Ch-07 (Pao - Engineering Analysis), страница 5

Apply the program CharacEq to find the characteristic equation for thevibration problem described in Problem 4.6. Apply the program MatxInvD to invert the matrix obtained in Problem4 and then apply the program EigenvIt to iteratively determine its smallesteigenvalue in magnitude and associated eigenvector.7.

Extend the vibrating system described in Problem 4 to four masses andfive springs and then implement the application of the programs MatxInvD and EigenvIt as described in Problems 5 and 6, respectively.8. Apply the programs CharacEq, Bairstow, and EigenVec to find thecharacteristic equation, eigenvalues, and associated eigenvectors for thematrix derived in Problem 4, respectively.9. Same as Problem 8 except for a four masses and five springs system.10.

An approximate analysis of a three-story building is described in Problem7 in the program EigenvIt. Derive the governing differential equations forthe swaymotions xi for i = 1,2,3 and then show that the stiffness matrix[K] and mass matrix [M] are indeed as those given there.CHARACEQ1. Apply Feddeev-Leverrier method to find the characteristic equation of thematrix:© 2001 by CRC Press LLC14 72583692. Apply Feddeev-Leverrier method to find the characteristic equation of thematrix: 5−10 −206010 73.

Apply Feddeev-Leverrier method to find the characteristic equation of thematrix: 2−10 −221432 94. Apply the program CharacEq for solving Problems 1 to 3.5. Apply Feddeev-Leverrier method to find the characteristic equation of thematrix:52006310 76. Apply poly.m of MATLAB to Problems 1 to 3 and 5.7. Find the roots of the polynomials found in Problem 6 by application ofroots.m of MATLAB.8. Apply plot.m of MATLAB for the polynomials obtained in Problem 6.9. Apply the function det of Mathematica to derive the characteristic equation for the matrix given in Problem 1.10.

Apply the function det of Mathematica to derive the characteristic equation for the matrix given in Problem 2.11. Apply the function det of Mathematica to derive the characteristic equation for the matrix given in Problem 3.12. Apply the function det of Mathematica to derive the characteristic equation for the matrix given in Problem 5.© 2001 by CRC Press LLCEIGENVEC1. Run the QuickBASIC version of the program EigenVec for the samplecase used in the FORTRAN version.2.

Apply the program EigenVec to find the eigenvector corresponding to theeigenvalue equal to 4.41421 for the matrix:20130 40303. Apply the program CharacEq to find the characteristic equation formatrix:14 736 10 258and then apply the program Bairstow to find the eigenvalues. Finally,apply the program EigenVec to find the eigenvectors.4. Apply the program CharacEq to find the characteristic equation for thematrix: 5−10 −206010 7and then apply the program Bairstow to find the eigenvalues.

Finally,apply the program EigenVec to find the eigenvectors.5. Apply the program CharacEq to find the characteristic equation for thematrix: 2−10 −221432 9and then apply the program Bairstow to find the eigenvalues. Finally,apply the program EigenVec to find the eigenvectors.6. The eigenvalues for the following matrix have been found to be equal to9.3726, 32 and 54.627:© 2001 by CRC Press LLC3216 0016 32 163216Find the associated eigenvector by applying the program EigenVec.7. The eigenvalues of the following matrix have been found to be equal to9.5492, 34.549, 64.451, and 90.451:5025002550250002550 0255025Find the associated eigenvector by applying the program EigenVec.8. Find the eigenvalue and associated eigenvector of the matrix:763600 0367236000367236000036 72 003672369.

Swaying motion of a three-story building is described in Problem 7 in theprogram EigenvIt. Use the data there to form the matrix [A] which isequal to [K]–1 [M].Apply the programs CharacEq and Bairstow to find all three eigenvaluesand then apply the program EigenVec to find the associated eigenvectors.10. Apply the function eig.m of MATLAB to find all eigenvalues of thematrices given in Problems 2 to 8.11. Apply the functions eigenvalues and eigenvectors of Mathematica tofind all eigenvalues of the matrices given in Problems 2 to 8.EIGENVIT1. Using an initial, guessed eigenvector {V} = [1 0 0]T, perform four iterativesteps to find the largest eigenvalue in magnitude and its associated normalized vector of the matrix: 2[A] = −10 −2© 2001 by CRC Press LLC21432 92.

