chapter 1 introduction … · a 3d model. 04 . gtu paper analysis (new syllabus) computer aided...

GTU Paper Analysis (New Syllabus)

Computer Aided Design (2161903) Department of Mechanical Engineering Darshan Institute of Engineering & Technology

Chapter 1 –Introduction

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Theory 1. What is graphic standard? Explain different CAD standards. 07 07

2. Write Bresenham’s line algorithm. Determine intermediate pixels for line starting from (1, 1) to (8, 5). 07

3. Explain DDA algorithm for line generation with its limitations. 07 07

4. Write a Breshnham’s algorithm for line having slop more than 45° 07 07

Explain Bresenham’s algorithm for generation of line with flow chart. 07

5. Explain IGES graphic standard in detail with structure. 03 07 07

6. State different commercial CAD software available and explain the features of any two CAD software in detail.

07

7. State the various stages for a design process, in which various CAD tools can be used to improve productivity.

03

8. Differentiate between Raster scan and vector scan displays. 03

9. State the various CAD software commercially available and explain the features used to model Hexagonal nut.

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10. Explain interactive computer graphics. 03

11. Calculate the memory requirement for the 24-bit true color system for the 1024 x 1024 pixel resolutions.

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12. State the various stages for a design process, in which various CAD tools can be used to improve productivity.

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13. Explain different types of coordinate systems available in CAD softwares. 04

Examples

1.

Determine following for an 8-plane raster display with resolution of 1280 x 1024 and a refresh rate of 60Hz (non-interlaced):

i. The size of graphical memory (refresh buffer memory).

ii. The time required to display a scan line & a pixel.

iii. The active display area of the screen if the resolution is 78 dpi (dots per inch).

07

2. Write steps required to plot a line whose slope is between 450 and 900, using Bresenham’s algorithm. 07

3. Determine the pixels for a straight line connecting two points (2, 7) and (15, 10) using Bresenham’s algorithm.

07

4. Using Bresenham’s line algorithm, find the Pixel value position of line between points (1,5) and (4,10). 07

5. Using DDA algorithm, find the Pixel value position of line between points (2,10) and (6,5) 07

6. Plot intermediate raster locations when scan converting a straight line from screen coordinate (2, 7) to screen coordinate (15, 10) using DDA algorithm.

07



Chapter 2– Curves and Surfaces

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Theory

1. With the help of neat sketches explain various types of surfaces. 07

2. Derive general parametric equation for Hermits cubic spline curve in matrix form. 07 07 07

3. Two Bezier curve sections A and B have order of 3 and 4 respectively. Derive the condition for 1st order (C1) continuity between these two sections.

07

4. What is parametric representation? A line having length 20 unit, passes through the point P1 (1, 2). It makes an angle 60˚ with X-axis. Determine the parametric equation of line.

07

5. Explain the following surfaces 1. Plane surface 2. Bezier surface 3. B-spline surface 4. Coons surface 07

6. Write parametric equation for Bezier curve. Briefly discuss its characteristics. 07

7. Explain analytic curves and synthetic curves with example. 07 04

8. For the position vectors P1 [3 7] and P2 [8 9], determine the parametric representation of line segment between them. Also determine the slope.

04

9. State the properties of Hermite Cubic Splines. How these curves are differing from Bezier curves? 03

10. Explain B-spline curve with figure. 04

11. What is Coons Patch? 03

12. Explain different types of surfaces with respect to modeling. 03 03

13. Develop vector equation of line in parametric form. 03

14. Explain feature based modeling. 04



15. Differentiate between Hermite Cubic Splines curves and Bezier curves. 03

16. What do you mean by Iso-parametric representations? Write the equation of a line in parametric form. 04

Examples

1. A Bezier curve is to be constructed using control points P0 (35, 30), P1 (25, 0), P2 (15, 25) and P3 (5, 10). The Bezier curve is anchored at P0 and P3. Find the equation of the Bezier curve and plot the curve for u= 0, 0.2, 0.4, 0.6, 0.8 and 1.

