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COMPUTER AIDED DESIGN AND ANALYSIS 1.0 INTRODUCTION Computer-aided design/computer-aided manufacturing is the use of computer to integrate design and manufacturing process for fabricating products that meet customer demand by optimizing all elements involved in the life cycle of the product. The aim of CAD/CAM is to minimize product design and changes and the time and cost required in taking the product from design concept to production and introduction of the product into the market. Design and manufacturing have been accomplished sequentially. Manufacturing engineers were given the detailed drawings and specifications of the product and were required to produce it. They often encountered difficulties because the design or product engineers never anticipate the production problems that could occur. This dilemma had been surmounted by the application of CAD/CAM. There now greater flexibility in manufacturing, improve productivity, increase product quality and uniformity. CAD/CAM has tools for prototyping a design and setting up virtual factory for manufacturing without retooling to facilitate fast delivery of process orders placed for different products. Therefore, CAD/CAM provides for just-in-time manufacturing. 1.1 COMPUTETR-AIDED DESIGN Computer-aided design is the use of computers to create or modify drawings and models. It can also be used for analysis of design data, electronic storage, and retrieval of design information. CAD is usually associated with interactive computer graphics, known as CAD system. CAD systems are computer programs using specialized computing hardware. The software normally comprises a number of different elements or functions that process the data stored in the database in different ways. CAD elements and their functions include the following: Model definition: adding geometric elements to a model of the form of a component. Model manipulation: to move, copy, delete, edit or otherwise modify elements in the design model. Picture generation: to generate images of the design model on a computer screen or on some hard-copy device. Uses interaction: to handle commands input by the user and to present output to user about the operation of the system. Database management: management of the files that make up the database. Applications: to generate information for evaluation, analysis or manufacture. Utilities: to modify the operation of the system in some way. These features may be provided by multiple programs operating on a common database, or by a single program encompassing all of the elements.

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1.2 COMPUTER-AIDED ENGINEERING Computer-aided engineering is a system of computerized analytical tools that share a common database with that of geometric models, permitting applications such as finite-element analysis of stresses, strains, deflections, and temperature distribution in structures and load-bearing members and the generation, storage, and retrieval of numerical control (NC) data. These systems are also use extensively in the design of integrated circuit and other electronic devices. CAE is a very powerful tool which offers industry the ability to save time and drudgery in creating drawings, produce better design and faster quotation and to provide automatic machine tool control from design produced on computer. 1.3 HISTORICAL DEVELOPMENT OF CAD/CAM CAD/CAM development was as a result of the need to meet the design and manufacturing requirements of aerospace industries after the Second World War. The manufacturing technology in late 40’s and early 50’s could not meet the design and manufacturing challenges which arose as a result of the need to develop sophisticated aircraft and satellite vehicles. This lead the US Air Force to approach MIT (Massachusetts Institute of Technology) to develop suitable control systems, drives and programming techniques for machine tools using electronic control. The first major innovation in machine control was the Numerical Control (NC). Early Numerical Control systems were basically hardwired systems, and were built with discrete systems or with later first generation integrated chips. Early NC machines used paper tape as an input device. Every NC machine was fitted with a tape reader to read paper tape and transfer the program to memory of the machine tool block by block. Mainframe computers were used to control a group of NC machines by mid 60’s. This configuration was then called Direct Numerical Control (DNC) as the computer bypassed the tape reader to transfer the program data to the machine controller. By late 60’s mini computers were being commonly used to control NC machines. In this phase, NC was soft wired with facilities of mass program storage, off-line editing and software logic control and processing. This development is called Computer Numerical Control. Since 70’s, numerical controllers are being designed using microprocessor, resulting in compact CNC systems. A further development to this technology is the distributed numerical control in which the processing of NC program is carried out in different computers to plant computers to the machine controller. Modern computer are built with 32 bit and 64 bit microprocessors. Manufacturing also started to use computers for several tasks such as inventory control, production planning and control, etc. CNC technology was adapted in the development of co-ordinate measuring machine’s which automated inspection. Robots were introduced to automate several tasks like machine loading, material handling, welding, assembly and painting. All these developments led to the evolution of flexible manufacturing cells and flexible manufacturing systems in the late 70’s. The advent of Computer-aide Design, on the other hand was to facilitate geometric modeling needs in automobile and aeronautical industries. The developments in computers, design workstations, graphic cards, display media and graphic input and output devices during the last

