About Frequency Steps

For a video that describes the fundamentals of natural frequencies and frequency extraction, see Natural Frequency Step in FEA for 3DEXPERIENCE.

If geometric nonlinearity is accounted for in the base state, a frequency step includes initial stress and load stiffness effects; these effects enable you to model small vibrations of a preloaded structure. Frequency steps are also linear perturbation procedures that can compute residual modes if requested.

Eigenvalue Extraction

The eigenvalue problem for the natural frequencies of an undamped finite element model is

(- ω^{2} M^{M N} + K^{M N}) ϕ^{N} = 0,

where $M^{M N}$ is the mass matrix (which is symmetric and positive definite); $K^{M N}$ is the stiffness matrix (which includes initial stiffness effects if the base state included the effects of nonlinear geometry); $ϕ^{N}$ is the eigenvector (the mode of vibration); and $M$ and $N$ are degrees of freedom.

When $K^{M N}$ is positive definite, all eigenvalues are positive. Rigid body modes and instabilities cause $K^{M N}$ to be indefinite. Rigid body modes produce zero eigenvalues. Instabilities produce negative eigenvalues and occur when you include initial stress effects. Abaqus/Standard solves the eigenfrequency problem only for symmetric matrices.

Eigenvalue Extraction Methods

You have two options for eigenvalue extraction methods: Lanczos and Automatic Multi-level Substructuring (AMS). The Lanczos solver with the SIM architecture is the default eigenvalue extraction method because it has the most general capabilities. However, the Lanczos method is generally slower than the AMS method. The increased speed of the AMS eigensolver is particularly evident when you require a large number of eigenmodes for a system with many degrees of freedom.

Structural-Acoustic Coupling

Structural-acoustic coupling affects the natural frequency response of systems. In Abaqus the AMS eigensolver and the Lanczos eigensolver can extract coupled modes to fully include this effect. The subspace eigensolver neglects the effect of coupling for the purpose of computing the modes and frequencies; the modes and frequencies are computed using natural boundary conditions at the structural-acoustic coupling surface. By default, the same is done for the AMS eigensolver; the coupling is projected onto the modal space and stored for later use.