07

2. Coordinates of four data points P0, P1, P2 and P3 are (2, 2, 0), (2, 3, 0), (3, 3, 0) and (3, 2, 0) respectively. Find the equation of Bezier curve and determine the coordinates of points on curve for u = 0, 0.25, 0.5, 0.75 and 1.0.

07 07

3.

A line is represented by the end points P1 (2, 4, 6) and P2 (-3, 6, 9). If the value of parameter u at P1 and P2 is 0 and 1 respectively, determine the tangent vector for the line. Also determine the coordinate of a point represented by; u equal to 0, 0.25, -0.25, 1 and 1.5. Also find the length and unit vector of line between two points P1 and P2.

07

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4. The vertices of a Bezier polygon are A0 [2, 2], A1 [3, 4], A2 [3, 4] and A3 [5, 4]. Determine 4 points on the Bezier curve.

07

5. Line passing through the end points P1 (2, 7, 3) and P2 (6.26, -9.78, 13) in the direction given by the unit vector 0.213i -0.839j +0.5k. Find the coordinate of the mid-point of the line.

07

6. Plot the Bezier curves having points, P0 (1, 3), P1 (5, 6), P2 (6, 0) and P3 (7,2). Plot for values u = 0, 0.2, 0.4, 0.6, 0.8, 1.0, if the characteristic polygon is drawn in sequence P0 – P1 – P2 – P3.

07

7. Plot the Bezier curve having endpoints P0 (0, 0) and P3 (7, 0). The other control points are P1 (7, 0) and P2 (7, 6). Plot values for u = 0, 0.1, 0.2, …, 1, if the characteristic polygon is drawn in the sequence P0 – P1 – P2 – P3.

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Chapter 3 – Mathematical representation of solids

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Theory

1. Explain constructive solid geometry (CSG). 07 04 04 05

2. Write limitations of a wire frame model. 02

3. Write short note on Constructive Solid Geometry (CSG) for solid modeling. 07

4. What do you understand by 2 ½ D model? Clearly distinguish it from 3-D model. 07

5. Compare CSG and B-rep techniques of solid modeling. 07

6. Differentiate between wireframe modeling and solid modeling technique. 04 07

7. Enlist the various methods of geometric modeling. Discuss wire frame modeling in detail. 07 07

8. Discuss steps involved in feature based modeling. List most commonly used feature operations in CAD systems.

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9. Differentiate between surface and solid modelling. State the limitations and applications of each of these modelling techniques.

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10. Explain same geometry different topology for solid modeling. 03



Chapter 4 – Geometric Transformations

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Theory

1. Explain two dimensional geometric transformations in details. Also give transformation matrix for

each.

07

2. Explain orthographic and oblique projections in details with suitable sketch. 07

3.

Write 3x3 transformation matrix for each of the following effects; (i). Scale the image to be twice as large and then translate it 1 unit to the left. (ii). Scale x direction to be half as large and the n rotate anticlockwise by 90O about origin. (iii). Rotate anticlockwise about origin by 90O and then scale the x direction by half as large. (iv). Translate down 0.5 unit, right 0.5 unit, and then rotate anticlockwise by 45O.

07

4. Derive the orthographic projection matrices for the Top view and Right Hand side view of a 3D model.

03

5. Find reflection matrix, when the axis of reflection is given by the equation y=5x. 04

6. Write 2D transformation matrix for Scaling, Rotation and Translation. 03

7. Prove that R(θ1).R(θ2) = R(θ1 + θ2) for geometrical transformation 04

8. Explain the concept of homogeneous coordinates and its use in representing geometrical transformation.

03

9. Derive the matrix for orthographic projection matrices for the Top view and Right Hand side view of a 3D model.

04



Examples

1.

A triangle PQR has its vertices at P (0, 0), Q (4, 0) and R (2, 3). It is to be translated by 4 units in X

direction, and 2 units in Y direction, then it is to be rotated in anticlockwise direction about the new

position of point R through 90o. Find the final position of the triangle.

07 07

2.

A triangle ABC has vertices as A (2, 4), B (4, 6) and C (2, 6). It is desired to reflect through an

arbitrary line L whose equation is y=0.5X+2. Calculate the new vertices of triangle and show

the result graphically.