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decades have been tremendous. This coupled with the development of operating system with graphic user interfaces and powerful interactive software packages for modeling, drafting, analysis and optimization provides the necessary tools to automate the design process. 1.4 CAD SYSTEMS DESIGN PROCESS The design process in CAD system consists of four stages as describe below: 1.4.1 GEOMETRIC MODELING: In geometric modeling, a physical object or any of its components is described mathematically. The designer first constructs a geometric model by giving commands through the CPU of the CAD system to create or modify lines, surfaces, solids, dimensions and text that are all together accurate and complete representation of 2D and 3D object. The model can be represented in three different ways; line, surface and solid. In line (wire-frame) representation, all edges are visible as solid lines. This image can be confusing for complex shapes. However, various colours are generally used for different parts of the object to ensure proper visualization. In surface model, all visible surfaces are shown, and in solid model, all surfaces are shown but the data describe the interior volume. There are three types of wire-frame models: 2D image shows the profile of the object or part. 2 ଵଶD image is obtained by translational sweep – that is, moving the 2D object along the z-axis. It

should be noted that a 2 ଵଶD for round object can be generated by simply rotating a 2D model

around the axis. 3D image is often referred to as solid model. It is created from ‘’sweep volume’’ or the use of solid geometry shapes called primitives. Solid modeling in 3d is the most advanced technique of geometric modeling. Although solid models have advantages, such as ease of design analysis and preparation for manufacturing the parts, they require more memory and processing time than the wire-frame and surface models. 1.4.2 ENGINEERING ANALYSIS After the design geometric features have been determined, the design is subjected to an engineering analysis. This analysis may involve stress, strain, deflection, vibration, heat transfer, temperature distribution or tolerance. One of the tools in CAD system used for engineering analysis is finite element method. In this method, the object is discritized into a large number of finite elements of triangular or rectangular shapes which form an interconnecting network of concentrated nodes. The entire object behaviour such as stress-strian, heat transfer and other characteristics can be calculated at of each node. Also the interrelating behaviour of all nodes in the entire object can be assessed. 1.4.3 DESIGN REVIEW AND EVALUTION This is an important stage to check for any interference between various components in order to prevent difficulties during assembly or use of the part, and whether moving members, such as

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linkages, are going to operate as intended. CAD systems contain kinematic packages which provide the capability to animate the motion of design component to identify potential problems associated with moving parts and other dynamic situations. Animation of object enables the designer to properly visualize the mechanism operation and guide against any interference with other components. During design review and evaluation stage, the part is precisely dimensioned and tolerance, as required for manufacturing.

1.4.4 AUTOMATED DRAFTING Automated drafting is concerned with the production of hard-copy engineering drawings directly from CAD system database for documentation and reference. Computer-aided design has some graphic features that are suitable for drafting process. These features include automatic dimensioning, generation of cross-hatched areas, scaling of the drawing, and the capacity to develop sectional views and enlarged views of particular part detailed. It also has the ability to rotate the part of to perform other transformation of the image (i.e oblique, isometric, or perspective view). The early use of computer in design has been justified by increase productivity in drafting function compared to manual drafting.

1.5 USES OF CAD Some of the uses of CAD are: 1. Creation of conceptual product models. 2. Modification of models to improve performance, aesthetics and ergonomics. 3. Display the product in different colours to select proper colour combination most appealing to customers. 4. Observe functional aspects of relative motion of various elements or assemblies. 5. Analyse model behaviour under the influence of load such as stress or thermal load and carry out the necessary design modifications to correct deficiencies in design. 6. Construction of detailed component drawings revealing full detail dimensions, tolerance, surface finishing requirements. 7. Store database for modification or manufacturing. 1.6 JUSTIFICATIONS FOR IMPLEMENTING CAD The following are reasons for implementing CAD system in design: 1. To increase the productivity of designer. CAD systems help designer to visualize the product and its member subassemblies and parts. This reduces the time required to synthesize, analyse and document the design. This reason also reduces the design costs and product development time. 2. To improve the quality of design. The engineering analysis of the design problem can be carried out thoroughly on a CAD system. All design alternatives can be investigated quickly.

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Design errors are reduced by the in-built accuracy of calculations and checks available in the system. This leads to improvement in quality and accuracy of design. 3. To improve documentation. The use of CAD system provides better engineering drawings, more standardization in the drawings, better documentation of the design, fewer drawing errors and grater legibility for the drawing. 4. To create a database for manufacturing. In the process of creating the database for the product design, such as geometry and dimension of parts, bill of materials, etc, much of the required database to manufacture is also created which can be accessed for computer integrated manufacturing (CIM) applications like CNC programming, robots, process planning, etc. 1.7 BENEFITS OF CAD SYSTEM TO DESIGN ENGINEER 1. Productivity improvement in design. CAD helps to increase design productivity by reducing the time for developing conceptual design, analysis and drafting. 2. Shorter lead time. CAD system enables the engineer to prepare a finished set of drawing and documentation in a relatively short time. Shorter lead times in design result in reduction of the elapsed time between customer order and delivery of finished product. 3. Flexibility in design. CAD offers the advantage of easy modification of design to accommodate customer’s specific requirements. 4. Improve design analysis. CAD system helps to optimize the design. The use of analysis software (i.e FEM and kinematic analysis) reduces the time and improves design accuracy and reduces the material use. 5. Easier creation and correction of engineering drawings. The application of solid modeling made it easier to understand the features of the components readily. Software like AutoCAD allows 3D view generation from a 2D model. Drawing can also be corrected and modified quickly as they are easily available on the CAD system. 6. Easier visualization of drawings. Engineering drawings can be visualized easily on the CAD system in several views. This enables the designer to present the details of the object to the customer in the desired fashion. 1.8 BENEFITS OF CAD SYSTEM IN MAUFACTURING 1. Tool and fixture design. 2. Computer-aided process planning (CAPP). 3. Preparation of assembly list and bill of materials. 4. Computer-aided inspection. 5. Coding and classification of components. 6. Production planning and control. 7. Production of programs for NC and CNC machines. 8. Assembly sequence planning.

