a type of pictorial drawing in which the three dimensions of a solid are not only shown in one view, but their actual sizes can be measured directly from it.

If a cube is resting on one of its corners on the HP, with its solid diagonal perpendicular to the VP, the front view of the cube is its isometric projection as shown in Fig. 16.2.

This text book is an endeavour to deal with the subject of Engineering Graphics and Drawing in such a way that students can understand the subject thoroughly. The subject of Engineering Drawing and (Computer) Graphics is a core course taught to the first year students of all disciplines of engineering in all engineering colleges and universities. Therefore, this text book is an attempt to help students grasp the basic concepts of engineering drawing clearly and easily.

The book has many distinguishing features. It covers the fundamental concepts of manual and computer aided drafting. It includes more than 400 solved examples and 275 practice exercises along with 250 short answer questions, i.e., objective type questions with answers.

The organisation of this text is done very clearly and logically. Each chapter is organised as basic theory, solved problems, exercises and objective type questions. Each chapter contains a large number of worked examples, the problems for which have been selected from examinations of different universities. This book comprises 19 chapters, starting with an introduction to drawing instruments and their uses, followed by Chapter 2 on various types of lines and their uses and layout of a drawing sheet. Chapter 3 attempts to give knowledge about the various methods and principles of dimensioning of a drawing. The description about sections and conventions is given in Chapter 4, followed by Chapter 5 on geometrical constructions. From here onward, various scales are described in Chapter 6 and then the concept of orthographic projection is described in Chapter 7. Chapter 8 of the text is dedicated to projections of points, followed by projections of lines in Chapter 9. Once students are clear about the basic concepts of projections of points and lines, then this book introduces them to the projections of planes and auxiliary projections in Chapters 10 and 11, respectively.

Chapter 12 discusses the projections of solids followed by a lucid explanation of the sections of solids in Chapter 13: this will imbibe an imagination of the sections view of a solid object or engineering component. Students of first year in almost all branches of engineering are also offered a course on manufacturing processes and practices, where they need to do some jobs on sheet metal. Chapter 14 covers development of surfaces that will help students develop understanding of a pattern and calculate the requirement of sheet metal for mass production.

Introduction

The term ‘intersection of surfaces’ is used when the surfaces of two solids intersect each other, such as of cylinders, cones, prisms, pyramids, etc. The term ‘interpenetration of solids’ is used when three-dimensional solids such as cylinders, cones, prisms, pyramids, etc., intersect each other. The term ‘intersection of surfaces’ is used for all situations, irrespective of the fact that the solids interpenetrating each other have plane or curved surfaces.

When two solids interpenetrate each other, their surfaces meet in a common line called the line or curve of intersection. This line of intersection of two surfaces may be joined by determining key points and joining these points in the correct order. The resulting line may be straight or curved depending upon the nature of the intersecting surfaces. Determination of the line of intersection between two solids is important, as their development into flat patterns is otherwise not possible. The term ‘development’ refers to the unfolded sheet from which the required objects can be formed without stretching the material, i.e., hoppers, pipe joints, etc. In view of the above mentioned applications, intersection of surfaces may be classified into three categories:

• (i) Intersection of two plane surfaces, viz. prism and prism, prism and pyramid.

• (ii) Intersection of two curved surfaces, viz. cone and cylinders, cylinder and cylinder, cone and cone.

• (iii) Intersection of a plane surface and a curved surface, viz. prism and cylinder, pyramid and cylinder, etc.

Methods of Determining Line of Intersection

The following two methods may be used to find the line of intersection:

• (i) Line Method or Piercing Point Method

• (ii) Cutting Plane Method

• (i) Line Method or Piercing Point Method:When two solids meet each other, elements on one solid intersect with the surface of the other solid. Points of intersection of these solids with the surface of the other solid are then located. These points are the vertices of the line of intersection. These points are more easily located from the view in which the lateral surface of the solid appears edgewise, i.e., a line. Then, these points from one view are projected on to the other view. The curve joining these points will be a line of intersection.

• (ii) Cutting Plane Method:This is the general method of finding the line of intersection of any two surfaces. In this method, the solids are assumed to be cut by a series of cutting planes.

Introduction

A line can be defined as the route or pathway of a point displaced from one position to another. A line can be a curved one or a straight one. A straight line is the shortest track or distance between two points. The projections of a straight line can be drawn by joining the respective projections of its end points. The position of a straight line in space may be defined by

• • the location of its end points or extremities from principal planes and its true length (TL).

• • the location of one of its end points and inclination of the line with the principal planes.

• • the location of its end points and distance between the end projectors, etc.

Position of a Straight Line

A line in space can be in various positions w.r.t. to the principal planes of projection as shown in Table 9.1:

(i) Line parallel to both HP and VP.

(ii) Line inclined to one plane and parallel to the other

• • Line inclined to the HP and parallel to the VP.

• • Line inclined to the VP and parallel to the HP.

(iii) Line perpendicular to one of the planes.

• • Line perpendicular to the HP and parallel to the VP.

