Questions and Answers of Systems Analysis And Design Using MATLAB

If , finda. y(n,m) ¼ x(n,m) h(n,m), circular convolution of x(n,m) and h(n,m).b. g(n,m) ¼ x(n,m)*h(n,m), linear convolution of x(n,m) and h(n,m). x(n, m) = [2 2 and h(n, m): -1 3
Convolve the following 2-D sequences using DFT, 2 4 f(n, m) = 2 3 2 and h(n,m) = 2 1 3 1 4
Compute g(n,m) ¼ f (n,m) * h(n,m) using a 4-point FFT, where [19 16 19 3 5 9 18 4 19 66 9 17 6 8 0 4 17 12 12 10 16 18 26 8 4 9 14 4 17 12 8 8 14 12 f(n,m) and hin, m)= === 18 18 1 5 17 17 6 10 15
In this example, we compare the processing speed of convolution in the spatial domain to that of the same in the frequency domain. Consider a gray scale image of size 512 * 512 pixels. We wish to
Find the HT of the 4 * 4 image f (n,m) given as 15 15 16 14- 16 17 15 13 f(n, m)= 14 11 12 15 13 12 15 13
Design a circularly symmetric 2-D low-pass FIR filter for an image of size 57 * 57 with a cutoff frequency of wc ¼ p4 w(n) 0.8 0.6 Uniform 1 0.4 Hamming Blackman 0.2 0 0 5 10 15 n 20 25 Hanning 30
Given a 4 * 4 sequenceupsample the sequence f (n,m) by a factor of 2 using linear interpolation. f(n,m) 2522 1313 314 2422 2 13 2
Resize the following 6 * 6 image f (n,m) to a 4 * 4 image. f(n,m) = 1 2 4 421 302 2 43 2 3 10 6 064 224322 5 3 2 103 20056 4 3 27 3
In this example, the LENA image is upsampled by a factor of 2 and downsampled by a factor 3. The linear interpolation filters are used for both the upsampling and downsampling operations. The
2.1 Consider a digital camera with a focal length of 50 mm, an f number of 5.6, and a CCD sensor pixel dimension of 16 mm 16 mm. Consider two objects at distances of 1 and 3 m from the camera,
2.2 Consider an out-of-focus camera with a blur radius of R. Assume that the object moves horizontally with uniform motion during the exposure period such that the resulting PSF on the film has a
2.3 Find the DFT of the checkerboard image defined by f(n,m) = 0.5+0.5(-1)+m n,m= 0, 1, ..., N-1
2.4 A circularly symmetric imaging system has a PSF given bya. Derive an expression for the OTF and MTF of this imaging system.b. At what spatial radial frequency does the system MTF fall to 1/ √2
2.5 Consider a linear, shift-invariant image degradation system with a PSF h(x, y) ¼ e2[jxjþjyj]. Suppose the input to the system is an image consisting of one line as shown in Figure 2.64. What
2.6 Suppose that an image f (x, y) undergoes planer motion, and let x0(t) and y0(t) be the time-varying components of motion in the x- and y-directions, respectively.The total exposure at any point
2.7 A PSF has no spatial frequencies greater than 400 cycles=mm. What values would you assign to the sampling interval Dx and the DFT length so as to obtain samples of the MTF in which aliasing is
2.8 Find the Fourier transform of a line oriented at an angle u as shown in Figure 2.65. xo y f(x, y) x
2.9 Find the Fourier transform of the 2-D image f (x, y) ¼ d(x2 þ y2 1). The image is shown in Figure 2.66. Display the transform as an image.
2.10 Consider the two sequences x(n1, n2) and h(n1, n2) shown in Figure 2.67.a. Determine y(n1, n2) ¼ x(n1, n2) * h(n1, n2), the linear convolution of x(n1, n2) and h(n1, n2).b. Develop a procedure
2.11 Determine the approximate number of multiplications required for the 2-D convolution of f (n,m) and h(n,m) for the three cases given below. Assume f (n,m) to be N N and h(n,m) to be L L.a.
2.12 Determine the convolution of fx(n,m) with h(n,m), where x(n,m)= 1 3 2 3 323 4 7 1 0 3058 20 26 and_h(n,m) = [2 12 6
2.13 An image of size 64 64 is to be filtered using a 5 5 FIR LPF. Finda. The number of real multiplications if the convolution is performed in the pixel domain.b. The number of real
2.14 The signal f (x) is band limited so that its spectrum, F(v) (i.e., Fourier transform), is zero outside the interval 2pB a. Expand the spectrum F(v) in a Fourier series on the interval2pB b. Show
2.15 Let P(x) ¼ xe0:5x2u(x) and let the number of quantizer levels be 8 (3 bit).a. What are the decision and reconstruction levels?b. Find the resulting MSE.