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1.9 INDUSTIAL APPLICATIONS OF CAD The potential applications of CAD are in the following areas: 1. Computer-aided process planning. 2. Coding of parts and retrieval. 3. Printed circuit design (PCB’s). 4. Finite element analysis. 5. Piping systems for process industries. 6. CNC programming. 7. Architecture – building design. 8. Kinematic and dynamic analysis of mechanisms. 1.10 CAD SYSTEM ARCHITECTURE/FEATURES CAD systems comprise of the following features: 1. Hardware: the computer and associated peripheral equipment. 2. Software: the computer program (s) running on the hardware. 4. Data: the data structure created and manipulated by the hardware. 4. Human knowledge and activities.

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2.0 CAD HARDWARE AND SOFTWARE

2.1 INTRODUCTION Computer-aided design hardware is made up of graphics workstations, graphics input and output devices. Computer-aided software consists of operating system that controls the basic house-keeping operations in the computer, software packages used for geometric modeling, and application software for design analysis and synthesis.

2.2 COMPUTER FUNDAMENTALS A computer is an electric machine which carried out mathematical and logical calculations and data processing function in accordance with a predetermined program of instructions. The physical components of a computer are called hardware, while the various programs are called software. Computers are classified based on operation principles as digital, analog and hybrid computers. A digital computer operates essentially by counting. The quantities are expressed as discrete digits or numbers. An analog computer operates by measuring rather than computing. Hybrid computer combines the features of both analog and digital computers. The higher number of computers used in modern days and for CAD systems are digital type.

Computers can also be classified based on application into special purpose and general purpose computers. The former are designed mainly to meet the requirement of a particular task or application while the latter are designed to meet the need of several applications. Based on sizes and capabilities, computers may be categorized as micro, min, main frame (host) and super computers. A micro computer is the smallest general purpose processing system whose central processing unit (CPU) is a microprocessor. It is a self-contained unit designed for use by one person at a time. The CPU is either 32 or 64 bit word length. Majority of micro computers used in CAD are PC/XT, PC/AT-286, PC/AT-386, PC/AT-486 and Pentium brand. The term XT and AT means extended technology and advanced technology respectively. Mini computers are 32 bit machines and can access 16 MB or more memory. It is designed to serve multiple users simultaneously. Most CAD/CAM turn key systems are based on standard mini computers. The significant feature of mini computers is the availability of virtual memory which allows execution of large programs in stages. This type of facility is very suitable for tasks requiring considerable storage as for finite element analysis. Main frame computers have large storage capacity and very high processing speed. Main frames have 32 or 64 bit word length. They support large number of terminals for use by different users

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simultaneously. They are used in applications requiring substantial data processing at high speeds. Super computers operate on speed between 20 and 400 mega flops (1 flop = 1 million floating points per second). They can be used for very complex CAD applications like solid modeling, kinematics, model analysis, finite element analysis and fluid flow simulation.

2.3 CAD HARDWARE SYSTEM Hardware is the physical and tangible components of a computer used to implement computer programs. Hardware includes the computer with its memory and CPU, disk or magnetic tape for storing programs, the display screens, and various devices for inputting data to computer like keyboards, digitizers and tablets, and the output devices which produce the ‘hard copy’ such as printers and plotters. 2.3.1 COMPUTER HARDWARE COMPONENTS The component of a computer hardware system consists of 1. Central processing unit 2. Input unit 3. Output unit 2.4 CENTRAL PROCESSING UNIT The central processing unit is the heart of a computer system which manipulates the data. The data are presented to CPU as electronic pulses and pauses, corresponding to 1s and 0s (binary digits). The smallest possible piece of information is referred to as ‘bit’. The system can receive a number of bits at the same time to make communication faster. There are 8, 16, 32 and 64 bit systems. In an 8 bit system for example, the CPU receives and send messages called ‘bytes’, such as 10110001, 00010000 etc, one at a time, through different communication lines. These lines form a communication bus on which other several devices are connected such as memory modules, input/output devices, and mass storage units, screens, printers, etc. The CPU consists of the following three parts: 1. Memory unit 2. Arithmetic and logical unit 3. Control unit 2.4.1 MEMORY UNIT The memory unit is used to hold data, instruction and results temporarily. This unit is divided into main memory and auxiliary memory. a) Main memory This is also called primary or internal memory. It is a physical part of the computer connected directly to the CPU. Primary memory is divided into three main categories. i) Main data storage: This includes magnetic cores and semiconductors. The magnetic core is a tiny hollow ferrite ring which magnetized by means electric current flowing through it. Its

