Product: ABAQUS/Standard
The tests in this section verify the complex eigenvalue extraction procedure in ABAQUS/Standard, which uses the subspace projection method. The procedure is tested for systems with symmetric stiffness matrices that include damping terms and for problems with friction, which introduces unsymmetry to the stiffness matrix.
Complex eigenvalue extraction for a system with a symmetric stiffness matrix, both with and without damping.
In both tests the model consists of a quadratic element of unit length. The nodes at one end (
) are constrained. The eigenvalue extraction is performed for the undeformed configuration.
The stiffness matrix in the first problem (pcfreq_ce4sf_real.inp) is symmetric and contains no damping. In the absence of damping contributions, the eigenvalues extracted by the complex eigensolver must have zero real components and the imaginary components (frequencies) must be the same as the frequencies obtained in the preceding frequency extraction step. In the second problem (pcfreq_ce4sf_real.inp) mass-proportional damping is introduced. The following relations can be derived for an underdamped system with mass-proportional damping: 
and 
, where 
and 
are the real and imaginary components of the complex eigenvalues, respectively; 
is the mass-proportional damping factor; and 
is the natural frequency of the undamped system. The complex eigenvalues obtained for this problem match the formulae above.
Complex eigenvalue extraction for a symmetric stiffness matrix without damping.
Complex eigenvalue extraction for a symmetric stiffness matrix with mass-proportional damping.
Complex eigenvalue extraction for a system with an unsymmetric stiffness matrix caused by a friction contribution.
The model consists of a ring with an inside radius of 1.0 and an outside radius of 2.0 and two plates positioned at both sides of the ring. The ring is modeled using a linear elastic material with a Young's modulus of 200, Poisson's ratio of 0.3, and density of 1.0. Contact pairs define contact between the side surfaces of the ring and the plates. The ring is meshed with 16 linear brick elements (element type C3D8). The plates are modeled with membrane elements (element type M3D4) for the models with deformable-to-deformable contact or with rigid elements (element type R3D4) for the problems with deformable-to-rigid contact.
The loading consists of two steps. In the first step the plates are moved a distance of 0.05 toward the ring to establish frictionless contact. In the second step the friction coefficient is increased to 0.3 and a rotational velocity is imposed on the ring. Because the complex eigensolver uses the subspace projection method, the natural frequencies must be extracted prior to the complex eigenvalue extraction step. The following problems with different contact models are considered:
deformable-to-deformable contact with small sliding (pcfreq_def_ss.inp),
deformable-to-deformable contact with small sliding, including friction-induced damping effects (pcfreq_def_ss_fdamp.inp),
deformable-to-rigid contact with small sliding (pcfreq_rg_ss.inp),
deformable-to-deformable contact with finite sliding (pcfreq_def_fs.inp), and
deformable-to-deformable contact with finite sliding in a restarted analysis (pcfreq_def_fs_res.inp).
An analytic solution is not available for this problem, so the results (the frequency of an unstable mode and the damping ratio) are compared only between the different models. As shown in the table that follows, the results for pcfreq_def_fs.inp, pcfreq_def_fs_res.inp, pcfreq_def_ss.inp, pcfreq_def_ss_fdamp.inp, pcfreq_def_ss_negdamp.inp, pcfreq_rg_ss.inp, and pcfreq_sst_3d.inp are in very good agreement. The differences in the results for pcfreq_sup_use.inp are due to the use of a substructure to model the elastic ring.
Deformable-to-deformable contact with small sliding.
Deformable-to-deformable contact with small sliding, including friction-induced damping effects.
Deformable-to-rigid contact with small sliding.
Deformable-to-deformable contact with finite sliding.
Deformable-to-deformable contact with finite sliding, restarted analysis.
Deformable-to-rigid contact with finite sliding, rotational velocity imposed in the steady-state transport step.
Axisymmetric mesh generation used in pcfreq_sst_3d.inp.
Substructure analysis.
Substructure generation file referenced in pcfreq_sup_use.inp.
Deformable-to-deformable contact with small sliding and velocity-dependent friction coefficient.