2.16 Design a 2-D 11 11 zero-phase FIR filter to approximate the desired frequency response shown in Figure 2.68. FIGURE 2.68 0.5 W -0.5 D D 0.5 W1 -0.5 Hd(1, 2) = { 1 (1, 2) E D 0 otherwise
2.17 Consider the 8 * 8 image f (n,m) shown below.a. Assume this image is filtered by a separable filter having the following horizontal and vertical frequency responsesWhat are the gray level values
2.18 Consider the 3 * 3 FIR filtera. What is the DC gain of this filter (H(0, 0))?b. What is the high-frequency gain H(p, p)?c. Is this a zero-phase filter?d. Is it separable?e. Find the output image
2.19 The histogram of a 32 * 32, 3 bit gray level image is given in Table 2.4.We wish to modify the gray scale of this image such that the histogram of the processed image is as close as possible to
2.20 Consider the following 3 bit gray level image.a. Find the histogram of this image.b. We wish to modify the gray scale of this image such that the histogram of the processed image is as close as
2.21 A motion-blurred image (uniform motion in the x-direction with x0(t) ¼ a tT) is observed in the presence of additive white noise with a power spectrum of Sn(vx, vy) ¼ N0.a. What is the
1. A vertical force F is applied to a two-bar truss as shown in the figure. Let cross-sectional areas of the members 1 and 2 be Ai and Az, respectively. Determine the area ratio A1/A2 in order to
2. The stress at a point P is given below. The direction cosines of the normal n to a plane that passes through P have the ratio nx: Hy-. nz = 3: 4: 12. Determine (a) the traction vector T(n);(b) the
3. At a point P in a body, Cartesian stress components are given by axx = 80 MPa, (Tw, = —40 MPa, crs = —40 MPa, and rjy = TyZ = t0- = 80 MPa. Determine the traction vector, its normal
4. If axx — 90 MPa, oy -45 MPa, Tx>, = 30 MPa, and azz = xxz = Tyz = 0, compute the surface traction T(n) on the plane shown in the figure, which makes an angle of 6 = 40° with the verti¬cal
5. Find the principal stresses and the corresponding principal directions stress for the following cases of plane stress.(a) crxv = 40 MPa, crro, = 0 MPa, = 80 MPa(b) or0- = 140 MPa, Oyy = 20 MPa,
6. If the minimum principal stress is —7 MPa, find itxv and the angle that the principal stress axes make with the xy axes for the case of plane stress illustrated.
7. Determine the principal stresses and their associated directions when the stress matrix at a point is given by"111'1 1 2 MPa _! 2 1
8. Let xJy'z' coordinate system be defined using the three principal directions obtained from Prob¬lem 7. Determine the transformed stress matrix [orjyy^ in the new coordinates system.
9. For the stress matrix below, the two principal stresses are given as 03 = —3 and crj = 2, respec¬tively. In addition, the two principal stress directions corresponding to the two principal
10. With respect to the coordinate system xyz, the state of stress at a point P in a solid is-20 0 0 0 50 0 0 0 50 MPa(a) m1, m2, and m3 are three mutually perpendicular vectors such that m1 makes
11. A solid shaft of diameter d = 5 cm, as shown in the figure, is subjected to tensile force P =13,000N and a torque T = 6,000N ¦ cm. At point A on the surface, what is the state of stress(write in
12. If the displacement field is given by u = x2 + ly1< v = -y2 - 2x(y - z)„ w = —z2 — 2xy(a) Write down 3x3 strain matrix.(b) What is the normal strain component in the direction of (1,1,1) at
13. Consider the following displacement field in a plane solid:u{x,y) = 0.04 - O.OLt + 0.006y v(x,y) = 0.06 + 0.009x + 0.012y(a) Compute the strain components sxv, ew,, and yJ7. Is this a state of
14. The displacement field in a solid is given by u = kx2 v = 2kxy2 w = k(x + y)z where k is a constant.(a) Write down the strain matrix.(b) What is the normal strain in the direction ofn={l, 1, I}7?
15. Draw a 2 x 2-inch square OABC on the engineering paper. The coordinates of O are (0, 0) and of B are (2, 2). Using the displacement field in Problem 13, determine the u and v displacements of the
16. Draw a 2 x 2 -inch square OPQR such that OP makes +73° to the x-axis. Repeat questions (a)through (d) in Problem 15 for OPQR. Give physical interpretations to your results.