• • Line perpendicular to the VP and parallel to the HP.

(iv) Line contained by one or both of the principal planes

• • Line contained by the HP.

• • Line contained by the VP.

• • Line contained by both HP and VP, i.e., in x-y line.

(v) Line inclined to both HP and VP.

(vi) Line contained by a profile plane (PP) or line contained by a plane, perpendicular to both HP and VP, i.e., in x-y line.

Line Parallel to Both HP and VP

Line AB, parallel to both HP and VP, is shown in Fig. 9.1 (a). Here ends A and B of the line AB are at equal distance from the HP and VP. When a line is parallel to any plane, its projections on that plane is a straight line of the same length. The elevation (front view) a'b’ and plan (top view) ab are both parallel to x-y line and their lengths represent the true length (AB of a line).

Fig. 9.1 (b) shows the orthographic projection of the line AB.

Introduction

There are a number of simple geometrical constructions with which a draughtsman or an engineer should be familiar, as these are essential in the preparation of engineering drawing figures. Though these geometrical constructions are generally studied in the lower classes, they are being discussed here because of their importance in engineering drawing.

Bisection of a Straight Line

Dividing a Line into Equal Parts

Method I

The points 1, 2, 3, etc., divide the line AB into six equal parts.

Method II

Introduction

The topics covered in this chapter are meant to introduce the readers to various types of lines, lettering and layout of sheets, which are used in engineering drawing.

Lines

A line is the basis of engineering drawing; a set of conventional symbols representing all the lines needed for different purposes may well be called an alphabet of lines. In engineering drawing each line has specific measurements and functions. The Bureau of Indian Standards (BIS) has recommended various types of lines and their applications in technical drawing. These are exhibited in Table 2.1.

Lettering

Lettering is an important feature of all engineering drawings, which is meant for indicating notes, dimensions, etc., on a drawing. Despite the fact that a drawing may be made accurately and neatly, poor lettering may lead to deteriorated appearance and sometimes make the drawing useless. Therefore, lettering on a drawing needs to be legible, neat in appearance and in appropriate style. The subsequent sections describe various types of letters.

Single Stroke Letters

BIS has recommended single stroke lettering for use in engineering drawing. In single stroke letters, the thickness of a letter is such that it is obtained in one stroke of the pencil. Horizontal line (stroke) is from left to right, and vertical or inclined lines (strokes) are from top to bottom.

Single stroke letters are of two types:

• • Upper case or capital letters

• • Lower case or small letters

Again both of them are further sub-divided into two categories:

• • Vertical

• • Inclined

In vertical lettering, stems are perpendicular to the line of lettering, whereas in inclined lettering, stems make a slope of 75° to the horizontal. Normally, letters are nominated by their height; therefore height of the letters enables us to classify the letters as

• • Lettering ‘A’

• • Lettering ‘B’

The standard ratios for line thickness (d) to height (h), d/h, as 1/14 and 1/10 are most prevalent as they result in a minimum number of thicknesses, as given in Table 2.2.

Introduction

The previous chapters of the book have described the fundamental concepts of manual drawing, using various drawing instruments and producing a hardcopy of the drawing. This traditional drawing, however, still remains the foundation for engineering communication, as the subject of computer aided drafting (CAD) has been developed on the fundamentals of traditional drawing. So the advent of CAD does not necessarily eliminate manual engineering graphics.

There are several reasons for implementing a computer aided design system. A few of them are:

• • To increase the productivity of the designer

• • To improve the quality of design

• • To improve communication

• • To create a database for manufacturing

Computer Graphics

The term ‘computer graphics’ is very generic in the sense that it has different meanings for different disciplines of study. In the operation of the graphics system by the user, a variety of activities take place, which can be divided into three categories: interaction with graphics terminal to create and alter images on the screen; construction of a model of something physical and of the image on the screen; and entering the model into computer memory. In working with the graphics system, the user performs these various activities in combination rather than sequentially.

Requirements for Computer Graphics

The basic requirements are a CAD workstation and requisite software. CAD workstation comprises a common PC with minimum Pentium configuration, a video display, a mouse or pen, and a plotter or printer. The graphics software can be divided into three modules: the graphic package, the application program and the application database. Various requirements of graphics software are:

• • It should be easy to use.

• • The package should operate in a consistent and predictable way, for the user.

• • Graphics programs should be efficient.

• • It should not be too expensive to make its use prohibitive.

This chapter explores the AutoCAD package for its basic drafting capabilities.

Getting Started with AutoCAD

Start AutoCAD by clicking on the Windows start button placed at the bottom left, then move the mouse to programs, click on AutoCAD as shown in Fig. 19.1 or it can directly be opened by double clicking on the desktop icon of AutoCAD.

Introduction

Freehand sketching is the technique of making drawing without the use of drawing instruments. It is one of the efficient ways employed to communicate ideas and designs. A designer records his ideas critically in the form of sketches which are later converted into drawings.

It is usually assumed that skill in sketching may be acquired more easily than proficiency in instrumental drawings. However, a lot of effort and practice is required to sketch two parallel lines, a circle and an ellipse, etc. Proficiency in sketching can be achieved only with constant practice.

