Complex Eigenvalue Extraction

The complex eigenvalue extraction procedure uses a projection method to extract the complex eigenvalues of the current system. The eigenvalue problem of the finite element model is formulated in the following manner:

where

$M^{MN}$

is the mass matrix (which is symmetric and, in general, is semi-positive definite);

$C^{MN}$

is the damping matrix;

$K^{MN}$

is the stiffness matrix (which can include initial stress stiffness and friction effects and, therefore, in general is unsymmetric);

$\mu$

is the complex eigenvalue;

${\varphi }_R^N$

is the right complex eigenvector;

${\varphi }_L^N$

is the left complex eigenvector and is defined as follows:

$${\left({\varphi }_L^N\right)}^{*}\left({\mu }^2{M}^{MN}+\mu C^{MN}+K^{MN}\right)=0;$$

where ${\left({\varphi }_L^N\right)}^{*}$ is a transpose conjugate left eigenvector

M and N

are degrees of freedom.

The complex eigenvalue extraction procedure in Abaqus/Standard uses a subspace projection method; thus, the eigenmodes of the undamped system with the symmetrized stiffness matrix must be extracted using the eigenfrequency extraction procedure prior to the complex eigenvalue extraction step. By default, the entire subspace is used as the base vector; this subspace can be reduced as described below. Abaqus/Standard always computes all the complex eigenmodes available in the projection subspace (taking into account any user-specified modifications to the subspace). The user-specified number of requested eigenmodes and frequency range for the complex eigenvalue extraction procedure do not influence the number of computed complex eigenmodes. It determines only the number of reported modes, which cannot be higher than the dimension of the projected subspace. To modify the number of computed eigenmodes, reduce the projection subspace as described below or change the number of eigenmodes extracted in the prior natural frequency extraction step accordingly. If you do not specify the number of requested complex modes or the frequency range, all the computed modes will be reported.

To take into account the unsymmetric effects, the unsymmetric matrix solution and storage scheme is used automatically for a complex eigenvalue extraction step. The unsymmetric effects will be disregarded if you specify that the symmetric solution and storage scheme should be used (see Defining an Analysis).

COMPLEX FREQUENCY
number of complex eigenmodes, frequency_min, frequency_max

Step module: Create Step: Linear perturbation: Complex frequency: Number of eigenvalues requested: All or Value, Minimum frequency of interest (cycles/time): value, Maximum frequency of interest (cycles/time): value

Abaqus/Standard automatically detects the contact nodes that are slipping due to velocity differences imposed by the motion of the reference frame or the transport velocity in prior steps. At those nodes the tangential degrees of freedom will not be constrained and the effect of friction will result in an unsymmetric contribution to the stiffness matrix. At other nodes in contact the tangential degrees of freedom will be constrained.

Friction at contact nodes at which a velocity differential is imposed can give rise to damping terms. There are two kinds of friction-induced damping effects. The first effect is caused by the friction forces stabilizing the vibrations in the direction perpendicular to the slip direction. This effect exists only in three-dimensional analysis. The second effect is caused by a velocity-dependent friction coefficient. If the friction coefficient decreases with velocity (which is usually the case), the effect is destabilizing and is also known as “negative damping.” For more details, see Coulomb friction. The complex eigensolver allows you to include these friction-induced contributions to the damping matrix.

COMPLEX FREQUENCY, FRICTION DAMPING=YES

Step module: Create Step: Linear perturbation: Complex frequency: Include friction-induced damping effects 

In complex eigenvalue extraction analysis damping can be defined by dashpots (see Dashpots), by “Rayleigh” damping associated with materials and elements (see Material Damping), and by quiet boundaries on infinite elements or acoustic elements. In addition, as described in Contact Conditions with Sliding Friction above, friction-induced damping can be included.

Structural damping, damping contributions from frequency-domain viscoelasticity, and all types of modal damping (except composite modal damping) are supported in complex eigenvalue extraction using the high-performance SIM architecture.

Motion, transport velocity, and acoustic flow velocity affect complex frequency analyses. Motion and transport velocity must be specified in a preceding steady-state transport general step, and their effects are included in the complex frequency step. The acoustic flow velocity has no effect in steady-state transport steps, and acoustic flow velocities specified in a steady-state transport step are not propagated to perturbation steps. The acoustic flow velocity must be specified in each linear perturbation step where it is desired.

Two methods are available to reduce the computational time in the stiffness projection phase of the analysis, but they should be used with caution. By default, the complex eigensolver projects the stiffness operator on the modal subspace. This is necessary when advanced mechanical effects, such as friction-induced effects and frequency-dependent properties, must be considered. However, for large modal subspaces, this operation can be expensive.