CH-01 (Pao - Engineering Analysis), страница 7

Apply the Gauss-Jordan elimination method to solve for x1, x2, and x3from the following equations:024−1  x1  1 3  x 2  = 17   x3  11924Show every normalization, elimination, and pivoting (if necessary) stepsof your calculation.7. Solve the matrix equation [A]{X} = {C} by Gauss-Jordan methodwhere:3242511   x1  −2  −1 x 2  =  −37   x3   3 Show every interchange of rows (if you are required to do pivoting beforenormalization), normalization, and elimination steps by indicating thechanges in [A] and {C}.8. Apply the program GauJor to solve Problem 7.9.

Present every normalization and elimination steps involved in solving thefollowing system of linear algebraic equations by the Gauss-Jordan Elimination Method:5x1 – 2x2 + x3 = 4–2x1 + 7x2 – 2x3 = 9x1 – 2x2 + 9x3 = 4010.11.12.13.14.Apply the program Gauss to solve Problem 9 described above.Use MATLAB to solve the matrix equation given in Problem 7.Use MATLAB to solve the matrix equation given in Problem 9.Use Mathematica to solve the matrix equation given in Problem 7.Use Mathematica to solve the matrix equation given in Problem 9.MATRIX INVERSION1. Run the program MatxInvD for finding the inverse of the matrix:3[A] = 02© 2001 by CRC Press LLC05020 32. Write a program Invert3 which inverts a given 3 × 3 matrix [A] by usingthe cofactor method. A subroutine COFAC should be developed for calculating the cofactor of the element at Ith row and Jth column of [A] interm of the elements of [A] and the user-specified values of I and J.

Letthe inverse of [A] be designated as [AI] and the determinant of [A] bedesignated as D. Apply the developed program Invert3 to generate allelements of [AI] by calling the subroutine COFAC and by using D.3. Write a QuickBASIC or FORTRAN program MatxSorD which willperform the addition and subtraction of two matrices of same order.4. Write a QuickBASIC or FORTRAN program MxTransp which willperform the transposition of a given matrix.5. Translate the FORTRAN subroutine MatxMtpy into a MATLAB m fileso that by entering the matrices [A] and [B] of order L by M and M byN, respectively, it will produce a product matrix [P] of order L by N.6. Enter MATLAB commands interactively first a square matrix [A] andthen calculate its trace.7.

Use MATLAB commands to first define the elements in its upper rightcorner including the diagonal, and then use the symmetric properties todefine those in the lower left corner.8. Convert either QuickBasic or FORTRAN version of the program MatxInvD into a MATLAB function file MatxInvD.m with a leading statementfunction [Cinv,D] = MatxInvD(C,N)9. Apply the program MatxInvD to invert the matrix:1[A] = 5836947 10 Verify the answer by using Equation 1.10.

Repeat Problem 9 but by MATLAB operation.11. Apply the program MatxInvD to invert the matrix:−9[A] =  −3−6−1−4−7−2 −5−8Verify the answer by using Equation 1.12. Repeat Problem 11 but by MATLAB operations.13. Derive [Rx] and verify that it is indeed equal to [Tx]T. Repeat for [Ry] and[Rz].14. Apply MATLAB to generate a matrix [Rz] for θz = 45° and then to use[Rz] to find the rotated coordinates of a point P whose coordinates beforerotation are (1,–2,5).© 2001 by CRC Press LLCFIGURE 7. Problem 18.15. What will be the coordinates for the point P mentioned in Problem 14 ifthe coordinate axes are rotated counterclockwise about the z-axis by 45°?Use MATLAB to find your answer.16. Apply MATLAB to find the location of a point whose coordinates are(1,2,3) after three rotations in succession: (1) about y-axis by 30°, (2)about z-axis by 45° and then (3) about x-axis by –60°.17.

Change m file Animate1.m to animate just the rotation of the front (F)side of the 4 2 3 brick in the graphic window.18. Write a MATLAB m file for animation of pendulum swing1 as shown inFigure 7.19. Write a MATLAB m file for animation of a bouncing ball1 using anequation of y = 3e–0.1xsin(2x + 1.5708) as shown in Figure 8.20. Write a MATLAB m file for animation of the motion of crank-pistonsystem as shown in Figure 9.21. Write a MATLAB m file to animate the vibrating system of a massattached to a spring as shown in Figure 10.© 2001 by CRC Press LLCFIGURE 8.

Problem 19.22. Write a MATLAB m file to animate the motion of a cam-follower systemas shown in Figure 11.23. Write a MATLAB m file to animate the rotary motion of a wankel camas shown in Figure 12.24. Repeat Problem 9 but by Mathematica operation.25. Repeat Problem 11 but by Mathematica operation.26. Repeat Problem 14 but by Mathematica operation.27. Repeat Problem 15 but by Mathematica operation.28. Repeat Problem 16 but by Mathematica operation.© 2001 by CRC Press LLCFIGURE 9.

Problem 20.© 2001 by CRC Press LLCFIGURE 10. Problem 21.FIGURE 11. Problem 22.© 2001 by CRC Press LLCFIGURE 12. Problem 23.1.7 REFERENCE1. Y. C. Pao, “On Development of Engineering Animation Software,” in Computers inEngineering, edited by K. Ishii, ASME Publications, New York, 1994, pp. 851–855.© 2001 by CRC Press LLC.