07

3.

A triangle ABC with vertices A (30, 20), B (90, 20) and C (30, 80) is to be scaled by factor 0.5 about a

point X (50, 40). Determine (i) the composition matrix and (ii) the coordinates of the vertices for a

scaled triangle.

07

4.

A triangle ABC with vertices A(0,0), B(4,0) and C(2,3) is Translated through 4 and 2 units along X and Y directions respectively and then Rotated through 90o in counterclockwise direction about the new position of point C. Find: (1) The concatenated transformation matrix and (2) The new position of triangle

07

5. A point P is translated by (4,6,0) rotated about x-axis by 45º CCW and then rotated about z- axis by 30º CCW. Obtain the concatenated homogeneous transformation matrix and final coordinates of a point P.

04

6. Triangle ABC has its vertices at A (0, 0), B (0, 4) and C (3, 2). Zoom this triangle 3 times and then hang it considering a free body using hook at point C with origin.

07

7.

A triangle ABC, having coordinate position of point A (15, 15) B (18, 12) and C (15, 20). Determine the new vertex position if the triangle is : 1. Scaled 0.5 times in X and 2 times in Y direction 2. If mirrored about a line y = 4x + 12.

07

8. Triangle ABC has its vertices at A (4, 2), B (8, 2) and C (6, 5). It is to be rotated anticlockwise about point C through 900. Find the new position of triangle.

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9.

Compare result in case of 2D transformation of triangle ABC. 1. Reflected about x-axis first followed by line y= -x 2. Rotated about origin at 2700. Coordinate of triangle ABC are: A (0, 0) B (3, 0) and C (0, 3).

07

10.

Calculate the concatenated transformation matrix for the following operations performed in the sequence as below: 21) Translation by 4 and 5 units along X and Y axis ii) Change of scale by 2 units in X direction and 4 units in Y direction iii) Rotation by 60° in CCW direction about Z axis passing through the point (4, 4). Find new coordinates when the transformation is carried out on a triangle ABC with A (4, 4), B (8, 4) and C (6, 8).

07

11. A triangle PQR with vertices P (2, 5), Q (6, 7) and R (2, 7) is to be reflected about a line x = 2y – 6. Determine, (i) The concatenated matrix and (ii) The coordinates of the matrices for the reflected triangle.

07



Chapter 5 – Finite Element Analysis

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Theory

1. What is shape function? Derive linear shape functions for 1-dimensional bar element in terms of natural coordinate. Also plot variation of shape functions within this element.

07

2. With the help of suitable examples explain condition of plane stress and plane strain. 07

3. List properties of global stiffness matrix [K]. 04 03

4. Write element stiffness matrix and element load vectors for a beam element. 03

5. What are the different types of elements used in FEA? Explain in brief. 07

6. Explain the concepts of FEM. Discuss the different steps involved in FEA in detailed. 07 07 07

7. Explain Penalty approach and Elimination approach for FEA. 04 04 07

8. What do you mean by primary and subsidiary design equation? Explain with example. 07

9. Explain Johnson method of optimum design with an example. 07

10. Discuss the advantages of finite element analysis. 03

11. With reference to finite element analysis, discuss the treatment of boundary condition using elimination approach.

07

12. Draw a sketch of following elements showing nodes:

(1) Quadrilateral (2) Six noded triangular (3) Tetrahedral

03

13. Discuss quadratic shape functions and their uses. 03

14. i. State the applications of FEA in the field of engineering. 04



ii. Write the properties of global stiffness matrix.

15. What do you mean by degree of freedom? Write the degree of freedom for structural, Heat transfer, fluid flow and magnetic applications

03

16. Draw and explain 2D element. 03

17. For a 1D linear element prove the relation between displacement, strain and stress σ = E B q. 04

18. Explain the statement “FEM operates from part to the whole”. 03

19. Explain plain stress and plain strains with figure. 04 03

20. Explain degree of freedom in FEA. Show degree of freedom associated with structural, electrical and heat transfer problem.

