Page 274 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 8.12 Description of Computer Programs 261

In the matrix iteration, the convergence is always to the largest eigenvalue,
and because the lower natural frequencies are generally desired, the stiffness
matrix should be decomposed. In the finite element problem, the mass matrix is
generally full, and, as shown in Sec. 8.7, matrices M and K can be interchanged
for the computation of either A or A.
The source programs are included on the disk and can be modified by
inserting the appropriate commands to examine intermediate results, for example,
the dynamic matrix [A\ or results after each iteration. These files have the
extension: for.
There are other computer programs that are available today. The House­
holder method, which relies first on Given’s method to tridiagonalize the A matrix,
requires no rotation matrix and solves for the eigenvalues and eigenvectors without
iteration. New software programs are constantly being developed and made
available for both IBM and Apple computers. Mathematica is one such system,
which is rapidly gaining favor due to its simplicity for the user.

8.12 DESCRIPTION OF COMPUTER PROGRAMS

Program RUNGA. RUNGA solves the differential equation mx + cx kx
= fit), with input values for m,c,k, and the forcing function fit) presented in
digital pairs, fftf) and t^, up to 20 values of t^. Function fit) is linearly interpo­
lated between input pairs. The output is given for a duration of about 2.5 periods
(30/47t), with increments of about 1/12 period (1 /4 tt). The output can be
numerical an d /o r as a rough graph.

Program POLY. POLY has three options:

1. Determines the coefficients to the polynomial resulting from the charac­
teristic determinant \M —AA^|. The user inputs the n X n matrices M and

K. The coefficients obtained can be sent to option 2.
2. Determines the roots of the polynomial
1

-n+\-^ ' - i - C ^ X + c, = 0
^ 2-^ '
The user inputs the (/7 + 1) coefficients c^. If these are the roots (eigenval­
ues) from option 1, these can be sent to option 3.

3. Option 3 determines the eigenvectors of M —\ K. The user inputs matri­
ces M and K and the eigenvalue A, for each eigenvector. The program
uses the Choleski decomposition to form the standard equation A - XL
Program ITERATE. ITERATE determines the eigenvalues and eigenvec­
tors using matrix iteration. The user inputs matrices M and K, and the standard
form is developed by Choleski decomposition. Subsequent modes can be deter­
mined with matrix deflation of the Gram-Schmidt orthonormalization.