An eigenfrequency study solves for the eigenfrequencies (natural frequencies) and the shapes of the corresponding eigenmodes.

When performing an eigenfrequency analysis, you can specify whether to look at the mathematically more fundamental eigenvalue, λ, or the eigenfrequency, f, which is more commonly used in a structural mechanics context. The relation between the two is

where i is the imaginary unit.

Because only the shape and not the size of the modes (eigenvectors) have physical significance, the computed modes can be scaled arbitrarily. You can select the method for scaling in the Eigenvalue Solver node of the solver sequence. If output of mass participation factors is required, then Scaling of eigenvectors must be set to Mass matrix. This means that the eigenmodes U are orthogonalized with respect to the mass matrix M so that

The mass (or ‘modal’) participation factor for mode i in direction j, rij, is defined as

Here, dj is a vector containing unity displacement in all degrees of freedom representing translation in direction j. The mass participation factor gives an indication of to which extent a certain mode might respond to an excitation in that direction.

The mass participation factors are available as a global variables, and these can be shown in a table using a Global Evaluation node under Derived Values in the Results branch, for example. The participation factor variables are available as predefined variables in the Solver submenu.

It is possible to compute eigenfrequencies for structures which are not fully constrained; this is sometimes referred to as free-free modes. For each possible rigid body mode, there is one eigenvalue which in theory is zero. The number of possible rigid body modes for different geometrical dimensions is shown in the table below.

In practice, the natural frequencies of the rigid body modes are not computed as exactly zero, but can appear as small numbers which can even be negative or complex. If rigid body modes are present in the model, then it is important to use a nonzero value in the Search for eigenfrequencies around text field in the settings for the Eigenfrequency study step. The value should reflect the order of magnitude of the first important nonzero eigenfrequency.

In a complex mode shape there are phase shifts between different parts of the structure, so that not all points reach the maximum at the same time under free vibration.

In a loaded structure, the natural frequencies may be shifted due to stress stiffening. With the Prestressed Analysis, Eigenfrequency study type you can compute eigenfrequencies taking this effect into account.