Modal frequency analysis

Modal frequency analysis is used to define the natural, undamped frequency response of a structure.

Theoretically it is similar to the buckling analysis, however the physical interpretation of the results is different.

The dynamic response of a structure may be represented by the equation: where:

• is the mass matrix,
• the damping matrix,
• the stiffness matrix,
• the vector of forcing functions which may depend on time and
• , and are respectively, the vectors of nodal accelerations, velocities and displacement.

If, instead of this general equation, we consider the equation of an undamped and unforced structure we obtain, by setting the damping matrix and vector of forcing functions to zero, the following equation:

This equation defines the basic response of the structure and may be used to find the resonant frequencies. To see this, note that a solution to the equation above may be written in the form: where is the frequency of vibration.

Substitution of the above equation with the equation before it gives:

That is, where

This is a standard eigenvalue problem. It can be solved using the subspace iteration method.

For a structure with n degrees of freedom there will be n eigenvectors. To each eigenvalue there corresponds an eigenvector which is often called a mode shape. In practice it is not necessary to determine all eigenvalues. Generally it is sufficient to find the lowest few eigenvalues as these dominate the response of the structure. The Stress analysis program allows you to enter the number of eigenvalues to be found. In addition when a number of eigenvalues are calculated a Sturm sequence check is performed to check that the eigenvalues are consecutive.