Sketching Materials

The following materials are required for making good sketches:

• (i) A fairly soft pencil, particularly HB or H

• (ii) A soft eraser

• (iii) A suitable paper

The pencil used for sketching should have a conical point. A good quality eraser should be used that will not spoil the paper. Depending upon the conditions of work and the purpose of a drawing, a variety of papers are used for sketching. Rectangular or square coordinate papers and isometric ruled papers are generally used by beginners. This helps the beginners in sketching straight lines and keeping a drawing to a good scale. As such, papers may not always be readily available; it is important to start sketching with plain paper too.

Uses of Sketches

These are as follows:

• (i) To help the designer in developing new ideas

• (ii) To convey the ideas of the designer to the draughtsman and management

• (iii) To provide a basis for discussion between engineers and workmen

• (iv) To serve as a teaching aid in the classroom

Sketching Straight Lines

Generally, the shape of an object consists of straight and curved surfaces which are represented by straight and curved lines respectively. The lines may be either horizontal, vertical or inclined. Horizontal lines are sketched with the motion of the wrist and the forearm. These are sketched from left to right. Vertical lines are sketched downwards with the movement of the fingers. Inclined lines (which are nearly horizontal) are sketched from left to right, whereas inclined lines (which are nearly vertical) are sketched downward.

The first step in sketching any straight line is to mark the end points of a line. A light line with one or series of strokes may then be tried between the end points. Fig. 18.1 shows the directions in which these lines must be sketched to attain the straightness.

Introduction

In engineering drawing, it is often required to make drawings that show the interior details of an object, which are not visible to the observer from outside. The method of showing the interior details by hidden lines makes the view confusing and difficult to understand, as shown in Fig. 4.1. To overcome this difficulty, complicated objects are assumed to be cut by an imaginary plane. The part of the object between the imaginary cutting plane and the observer is assumed to be removed, as shown in Fig. 4.2. The exposed or cut surfaces are identified by section lines or hatching lines. The view obtained is called the sectional view, as shown in Fig. 4.3. Hidden lines and details behind the section are usually omitted from the sectional view, unless they are required for clarity. The sectional view is very important in daily life, as it helps in manufacturing and explaining the construction of complicated machines and their parts.

Cutting Plane or Sectional Plane

The imaginary plane by which an object is assumed to be cut is called the cutting plane or the sectional plane. The cutting plane used in sectioning is indicated by a line in a view adjacent to the sectional view. This line is called the cutting plane line. It is a long chain that is thin, but is thick at the ends. Arrowheads, put at the ends, indicate the direction in which the cut away object is viewed, i.e. the direction of sight as shown in Fig. 4.4. At the end of the cutting plane line, capital letters such as X – X, Y – Y or A – A are often marked to identify the cutting plane line with corresponding section. When the cutting plane line coincides with a centre line, the cutting plane line takes precedence over a centre line. In case of symmetrical objects, the cutting plane line is omitted, unless it is needed for clarity.

Section Lines or Hatching Lines

The lines used to represent the material which has been cut by a cutting plane are called section lines. These are also called hatching lines. Section lines are thin, equally spaced, parallel and inclined lines drawn in the area of the view where a cutting plane cuts the material of the object.

Introduction

Every drawing consists of the views necessary to describe the shape of an object, and must provide its exact length, width, height, size, position of holes and any other details required for the manufacturing of the object. Providing this information on a drawing is called dimensioning.

Types of Dimensions

A drawing usually requires two types of dimensions:

• (i) Size or functional dimension

• (ii) Location or datum dimension

• (i) Size or functional dimension: It gives the size of a piece or component as shown in Fig. 3.1. It is usually represented by the letter S.

• (ii) Location or datum dimension: It fixes the relationship of the component parts (holes, slots, etc.) of a piece of structure. It may be from centre to centre, surface to surface or surface to centre, as shown in Fig. 3.1. Generally it is denoted by the letter L.

Elements of Dimensioning

The elements of dimensioning include the extension or projection line, dimension line, leader line, arrowhead and the dimension itself. The various elements of dimensioning are shown in Fig. 3.2.

Execution of Dimensions

The points given below have to be always followed while executing dimensions:

Placing Dimensions

Dimensions may be placed according to either of the following two systems:

Deterministic evolution is a hallmark of classical mechanics. Given a set of exact initial conditions, differential equations evolve the trajectories of particles into the future and can exactly predict the location of every particle at any instant in time. So what happens if our uncertainties in the initial position or velocity of a particle are tiny? Does that mean that our uncertainties about the subsequent motion of the particle are necessarily tiny as well? Or are there situations in which a very slight change in initial conditions leads to huge changes in the later motion? For example, can you really balance a pencil on its point? What has been learned in relatively recent years is that, in contrast to Laplace’s vision of a clock-like universe, deterministic systems are not necessarily predictable. What are the attributes of chaos and how can we quantify it? We begin our discussion with the notion of integrability, which ensures the absence of chaos.