03

21. Draw a sketch of following elements showing nodes: (i) Quadrilateral (ii) Six noded triangular (iii) Tetrahedral

03

22. Explain penalty approach used in FEA with an example. 04

23. Explain in details : The general procedure of Finite Element Method. 07

24. List various engineering application of FEA. 03

25. What do you mean by thermal effects of temperature? How it is included in calculation for 1-D elements?

04

26. What is shape function? Derive linear shape functions for 1-dimensional bar element in terms of natural coordinate. Also plot variation of shape functions within this element.

07

27. List properties of global stiffness matrix [K]. 03

28. Write element stiffness matrix and element load vectors for a beam element. 03

29. What are axisymmetric elements? Explain. 04



Examples

1.

Consider the bar shown in figure-1. An axial load F=35 kN is applied as shown. Using penalty approach for handling boundary conditions, determine nodal displacements and support reactions. Take E=200 Gpa. For all elements. Length of each element is in mm.

07

2.

Consider the bar as shown in figure-2. Determine the nodal displacements and element stresses, if the temperature rises from 20’C to 60oC. Take P=300kN, E1=70Gpa, A1=900 mm2, Coefficient of thermal expansion, α1=23 x 10-6 per ‘C ; E2=200Gpa, A2=1200 mm2, Coefficient of thermal expansion, α2=11.7 x 10-6 per ‘C.

07

3. Evaluate the shape functions N1, N2 and N3 at the interior point P (3.85, 4.8) for the triangular element shown in figure-3. Also determine Jacobean of the transformation J for the element.

07



4.

A stepped shaft is shown in figure-1. Using Elimination Approach, determine the stresses and nodal displacements for each element. Assume uniform material for the complete shaft having a modulus of elasticity as 200 Gpa and axial force F as 35kN. Length of each element is in mm.

07

5 A four bar truss is as shown in figure-4. Assuming that for each element, the cross-sectional area is 400 mm2 and modulus of elasticity is 200 GPa, determine the nodal displacements. Length of each element is in mm.

07



6

A stepped metallic bar with circular cross section consists of two segments. Length & cross section area of first segment is 350 mm & 275 mm2 respectively. Length & cross section area of second segment is 250 mm & 175 mm2 respectively. Assume modulus of elasticity is 200 Gpa. If one end of the bigger segment is fixed and if an axial tensile force acting on the free end of the smaller segment is 700 kN, find: (1) Nodal displacements, using global stiffness matrix. (2) Elemental Stresses, (3) Support Reaction.

07

7

A two-step as shown in figure is subjected to thermal loading conditions. The length of left step is 250 mm & length of right step is 350 mm. An axial load P = 200 x 103 N applied 20°C to the end. The temperature of the bar is raised by 50°C. Calculate: (13) Element stiffness matrix (ii) Global stiffness matrix Consider E1 = 70 x 103 N/mm2, E2 = 200 x 103 N/mm2, A1 = 700mm2, A2 = 1000 mm2, α1 = 23 x 10-6 per °C and α2 = 11.7 x 10-6 per °C.

07



8

Fig.1 shows the compound section fixed at both ends. With the help of FEA estimate the reaction forces at the supports and the stresses in each material when a force of 200 KN is applied at the change of cross section.

07

9

A system of a rigid cart connected by three linear springs as shown in Fig.2. The force of 60 N is acting on cart as shown in figure. Determine the following: (1) Use finite element concept to assemble the elemental stiffness matrices of three linear springs into global stiffness matrix. (2) Write global load vector. (3) Find Nodal solution.

07

10 For one dimensional element shown in Figure 1, temperature at node 1 is 100 0C and at node 2 is 40 0C. Evaluate shape function associated with node 1 and node 2. Calculate temperature at point P. Assume linear shape function.

04



11

A stepped metallic bar, made of aluminum (E1 = 70 x 103 N/mm2) and steel (E2 = 200 x 103 N/mm2), is subjected to the axial force of 5000 N, as shown in figure 2. It is attached to rigid wall at node 1. Determine nodal displacements using finite element analysis.