17. For steel, the following material data are applicable: Young's modulus E = 207 GPa and shear modulus G — 80 GPa. For the strain matrix at a point shown below, determine the symmetric 3x3 stress
18. Strain at a point is such that Ea. = = 0, ^ = -0.001, = 0.006, and evz = ^ = 0. Note:You need not solve the eigen value problem for this question.(a) Show that n1 = i + j and n2 = —i + j are
19. Derive the stress-strain relationship in Eq. (1.60) from Eq. (1.55) and the plane stress conditions.
20. A thin plate of widthb, thickness t, and length L is placed between two frictionless rigid walls a distance b apait and is acted on by an axial force P. The material properties are Young's
21. A solid with Young's modulus E = 70GPa and Poisson's ratio = 0.3 is in a state of plane strain parallel to the ry-plane. The in-plane strain components are measured as follows:e.v.v = 0.007, Eyy
22. Assume that the solid in Problem 21 is under a state of plane stress. Repeat (b) through (f).
23. A strain rosette consisting of three strain gages was used to measure the strains at a point m a thin-walled plate. The measured strains in the three gages are = 0.01, eB = -0.0006, and£c =
24. A strain rosette consisting of three strain gages was used to measure the strains at a point m a thin-walled plate. The measured strains in the three gages are: eA = 0.016, = 0.004, and gc =
25. A strain rosette consisting of three strain gages was used to measure the strains at a point in a thin-walled plate. The measured strains in the three gages are: £a = 0.008, sB = 0.002, and sc =
26. The figure below illustrates a thin plate of thickness t. An approximate displacement field, which accounts for displacements due to the weight of the plate, is given by u(x,y)=^(2bx-x2
27. The stress matrix at a particular point in a body is-2 1 -3"1 0 4 x 107Pfl 1CO Determine the corresponding strain if E = 20 x 1010 Pa and v = 0.3.
28. For & plane stress problem, the strain components in the x-y plane at a point P are computed as evr = Eyy = .125 x 10~2, exy = .25 x lO"2(a) Compute the state of stress at this point if Young's
29. The state of stress at a point is given by"80 20 40"a] = 20 60 10 MPa
40 10 20(a) Determine the strains using Young's modulus of 100 GPa and Poisson's ratio of 0.25.(b) Compute the strain energy density using these stresses and strains.(c) Calculate the principal
30. Consider the state of stress in problem 29 above. The yield strength of the material is 100 MPa.Determine the safety factors according to the following: (a) maximum principal stress criterion,(b)
31. A thin-walled tube is subject to a torque T. The only non-zero stress component is the shear stress rA-y, which is given by tVj, = 10,000 T(Pa), where Tis the torque in N.m. When the yield
32. A thin-walled cylindrical pressure vessel with closed ends is subjected to an internal pressure p = 100 psi and also a torque T around its axis of symmetry. Determine T that will cause yield¬ing
33. A cold-rolled steel shaft is used to transmit 60 kW at 500 rpm from a motor. What should be the diameter of the shaft if the shaft is 6 m long and is simply supported at its end? The shaft also
34. For the stress matrix below, the two principal stresses are given as ai =2 and aj = —3, respec¬tively. In addition, two principal directions corresponding to the two principal stresses are
35. The figure below shows a shaft of 1.5-tn diameter loaded by a bending moment Mz = 5,000 lb ¦ in, a torque T = 8,0001b • in, and an axial tensile force N = 6,0001b. If the material is ductile
36. A 20-mm-diameter rod made of a ductile material with a yield strength of 350 MPa is subject to a torque of T = 100 N • m and a bending moment of M = 150 N • m. An axial tensile force P is
37. A circular shaft of radius r in the figure has a moment of inertia I and polar moment of inertia J.The shaft is under torsion Tz in the positive z-axis and bending moment Mv in the positive
38. A rectangular plastic specimen of size 100 x 100 x 10mm3 is placed in a rectangular metal cavity. The dimensions of the cavity are 101 x 101 x 9mm3. The plastic is compressed by a rigid punch
1. Three rigid bodies, 2, 3, and 4, are connected by four springs, as shown in the figure. A horizon¬tal force of 1,000 N is applied on Body 4, as shown in the figure. Find the displacements of the
2. Three rigid bodies, 2, 3, and 4, are connected by six springs as shown in the figure. The rigid walls are represented by 1 and 5. A horizontal force F3 = 1000N is applied on Body 3 in the
3. Consider the spring-rigid body system described in Problem 2. What force F2 should be applied on Body 2 to keep it from moving? How will this affect the support reactions?Hint Impose the boundary
4. Four rigid bodies, 1, 2, 3, and 4, are connected by four springs as shown in the figure. A hori¬zontal force of 1,000 N is applied on Body 1 as shown in the figure. Using FE analysis, (a) find
5. Determine the nodal displacements and reaction forces using the direct stiffness method. Calcu¬late the nodal displacements and element forces using the FE program.F, = 50 N *, = 50 N/cm k2 = 60
6. In the structure shown below, rigid blocks are connected by linear springs. Imagine that only horizontal displacements are allowed. Write the global equilibrium equations [K]{Q} = {F}after
7. A structure is composed of two one-dimensional bar elements. When 10N force is applied to node 2, calculate displacement vector {Q}r = {uy, 112, 113} using the finite element method.
