Page 275 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

262 Computational Methods Chap. 8

Program CHOLJAC (Choleski-Jacobi). CHOLJAC has three options:
1. Determines the three matrix products M ^ K. The user inputs the n X n
matrices M and K.
2. Determines the eigenvalues and eigenvectors of M - kL It uses Jacobi
diagonalization.
3. Determines the eigenvalues and eigenvectors of M - AA^. It uses Choleski
decomposition and Jacobi diagonalization.

REiFERENCES

[1] Crandall, S. H. Engineering Analysis, A Survey of Numerical Procedures. New York:
McGraw-Hill Book Company, 1956.
[2] Ralston, A., and Wilf, H. S. Mathematical Methods for Digital Computers, Vols. I
and II, New York: John Wiley & Sons, 1968.
[3] Salvadori, M. G., and Baron, M. L. Numerical Methods in Engineering. Englewood
Cliffs, N.J.: Prentice-Hall, 1952.
[4] Bathe, K.-J., and Wilson, E. L. Numerical Methods in Finite Element Analysis.
Englewood Cliffs, N.J.: Prentice-Hall, 1976.
[5] Meirovitch, L. Computational Methods in Structural Dynamics. Rockville, MD:
Sijthoff & Noordhoff, 1980.

PROBLEMS

8-1 For the system shown in Fig. P8.1, the flexibility matrix is
0.5 0.5 0.5
[a] = T 0.5 1.5 1.5
0.5 1.5 2.5
Write the equation of motion in terms of the flexibility and derive the characteristic
equation
- 5A^ 4- 4.5A - 1 = 0
Show agreement with the characteristic equation in Sec. 8.1 by substituting A = 1/A in

the foregoing equation.
| —AAAAr-| 2m j-AAAAr-| rn AAAAr-[~^^

*"-^1 ^^2 ^^^3 Figure P8.1.

8-2 Use the computer program POLY to solve for A, and </>,, and verify the and (/> ,
given in Sec. 8.1.