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polarity is determines by the direction of current flow. Hence, a single core can be made to represent binary 0 and 1. Semiconductor memory consists of large number of transistors fixed on silicon chips in high densities. Transistor is a two way device that can be made to conduct or not to conduct, thus representing a binary 1 or 0. In modern days, celeron chips are being used for main memory. ii) Control storage: It helps the CPU in performing its functions. iii) Local storage: It is the high-speed working registers used in the arithmetic and logical operations. b) Auxiliary memory Auxiliary memory is also referred to as secondary storage. It is physically external device to the computer and is not connected directly to the CPU. There are three types of secondary storage: i) Sequential access storage: This implies reading one particular record in the file, and all records preceding it must also be read. Sequential access storage is called Read Only Memory (ROM). This memory is non-volatile, i.e not erased when the computer system is turned off. ii) Direct access storage: individual block or records have a unique address based on physical location. Access is accomplished by direct access to reach general vicinity plus sequential searching. Counting or waiting to reach the final location. Disk unit such as magnetic or optical disc are direct access devices. iii) Random access storage: This is one in which the time to access a given location is independent of the sequence of prior accesses and is constant. The main memory systems are of random access memory (RAM) type. This memory is erased when the computer system is turned off. 2.4.2 ARITHMETIC LOGIC UNIT Arithmetic and logic unit is used to perform all arithmetic and logic operations such as addition, subtraction, multiplication, division, and comparison, etc. The control unit collects data from its location in the memory and loads it in the arithmetic unit. The calculation is performed in the arithmetic and logic unit, and the results are then stored in the memory or retained in the arithmetic unit for further calculation. This unit is essentially a microprocessor. 2.4.3 CONTROL UNIT Control unit coordinates the activities of all other units in computer system. It obtained instructions from the memory, decodes them and direct the various unit to perform specific functions. The two major functions of this unit are: 1. It controls the transfer of data and information between various units.

2. It initiates appropriate functions by the arithmetic unit conceptually.

2.5 PHYSICAL CHARACTERISTICS OF MEMORY UNIT 1. Volatile memory: the content of this memory decay naturally unless refreshed or is lost when electric power is switched off.

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2. Non-volatile memory: The information once recorded remains without deterioration until deliberately charged, e.g magnetic surface memory 3. Non-erasable memory: the memory that cannot be erased, except by destroying the storage unit. Example is semiconductor memory. 4. Random Access Memory (RAM): RAM means that any word in the memory may be accessed without having to go through all other word to get it. This memory is more suitable to microprocessor. The disadvantage of RAM is that it is volatile. 5. Read Only Memory (ROM): In ROM, the information is ‘burnt’ into the chip at the time of manufacturing. It cannot be changed and refresh information cannot be ‘written’ into it. The information can be ‘read’ and transferred for use elsewhere. When electric power is switch off the bit patterns in the memory are not lost. 6. Programmable Read Only Memory (PROM): It is programmed to record information using a facility known as PROM-Programmer. However, once the chip has been programmed the recorded information cannot be altered. 7. Erasable Programmable Read Only Memory (EPROM): in this, information can be erased and the chip programmed a new to record different information. Erasing is achieved by exposing the chip to ultraviolet light. 8. Electrical Erasable ROM (EEROM): This can be programmed or erased by electrical signal. This provides an easy method to load and store information temporarily or permanently on ROM. This can also work as a back-up to RAM. 2.6 MASS STORAGE DEVICES A computer system has a number in-built mass storage device. The reason for external storage is to retain data and programs for further use. Large number of files containing information can be stored on external media. The popular known external media used with computer systems are: 1. Hard disc 2. Floppy disc 3. Compact disc ROM (CD-ROM) 4. Cartridge magnetic tape, digital audiotape (DAT) 2.7 GRAPHIC DISPLAY MEDIA Graphics display is used to display programs and drawings in CAD systems. Examples are monitor or video display unit. The display of engineering graphics is based on either cathode ray tube (CRT) or plasma display technology. In cathode ray tube, the heated cathode emits high stream of electrons which are formed into beam by an aperture in control grid surrounding the cathode. The electrons are then accelerated and focused to a point on the display surface. The electron beam is then swept across the phosphor-coated face of the tube line-by-line. The beam current is regulated to increase its intensity in order that brighter or darker points are created along each swept line.

