Page 538 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Appen. F Computer Programs 525

If the eigenvectors are also desired, the eigenvalues are sent to option (3),
where the Gaussian elimination method is used on the equation {M - \K)(f) = i)
for the eigenvectors cf).
When the coefficients c, are initially available, option (2) can be used
directly for the eigenvalues A.
When M, K, and A are initially available, option @ can be used directly to

determine the eigenvectors cj).

Iteration
In the flow diagram for the iteration method (see Fig. F-3), the method input block
shows matrix order N, mass matrix M, and stiffness matrix K. The equation of
motion is expressed in the form K^M X = AX, and the stiffness matrix K = Q^Q
is first decomposed by the Cholesky method for the determination of Q, Q \ and

Q~^ and the dynamic matrix A = K^M = Q~^Q~ ^M, which in this case is
generally unsymmetric. The sweeping matrix S is introduced as a unit matrix / for
the first mode.
The iteration procedure follows in the block ASX^^^ = A,W,, which is
normalized in the next block and tested for convergence in the decision block and
looped back for further iteration. When the difference |A,^, —A,| reaches a value

smaller than the tolerance, the first mode A, and its eigenvector cf) is complete, and
the calculation is sent back to the left loop for the determination of the sweeping
matrix and iteration for the second mode, etc.

CHOLJAC

The program CHOLJAC offers three options. See Fig. F-4. Option (D determines
the product M ^ K o f any two square matrices M and K . The user inputs the
N X N matrices M and K .
Option (2) determines the eigenvalues and eigenvectors of /I - A/, where Ä
is the symmetric dynamic matrix. The user inputs the matrix A and Jacobi
iteration is applied to diagonalize the matrix A . The eigenvalues Ä and the
j
eigenvectors c) are outputted.
Option (3) starts with the input of the mass and stiffness matrices M and K .
By using Cholesky decomposition and Jacobi diagonalization, the eigenvalues and

eigenvectors of (M —XK)(t) are determined. The program decomposes the first
matrix inputted, which for the flow diagram shown is the stiffness matrix. The
eigenvalues are then proportional to the reciprocal of the natural frequencies oj~.