07

12

Figure 3 shows two springs connected in series, having stiffness 12 and 8 N/mm respectively. One end of the assembly is fixed and a force of 60 N is applied at the end. Using finite element method; (i). Derive global stiffness matrix (ii). Derive global load vector (iii). Find displacement of all the nodes

07

13. Consider the bar shown in figure-1 below. Determine the nodal displacement and element stresses. 04



A1=250 mm2 A2=150 mm2 E1= 200Gpa E2=70Gpa L1=250mm L2=250 mm

14.

Formulate the finite element model using 1D-bar element for the system shown in figure-2 below. Area at the junction shown below is AJ =250 mm2, at the left end is equal to AL=750 mm2 and at the right end is equal to AR=500 mm2. Length up to junction from any end is 200 mm. Load P=500kN is acting at the junction. Young’s modulus of elasticity E= 200 Gpa. The temperature of the system is raised by 40°C. Co-efficient of thermal expansion is 11×10-6 per °C. Assemble the stiffness matrix & force vector.

07

15. Find the nodal displacements and elemental stresses for the axially loaded stepped bar as shown in figure-3 below using Penalty approach. E=200Gpa, A1= 200 mm2, A2 =180 mm2 and Δx= 0.1 mm.

04



16. A thin plate as shown in figure-4 has a uniform thickness of 10 mm and modulus of elasticity is 200 Gpa. The plate is subjected to a point load P = 500 N as shown in figure. Model the problem with two elements and find stresses in each element.

07



17.

For the one dimensional fluid flow problem as shown in figure 5 with velocity known at right end, determine velocities at nodes 1 and 2. Let Kxx=2cm/s

07

18. Evaluate the shape functions N1, N2 and N3 at the interior point P for the triangular element as shown in figure 6.

04



19.

A three element truss as shown in figure 7 has modulus of elasticity E= 200 Gpa. The area of each element is 50 mm2. The length L1= 1000 mm and L2=750 mm. The load P1=25 KN and P2 = 30 kN are applied as shown. Determine the nodal displacements, reaction forces and elemental stresses.

07



20. A 90 m long 1D element is having linear shape function if the temperature at node 1 is -500 C and at node 2 is 700 C, find the temperature at a point 25 m away from node 1.

04

21.

Find the Jacobian matrices for triangle shown in Fig.1

03

22.

Modeled the tapered bar shown in figure 2 by considering it is made of 2 elements and determine deflection at both end and in middle of the bar. Assume modulus of elasticity as 200 Gpa.

07 07

23. Derive the global stiffness matrix for the system of spring shown in fig 3 03



24.

For the loading system as shown in figure 4, find out displacement, stress and support reaction. Assume modulus of elasticity 80 x 103 N/mm2.

07

25. An axial load of 20 KN applied on the bar as shown in figure 2. Using finite element method find the nodal displacement, stresses in each section and the reaction forces.

07



26. A 25 m long 1D element is having linear shape function if the temperature at node 1 is 500C and at node 2 is -200C, find the temperature at a point 5 m away from node 1.

04

27.

In a triangular element, temperatures at nodes A, B and C are 100, 200 and 300 0C respectively. The coordinates of nodes are A (0, 0), B (10, 0) and C (5, 8). Estimate the shape functions associated with nodes and find temperature at point P (5,6). Refer figure 3.

04

28. For two member Struss as shown in figure 4, derived global stiffness matrix and determine deflection at node 2. Consider area A = 200 mm2 and E = 200 Gpa for each element.

07



29.

Determine the temperature at x = 40 mm (Figure 1), if the temperature at nodes Ti = 120 °C, Tj = 80 °C and xi = 10 mm and xj = 60 mm. Consider linear shape function.

04

30.

Consider the bar shown in Figure 2. An axial load P = 200 x 103 N is applied as shown. Using the penalty approach for handling boundary conditions, (a) Determine the nodal displacements (b) Determine the stress in each material. (c) Determine the reaction forces.

07



A1 = 2400 mm2 A2 = 600 mm2

E1 = 70 x 109 N/m2 E2 = 200 x 109 N/m2

chapter 1 introduction … · a 3d model. 04 . gtu paper analysis (new syllabus) computer aided...

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