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In plasma display, a thin (25 mm) glass enclosure is filled with inert gas such as neon. There are two sets of electrodes at right angle to each other that do not touch at the point where they appear to cross, being separated by a critical distance. Plasma displays are flat, possess high light output, low power consumption and are relatively expensive.

2.8 INPUT DEVICES Input devices are used for providing instructions to the computer. An input device captures data and makes them available to the graphics program. The program processes the input data and determines the resulting effect on the visual display. The main graphical devices used in CAD systems are: 1. Light pens 2. Analogue devices 3. Keyboard devices 2.9 OUTPUT DEVICES 1. Printers: A printer is an output device that converts data from computer and prints it in readable form on paper. 2. Plotters: A plotter automatically generates copy of file stored on the tape or disk. It uses paper, vellum or film.

2.10 CAD SYSTEM SOFTWARE CAD system software is a set of programs use to operate the computer graphics system and implement certain specialized functions. It includes programs to generate images or drawings, manipulate the drawing, and accomplish several types of interaction between the user and the system. The CAD software for a particular computer system is dependent on the type of hardware used in the system. TYPES OF SOFTWARE There two major types of software require in CAD system. They are: 1. System software: This includes the operating system for controlling the operation of the computer and the associate software such as editors and filing systems. The operating system is the main system software element in CAD system. It acts as interface between the user, the computer hardware and software. The modern operating systems are made up of a collection of integrated program modules which collectively serve different purpose like hardware management, user interfacing, and the provision of user facilities. The operating system also controls the peripheral devices. 2. Application software: This is CAD package designed for a specific problem. The most

common computer modeling software packages are AutoCAD, ProE, and SolidWorks.

Application software (package) converts CAD software into useful system.

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Some computer software packages perform specific engineering analysis and/or simulation tasks that assist the designer, but they are not considered a tool for the creation of the design that CAD is. Such software fits into two categories: engineering based and non-engineering-specific. Some examples of engineering-based software for mechanical engineering applications—software that might also be integrated within a CAD system—include finite-element analysis (FEA) programs for analysis of stress and deflection, vibration, and heat transfer (e.g., Algor, ANSYS, and MSC/NASTRAN); computational fluid dynamics (CFD) programs for fluid-flow analysis and simulation (e.g., CFD++, FIDAP, and Fluent); and programs for simulation of dynamic force and motion in mechanisms (e.g., ADAMS, DADS, and Working Model). Examples of non-engineering-specific computer-aided applications include software for word processing, spreadsheet software (e.g., Excel, Lotus, and Quattro-Pro), and mathematical solvers (e.g., Maple, MathCad, MATLAB, Mathematica, etc).

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3.0 COMPUTER GRAPHICS 3.1 INTRODUCTION Computer graphics is defined as the creation, storage, and manipulation of drawings or pictures with the aid of computer. Computer graphics is an effective means of communication between people and computers. That means, user communicates data and command to the computer through input terminal and the computer communicates with the user via a Cathode Ray Tube (CRT). In CAD system, computer graphics is also used to denote users oriented system in which computers are employ to create, transform and display data in form of pictures or symbols. These pictures are modified, enlarged, reduced in size, move to another on the screen, rotated and transform. 3.2 FUNCTIONS OF COMPUTER GRAPHICS 1. Modeling deals with the description of an object in terms of its spatial coordinates lines, areas, edges, surfaces and volume. 2. Storage is concerned with model storage in the memory of the computer. 3. Viewing involves the use of computer to view model from a specific angle and presents on its screen what it sees. 4. Manipulation is used to construct model from basic primitives in combination with Boolean algebra. 3.3 2D TRANSFORMATIONS Computer graphics display screen is limited in size because there are fixed number of pixels on the display screen. This posed a problem while trying to produce real drawing using computer. Such difficulty can be overcome by the application of transformation techniques such rotating the object to give a clear picture of its shape. Transformation in a single plane is called 2D transformation. There are five types of transformation routines: 1. Translation 2. Scaling 3. Rotation 4. Reflection 5. Shear 3.3.1 TRANSLATION Any graphical entity can be translated or moved in X and Y direction by using this routine. The basic equations used in this routine are X’ = X+Tx Y’ = Y+Ty

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where (X’,Y’) are the new coordinates after translation and (X,Y) are the old coordinates before translation. Tx and Ty are the distance to be translated in x and y direction respectively. The equation can be written in matrix form as

1010001

11''

yx TTYXYX

Where

1010001

yx TTis called translation matrix.

