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.

where K is the stiffness matrix, M is the mass matrix, u is the eigenmode displacement vector, and ω = 2πf is the angular frequency. In case of damping, the eigenvalue equation is expanded to

where C is the viscous damping matrix, and K can be complex valued.

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 Scaling of eigenvectors is set to Mass matrix, the eigenmodes u are orthogonalized with respect to the mass matrix M so that

To compute modal participation factors, a Participation Factors node must be present under Definitions in the current component. When you add an Eigenfrequency study from the Add Study window, such a node is automatically created.

You can also add it manually under Definitions>Variable Utilities. If you do that after an eigenfrequency study has been run, you need to do an Update Solution in order to get access to the variables containing the participation factors.

The modal participation factors are available as global variables, and these can for example be displayed in a table using a Global Evaluation node under Derived Values in the Results branch. The participation factor results are available as predefined variables in the Definitions submenu for the component. In Table 2-1, the variables created from a Participation Factors node is listed (assuming the default tag mpf1).

If a Participation Factors node is present in the model when an eigenfrequency study is run, an evaluation group named Participation Factors is automatically generated. It contains a table with the translational and rotational modal participation factors for all computed eigenfrequencies.

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.

where the approximate expression is valid with high accuracy (within 2%) as long as the damping is less than 0.2.

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.

To do a prestressed analysis, Include geometric nonlinearity must be selected in the Eigenfrequency study step. This is automatic when you add the Eigenfrequency, Prestressed study type.

If the material data (stiffness or damping) is frequency dependent, the eigenvalue problem will become nonlinear. The most common case is when there are viscoelastic materials in the structure. In this case, the eigenvalue equation Equation 2-1 becomes

The eigenvalue solver as such assumes that the matrices involved are constant, so they must be evaluated at a certain frequency, the linearization point.

In order to get a correct solution to Equation 2-3, the linearization point ωL must be close to the actual eigenvalue ω. This is in general possible only for one single eigenfrequency at a time. You must solve this problem either by manual iteration, or by using some type of scripting, for example through a model method.