8. Use FEM to determine the axial force P in each portion, AB and BC, of the uniaxial bar. What are the support reactions? Assume E — 100 GPa, area of cross sections of the two portions AB and BC
9. Consider a tapered bar of circular cross-section. The length of the bar is 1 m, and the radius varies as r(x) = 0.050 - 0.040a:, where r and x are in meters. Assume Young's modulus = 100MPa.Both
10. The stepped bar shown in the figure is subjected to a force at the center. Use FEM to determine the displacement at the center and reactions and flR.Assume: E = 100 GPa, area of cross sections of
11. Using the direct stiffness matrix method, find the nodal displacements and the forces in each element and the reactions.2000 lb/in 1500 lb/in.400 lb 500 lb/in 4 |
12. A stepped bar is clamped at one end and subjected to concentrated forces as shown.Note-. The node numbers are not in usual order!5kN 1 m m -2 m 1 m 2.7 Exercise 101 Assume: E = 100 GPa, section =
13. The uniaxial bar FE equation can be used for other types of engineering problems, if the proper analogy is applied. For example, consider the piping network shown in the figure. Each section of
14. For a two-dimensional truss structure, as shown in the figure, determine displacements of the nodes and normal stresses developed in the members using the direct stiffness method. Use E =30 x 106
15. For a two-dimensional truss structure, as shown in the figure, determine displacements of the nodes and normal stresses developed in the members using a FE program in Appendices. Use E = 30 x
16. The truss structure shown in the figure supports force F at Node 2. FEM is used to analyze this structure using two truss elements as shown.iz, cl c(a) Compute the transformation matrix for
17. The truss structure shown in the figure supports the force F. FEM is used to analyze this struc¬ture using two truss elements as shown. Area of cross-section (for all elements) — A = 2 in2,
18. Use FEM to solve the plane truss shown below. Assume AE = 106N, L = 1 m. Determine the nodal displacements, forces in each element, and the support reactions.
19. The plane truss shown in the figure has two elements and three nodes. Calculate the 4 x 4 ele¬ment stiffness matrices. Show the row addresses clearly. Derive the final equations (after
20. Use FEM to solve the two plane truss problems shown in the figure below. Assume AE = 106 N, L = 1 m. Before solving the global equations [K]{Q} = {F}, find the determinant of [K]. Does [K] have
21. Determine the member force and axial stress in each member of the truss shown in the figure using one of FE analysis programs in the Appendix. Assume that Young's modulus is 104 psi and all
22. Determine the normal stress in each member of the truss structure. All joints are ball-joint and the material is steel, whose Young's modulus is E = 210 GPa.
23. The space truss shown has four members. Determine the displacement components of Node 5 and the force in each member. The node numbers are numbers in the circle in the figure. The dimensions of
24. The uniaxial bar shown below can be modeled as a one-dimensional truss. The bar has the following properties: L = 1 m, A = 10~4m2, E = 100 GPa, and a = IQ-A/0C. From the stressfree initial state,
25. In the structure shown below, the temperature of Element 2 is 100° C above the reference tem¬perature. An external force of 20,000 N is applied in the x-direction (horizontal direction) at Node
26. Use FEA to determine the nodal displacements in the plane truss shown in Figure (a). The temperature of Element 2 is 100oC above the reference temperature, i.e., Ar'2' = 100oC. Com¬pute the
27. Repeat Problem 26 for the new configuration with Element 5 added, as shown in Figure (b).
28. Repeat Problem 26 with an external force added at Node 3, as shown in Figure (c).
29. The properties of the members of the truss in the left side of the figure are given in the table.Calculate the nodal displacement and element forces. Show that force equilibrium is satisfied as
30. Repeat Problem 29 for the truss on the right side of the figure. Properties of Element 6 are same as those of Element 5, but AT = 0 "C.
31. The truss shown in the figure supports the force F = 2,000 N. Both elements have the same axial rigidity of AE = 107 N, thermal expansion coefficient of a = 10~6/OC, and length L = 1 ra.While the

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