3.3.2 SCALING This routing is used to enlarge or reduce the object. The equations used are X’ = X.Sx Y’ = Y.Sy where Sx and Sy are the scaling factors in x and y direction respectively. For enlarging an object the, the scaling factor must be grater than one and for object reduction the value is less than one. The equation can be written in matrix form as

1000000

11'' y

x

SS

YXYX

Where

1000000

y

x

SS

is called scaling matrix.

When scaling is effected with respect to some fixed point (x0,y0), translate point (x0,y0), to origin, perform scaling, translate the point back to its original position. Hence the final scaling matrix can be obtained by multiplying the 3 matrix as follows:

1)1()1(0000

1010001

1000000

1010001

000000 ySxSS

S

yxS

S

yx yx

y

x

y

x

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3.3.3 ROTATION Rotation of any point is effective with respect to some fixed point. We assume anticlockwise to be positive and clockwise rotation as negative. X = r cos Y = r sin X’ = r cos ( ) = sinsincoscos r = sin)sin(cos)cos( rr sincos' YXX Similarly, X’ = r sin ( )

cossin' YXY In matrix form, the equation is can be written as

1000cossin00cos

11''

YXYX

Concatenation Rule Sequence of transformation can be combined into one by the concatenation process. When the three transformation matrices are to be concatenated, then the following rule is applied: A.B.C = A.(B.C) = (A.B).C Where A, B and C are transformation matrices. 3.3.4 REFLECTION The use of reflection helps to obtain mirror image of an object. Reflection is always carried out about an axis called axis of reflections. If the axis of reflection is x-axis then the x-coordinate of the point remains unaltered while the y-coordinate flips (i.e changes sign). Similarly, when the axis of reflection is y-axis, the y-coordinate of the reflection remains the same while the x-coordinate changes its sign. Reflection about x-axis: let the reflection of point P about x-axis be P’

The reflection matrix is given by

100010001

xR

The coordinates points (x’,y’) becomes

xRYXyx 11''

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Reflection about y-axis:

100010001

yR

The coordinates points (x’,y’) becomes

yRYXyx 11''

Reflection matrix when the axis of reflection is a line ( y = mx) passing through the origin The following steps are involved: 1. Rotate the line: that is axis of reflection y = mx in clockwise direction by angle m1tan to align it with x-axis. The rotation matrix is given by

1000)cos()sin(0)sin()cos(

1

A

2. Reflection about x-axis is given by

100010001

2A

3. Rotation of the line counter-clockwise by angle so that the original position of the object is restored.

1000cossin0)sin(cos

3

A

The resultant matrix is given by 321 .. AAAR Reflection matrix when the axis of reflection is a line ( y = mx+c) passing through the origin The following steps are involved: 1. Translate the line so that it passes through the origin. 2. Rotate the line so as to align it with one of the axis. 3. Reflection about the axis in (2) is obtained. 4. The line is rotated back to its original inclination. 5. The line is translated back to its original position. The resultant matrix is obtained by concatenating the above five transformation matrices.

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3.3.5 SHEAR Shearing transformation produces distortion of an object or image. We have two types of shear namely x-direction shear and y-direction shear. In the x-direction shear, the x-coordinate will change while the y-coordinate remains unchanged. If Shx is the shearing factor in the x-direction, then the basic equations are

yShxx x 1

yy 1 In y-direction shear, the y-coordinate will change while the x-coordinate will remain same. If Shy is the shearing factor in the y-direction, then the basic equations are given as

xx 1 xShyy y 1

3.4 EXAMPLES 1. Find the final position of the line having the end points (3,5) and (10,5) when it is translated by three units in x-direction and the rotated by 30o in clockwise direction. Solution Translation matrix when the line is translated by 3 unit in x-direction

103010001

1010001

1

yx TTT

Rotation matrix when the line is rotated by 30o in clockwise direction

1000)30cos()30sin(0)30sin()30cos(

1000)cos()sin(0)sin()cos(

1

R

=

1000866.05.005.0866.0

The resultant matrix is

1000866.05.005.0866.0

103010001

11 RTR

=

15.1598.20866.05.005.0866.0

The final position of point (3,5) is

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RYXyx 11''

133.1696.715.1598.20866.05.005.0866.0

1531'' 11

yx

Similarly, the final position of point (10,15) is

117.2758.1315.1598.20866.05.005.0866.0

15101'' 22

yx

2. A triangle having vertices (1,10), (5,2) and (8,4) is translated by 3 units in y-direction then rotated by 45o in counter-clockwise direction, the it is scaled by 3 units in x-direction. Find the position of the triangle. Solution 1. Translation matrix when the triangle is translated by 3 units in y-direction