Using an initial, guessed eigenvector {V} = [1 0 0]T, perform four iterativesteps to find the largest eigenvalue in magnitude and its associated normalized vector of the matrix:2[A] = 0110 40303. Apply the program EigenvIt to find the largest eigenvalue in magnitudeand its associated normalized eigenvector of the matrix:51[A] = 4.5.6.7.33Next, apply the program MatxInvD to find the inverse of [A] which isto be entered as input for program EigenvIt to iterate the smallest eigenvalue in magnitude and its associated normalized eigenvector for [A].Compare the results with the analytical solution of smallest = 2 and largest = 6.Apply the program MatxInvD to find the inverse of the matrix [A] givenin Problem 1 and then apply the program EigenvIt to find the smallesteigenvalue in magnitude and its associated normalized eigenvector of [A].For checking the values of smallest obtained here and largest obtained inProblem 1, derive the characteristic equation of [A] by use of the programCharacEq and solve it by application of the program Bairstow.Same as Problem 4 but for the matrix [A] given in Problem 2.Apply poly.m, roots.m, polyval.m, plot.m, and xlabel and ylabel toobtain a plot of the characteristic equation of the matrix [A] given inProblem 1, shown in Figure 7, to know the approximate locations of thecharacteristic roots.For a 3-floor building as sketched in the left side of Figure 8, an approximate calculation of its natural frequencies can be attempted by using alumped approach which represents each floor with a mass and the stiffnesses of the supporting columns by a spring as shown in the right sideof Figure 8.

If the swaying motion of the floors are expressed as xi =Xisint for i = 1,2,3 where is the natural frequency and Xi are theamplitudes, it can be shown that and {X} = [X1 X2 X3]T satisfy thematrix equation [K]{X} = 2[M]{X}, in which the mass matrix [M] andstiffness matrix [K] are formed by the masses and spring constants asfollows:m1[M] =  0 0© 2001 by CRC Press LLC0m2000m 3 and k1[K] = − k 2 0−k2k1 + k 2−k30 −k3 k 2 + k 3 FIGURE 7. Problem 6.FIGURE 8. Problem 7.© 2001 by CRC Press LLC8.9.10.11.12.To find the lowest natural frequency min, the program EigenvIt can beapplied to obtain the max from the matrix equation [A]{X} = {X} wherethe matrix [A] is equal to [K]–1[M] and = 2.

min is equal to 1/max.Determine the numeric value of min for the case when m1 = 8x105, m2 =9x105, and m3 = 1x106 all in N-sec2/m, and k1 = 3x108, k2 = 4x108, andk3 = 5x108 all in N/m.Referring to Figure 2 in the program EigenVec, iteratively determine themaximum and minimum principal stresses and their associated principalplanes at a point where the two-dimensional normal and shear stressesare x = 50, y = –30, and xy = yx = –20 all in N/cm2. Compare the resultswith those obtained in the program EigenVec.Same as Problem 8, except for a three-dimensional case of x = 25, σy = 36,z = 49, xy = yx = –12, yz = zy = 8, and zx = xz = –9, all in N/cm2.Apply MATLAB to invert the matrix [A] given in Problem 1 and thenapply EigenvIt.m to iterate the eigenvalue which is the smallest in magnitude and also the associated eigenvector.Same as Problem 10 but for the matrix [A] given in Problem 2.Apply Mathematica to solve Problems 10 and 11.7.7 REFERENCES1.

W. F. Riley and L. Zachary, Introduction to Mechanics of Materials, Wiley & Sons,Inc., New York, 1989.2. K. N. Tong, Theory of Mechanical Vibration, Wiley & Sons, Inc., New York, 1960.3. Y. C. Pao, “A General Program for Computer Plotting of Mohr’s Circle,” Computersand Structures, V. 2, 1972, pp. 625–635. This paper discusses various sources of howeigenvalue problems are formed and also methods of analytical, computational, andgraphical solutions.4. Y.

C. Pao, “A General Program for Computer Plotting of Mohr’s Circle,” (for twodimensional cases), Computers and Structures, V. 2, 1972, pp. 625–635.5. F. B. Seely and J. O. Smith, Advanced Mechanics of Materials, Second Edition, JohnWiley, New York, 1957, pp. 59–64.6. F. B. Hilebrand, Methods of Applied Mathematics, Prentice-Hall, Englewood Cliffs,NJ, 1960.7. S. Perlis, Theory of Matrices, Addison-Wesley Publishing Company, Reading, MA,1952.© 2001 by CRC Press LLC.