Eigenfrequencies and eigenmodes

The solution of this problem must be preceded by the first four steps of the linear elastostatic problem solution, which include the calculation of element stiffness matrices $\mathbf{K}$. The following conditions apply:

• The same elements can be used as in linear elastostatics.
• Analytical prescription of symmetry or periodicity cannot be used.

The right-hand side does not matter in this case; therefore, only quantities specifying material properties and displacement boundary conditions can be considered in the first load case (AS 1) in the name.i2 file. It is recommended to check whether rigid body motion is prevented, meaning no pivots smaller than $\mathtt{PIVOT}$ are reported in the name.o4 file.

Input data are written to the text file name.iM. The program generates consistent (thus positive-definite) mass matrices $\mathbf{M}$ of elements.

Input data are written to the text file name.iE, where $\mathtt{KEVP}=0$. The program uses the subspace iteration method to calculate $\mathtt{NROOT}$ eigenpairs (eigenvectors and eigenvalues) of the generalized eigenvalue problem $$\sum(\mathbf{K}-\lambda_i\mathbf{M})\mathbf{v}_i=0, \quad i=1,\dots,\mathtt{NROOT},$$ where $\mathbf{v}_i$ and $\lambda_i$ are the $i$-th eigenvector and eigenvalue, and $\sum(\dots)$ denotes global (not element) matrices.

Input data are written to the text file name.iF. For each of the $\mathtt{NROOT}$ eigenpairs calculated in the previous step the program stores two records in the binary file name.FRQ. Odd records contain the normalized eigenvectors $\mathbf{v}_i$ while even records contain their corresponding eigenfrequencies $f_i$, where $$\lambda_i=(2\pi f_i)^2=\omega_i^2.$$

Input data are written to the text file name.i5, where the problem type key is $\mathtt{KPROB}=1$. The name.FRQ file must be renamed or copied to name.S.

The solution of this problem must be preceded by the calculation of element stiffness matrices $\mathbf{K}$ (i.e., the first three steps of the linear elastostatic problem solution). The following conditions apply:

• The same elements can be used as in linear elastostatics.
• Analytical prescription of symmetry or periodicity cannot be used.

The right-hand side does not matter in this case; therefore, only quantities specifying material properties and displacement boundary conditions can be considered in the first load case (AS 1) in the name.i2 file.

Input data are written to the text file name.iM. The program generates consistent (thus positive-definite) mass matrices $\mathbf{M}$ of elements.

Input data are written to the text file name.iR. The program factorizes the matrix $$\sum(\mathbf{K}+\mathtt{SHIFT}\,\mathbf{M}),$$ where $\sum(\dots)$ denotes global (not element) matrices. Choosing $\mathtt{SHIFT}>0$ makes it possible to achieve positive definiteness of this matrix in case $\sum(\mathbf{K})$ is only positive semidefinite.

Input data are written to the text file name.iE, where $\mathtt{KEVP}=0$. The program uses the subspace iteration method to calculate $\mathtt{NROOT}$ eigenpairs (eigenvectors and eigenvalues) of the generalized eigenvalue problem $$\sum(\mathbf{\tilde K}-\lambda_i\mathbf{M})\mathbf{v}_i=0, \quad i=1,\dots,\mathtt{NROOT},$$ where $\mathbf{v}_i$ and $\lambda_i$ are the $i$-th eigenvector and eigenvalue, and $\sum(\dots)$ denotes global (not element) matrices. When $\mathtt{SHIFT}>0$, the program works with the matrix $\sum(\mathbf{\tilde K})=\sum(\mathbf{K}+\mathtt{SHIFT}\,\mathbf{M})$ factorized by the HFRO program, whereas for $\mathtt{SHIFT}=0$ it uses the matrix $\sum(\mathbf{K})$ factorized by the FEFS program. The eigenvalues $\lambda$ are determined from the relation $\lambda=\lambda_\mathtt{SHIFT}-\mathtt{SHIFT}$, so that a free (unconstrained) body has zero as a multiple eigenvalue.

Input data are written to the text file name.iF. For each of the $\mathtt{NROOT}$ eigenpairs calculated in the previous step the program stores two records in the binary file name.FRQ. Odd records contain the normalized eigenvectors $\mathbf{v}_i$ while even records contain their corresponding eigenfrequencies $f_i$, where $$\lambda_i=(2\pi f_i)^2=\omega_i^2.$$

Input data are written to the text file name.i5, where the problem type key is $\mathtt{KPROB}=1$. The name.FRQ file must be renamed or copied to name.S.