130010001

1010001

1

yx TTT

2. Rotation matrix when the triangle is rotated by 45o in counter-clockwise direction

1000707.0707.00707.0707.0

100045cos45sin045sin45cos

1000cossin0sincos

1

R

3. Scaling matrix when the triangle is scaled by 3 units in x-direction

100000003

1000000

1 y

x

SS

S

The resultant matrix is

100010003

1000707.0707.00707.077.0

130010001

111 SRTR

=

1121.2363.60707.0121.20707.0121.2

The final position at point (1,10) is

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1898.9452.251121.2363.60707.0121.20707.0121.2

11011'' 11

yx

The final position of point (5,2) is

107.701121.2363.60707.0121.20707.0121.2

1251'' 22

yx

The final position at point (8,4) is

1605.10121.21121.2363.60707.0121.20707.0121.2

1481'' 33

yx

3.5 3D TRANSFORMATION In 3D transformation, the x, y and z coordinates of a point are considered. 3.5.1 The translation matrix is given as

1010000100001

zyx

T

TTT

R

Where Tx, Ty and Tz are translation in x, y and z direction respectively. 3.5.2 Scaling transformation matrix is

1000000000000

z

y

x

s SS

S

S

Where Sx, Sy and Sz are the scaling factor in x, y and z direction respectively. 3.5.3 Rotation matrix: Rotation matrix about z-aix is given by

1000010000cossin00sincos

zR

Rotation matrix about the x-axis

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10000cossin00sincos00001

xR

Rotation matrix about the y-axis is given by

10000cos0sin00100sin0cos

yR

3.5.4 Reflection

Reflection in 3D can be obtained with respect to any plane. The plane of reflection can be standard plane such as XY, YZ and ZX etc. For example, if the reflection is with respect to XY plane, the z-coordinate of the point flips. Thus the corresponding reflection matrix is

1000010000100001

fR

The reflection about any plane may be obtained via the process of concatenation. 3.6 EXAMPLES 1. A sphere having centre (10,10,10) and radius 8 units is translated by 3 units in x-direction and 5 units in z-direction. Then it is rotated by 45o in anticlockwise direction about y-axis. Find the new centre of the sphere. Solution Translation matrix in x and z directions

1503010000100001

1010000100001

1

zyx TTT

T

Rotation matrix about y-axis

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1000045cos045sin0010045sin045cos

10000cos0sin00100sin0cos

1

R

=

10000707.00707.000100707.00707.0

The resultant transformation matrix is 11 RTT

10000707.00707.000100707.00707.0

1503010000100001

T

=

1414.10414.10707.00707.000100707.00707.0

The new centre point is

1414.10414.10707.00707.000100707.00707.0

11010101111 zyx

= 1414.110414.1 Hence the new centre is (-1.414 10 1.414) 3.7 ASSIGNMENT 1. A line having end points (3,3) and (5,5) is reflected about a line with equation y = 2x + 3. Find the final position of the line. 2. Determine a 3 x 3 homogeneous transformation matrix to change a square into rectangle with scaling factor Sx = 1 and Sy = 2 and rotating the rectangle by 90o in anticlockwise direction, maintaining the centre of rectangle at the centre of original square. 3. A pyramid is defied by the coordinate A(0, 0, 0), B(2, 0, 0), C(0, 1, 0) and D(0, 0, 1) is rotated by 45o about the line L that has direction V = j + k and passes through point C(0, 1, 0). Find the coordinate of the rotated figure.

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4. The edges of a cube have the coordinates A(0, 0, 0); B(0, 0, 15); C(0, 15, 0); D(0, 15, 0); E(15, 0, 0); F(15, 0, 15); G(15, 15, 0); and H(15, 15 ,15). The cube is rotated from its above initial position by angle 45o about x-axis and then translated by (15, 15, 15) units. Determine the coordinates of the edges in the final position. 3.8 PROJECTIONS Projection deals with the representation of curves, surfaces and solids on a plane. Projection is obtained by drawing straight lines from various points on the contour of the object to meet a plane usually referred to as projection plane. The figure is formed by joining in correct sequence the point of intersection on the plane. In computer graphics, the concept of projection is used in viewing 3D object. Forms of projections The two main types of projections include parallel and perspective projections. In parallel projection, the projectors are parallel to each other so the hypothetical centre of projection is at infinity. Whereas in perspective projection there is a finite distance between centre of projection and plane projection. Centre of projection is specified by homogeneous coordinates of the form (x, y, z, 1). In parallel projection, the direction of view has to be specified by a vector which can be determined by subtracting two points Pd = (xA, yA, zA, 1) – (xa, ya, za, 1) = (a, b, c, 0). If the centre of projection in the case of perspective viewing is shifted to infinity, it becomes parallel projection.

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4.0 GEOMETRIC MODELING TECHNIQUES 4.1 INTRODUCTION Computer representation of component geometry using software package is called geometric model. Geometric modeling techniques include wire frame modeling, surface modeling and solid modeling. 4.2 WIRE FRAME MODELING In this type of modeling scheme, the object is represented by its edges. The wire frame model of an object appears as if it is made of wires. Wire frame models are simple and easy to create, and require relatively little computer time and memory. They do not provide complete description of the object. They also contain little information about the surface and volume of the part. Although wire frame modeling is ambiguous in understanding the object, this has been the method traditionally used in the 2D representation of the object, where orthographic views such as plane, elevation, end view, etc are employed to described the object graphically. Comparison between 2D and 3D wire frame models 2D models 3D wire frame models 1. End (vertices) of lines are represented by their X and Y coordinates.

End of lines are represented by their X, Y and Z coordinates.

2. Curved edges are represented by circles, ellipses, splines, etc.

Curves surfaces are represented by suitably spaced generators.

3. Additional views and sectional views are required to represent a complex object with clarity.

Hidden line or hidden surface elimination is a must to interpret complex components correctly.

4. 3D image reconstruction is tedious 2D views as well as various pictorial views can be generated easily.

5. It uses only one global coordinate system It may require the use of several coordinate systems on different faces of the components.

4.3 SURFACE MODELING In geometric surface modeling, a component is represented by its surfaces which in turn are represented by their vertices and edges. For example, six surfaces are combined together to create a cuboid. Surface modeling systems are useful to calculate surfaces intersections and surface areas. The systems are capable of producing shaded images and removing hidden lines automatically. Surface model can be constructed using a large variety of surface feature or standard surface convections (e.g box, pyramid, wedge, dome, sphere, cone, torus, dish and mesh) available in CAD systems. The plane is the most basic feature to represent a surface element. More complex

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shapes can be defined by tabulated cylinders, ruled surface, surface of revolution, surface sculptured, sweep surface and fillet-surfaces. Since surface models do not represent the solid nature parts because they contain no information describing what is within the part interior. They cannot be used as a basis in engineering analysis programmes such as finite element and modal analysis for stress and strain predictions. However, surface models are used where complex 3D geometries must be designed, especially where the application has to do with the exterior shell of objects such as sheet metal and thin moulded plastic parts. Surface modeling has been popularly use in aerospace and automobile industries. Surface model require more computational time and memory compared to wire frame. 4.4 SOLID MODELING Solid models are true 3D representation of physical object. This is due to the fact that solid model is bounded, homogeneously three dimensional and finite. Solid modeling is an accurate geometric description which includes both the external surface and internal structure of the object. It allows the designer to determine information such as the object mass properties, interference, and internal cross sections. Solid models differ from wire frame and surface models in the kind of geometric information they provide. Wire frame models only show the edge geometry of an object. They give no information about what is inside an object. Surface models provide surface information, but lack information about object internal structure. Solid models provide complete geometric descriptions of objects. Solid models can be used in design analysis to yield reliable results. Solid models apart from geometric information also provide important data such as volume, mass, properties and centre of gravity. The designer can also export models created to other applications for finite element analysis (FEA), rapid prototyping and other special engineering applications. 4.5 METHODS OF SOLID MODELING 1. Spatial Enumeration: in this form of 3D volumetric model, a section of 3D space is described by matrix of evenly spaced cubic volume element. 2. Cell Decomposition: This is a hierarchical adaptation of spatial enumeration. 3D space is sub-divided into cells. Cells could be of different sizes. These simple cells are glued together to describe a solid object. 3. Boundary Representation: The solid represented by its boundary which consists of a set of faces, a set of edges and a set of vertices as well as their topological relations. 4. Sweep technique: In this method, a planar shape is moved along a curve. Translational sweep can be used to create prismatic objects and rotational sweep could be used for axisymetric components. 5. Primitive Instancing: this modeling scheme provides a set of possible object shapes which are described by a set of parameters. Instances of object shape can be created by varying these parameters.

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6. Constructive Solid Geometry (CGS): Primitive instancing are combined using Boolean set operations to create complex objects. Most modeling packages use any one of the following modeling techniques: i. Constructive solid geometry (CSG) ii. Boundary representation iii. Hybrid method which is a combination of the two above. 4.6 CONSTRUCTIVE SOLID GEOMETRY (CSG) In a CSG model, physical objectives are created by combining basic elementary shapes known as primitives like block, cylinders, cones, pyramids and spheres. The Boolean operations such as union, difference and intersection are used to carry out this operation. For example, assuming we employed two primitives, a block and a cylinder which are located in space. A “union” operation (AUB) will combine the two to convert them into a new solid. The difference operation (A - B) will create a block with a hole. An intersection operation (AnB) will yield the portion common to the two primitives. 4.7 Boundary Representation Boundary representation is created based on the fact that a physical object is enclosed by a set of faces which themselves are closed and orientable surfaces. In this model, face in bound by edges and each is bounded by vertices.