Products: Abaqus/Standard Abaqus/AMS
The tests in this section verify the mode-based steady-state dynamic analysis procedure supporting nondiagonal damping (structural, viscous, material, and global damping) using the AMS eigensolver in Abaqus/Standard. As a reference solution, the results obtained by the subspace-based steady-state dynamic analysis procedure using the Lanczos eigensolver are used. Some tests are compared to the steady-state direct method.
Mode-based steady-state dynamic step using the eigensolution computed by the AMS eigensolver for a system with material damping, global damping, and the damping controls option.
The two-dimensional model consists of a linear element of unit length with material damping. The nodes at the bottom (y = 0.0) are constrained, and real and imaginary parts of the concentrated loads are applied to the nodes at the top (y = 1.0) . The three-dimensional model is used for testing the selecting eigenmodes and selective modal recovery features.
The nodal variables at the requested frequency obtained by both the mode-based steady-state dynamic analysis procedure using the AMS eigensolver and the subspace-based steady-state dynamic analysis procedure using the Lanczos eigensolver are identical.
Mode-based steady-state dynamic analysis using the AMS eigensolver (CPE4).
Subspace-based steady-state dynamic analysis using the Lanczos eigensolver (CPE4).
Mode-based steady-state dynamic analysis using the Lanczos eigensolver (CPE4) including global damping and damping controls.
Mode-based steady-state dynamic analysis using the AMS eigensolver (C3D8). Global damping and damping controls tested.
Subspace-based steady-state dynamic analysis using the Lanczos eigensolver (C3D8).
Mode-based steady-state dynamic step using the eigensolution computed by the AMS eigensolver for a system with discrete material damping (connector damping and dashpot). Global damping and damping controls options are tested here.
The simple one degree of freedom model consists of three components: a spring, a mass, and a dashpot. Left-hand sides of the spring and the dashpot are connected to the ground, and the mass element is attached to the right-hand sides of the spring and the dashpot. A unit concentrated load is applied to the mass element in the direction of degree of freedom 1.
The connector model consists of three Cartesian-type connectors that are sequentially connected together. It has two degrees of freedom, and complex connector loads are applied on the two middle nodes.
The results from the mode-based steady-state dynamic analysis procedure using the AMS eigensolver and the subspace-based steady-state dynamic analysis procedure using the Lanczos eigensolver for the spring-mass-damper system are identical in the frequency range of interest.
For the connector model, the results from the mode-based steady-state dynamic analysis procedure using the AMS eigensolver and the subspace-based steady-state dynamic analysis procedure using the Lanczos eigensolver are identical in the frequency range of interest.
Mode-based steady-state dynamic analysis using the AMS eigensolver for a spring-mass-dashpot model with one degree of freedom.
Subspace-based steady-state dynamic analysis using the Lanczos eigensolver for a spring-mass-dashpot model with one degree of freedom.
Mode-based steady-state dynamic analysis using the AMS eigensolver for a three-dimensional connector element model with connector damping. Global damping and damping controls tested.
Subspace-based steady-state dynamic analysis using the Lanczos eigensolver for a three-dimensional connector element model with connector damping. Global damping and damping controls tested.
Mode-based steady-state dynamic analysis using the AMS eigensolver tested with global damping and damping controls.
Mode-based steady-state dynamic step for a system with frequency-dependent viscoelastic material and property evaluation feature in the frequency extraction step.
The two-dimensional model is a simple cantilever beam model with 12 CPS4 elements. Left-end nodes of a cantilever beam are fixed, and 1.0 GPa is applied to the top surface of the cantilevered beam. Frequency-domain viscoelastic material is defined in a tabular form.
The results from the mode-based steady-state dynamic analysis procedure at about every 10 Hz are compared with the results from the subspace-based steady-state dynamic analysis procedure with the Lanczos eigensolver, as shown in the table below.
| Frequency | SSD with AMS | SSD, SP with Lanczos | ||
|---|---|---|---|---|
| Magnitude | Phase | Magnitude | Phase | |
| 9.08 Hz | –2.714 | 1.5522e-03 | 2.714 | 180.0 |
| 19.18 Hz | 5.580 | 179.9 | 5.581 | 179.9 |
| 29.29 Hz | 7.326 | 0.2724 | 7.235 | 0.2676 |
| 39.39 Hz | 1.751 | 8.055e-02 | 1.751 | 7.9158e–02 |
| 49.49 Hz | 0.9103 | 6.2855e-04 | 0.9103 | 3.8876e-02 |
| 59.59 Hz | 0.5928 | 8.5927e-03 | 0.5928 | 8.4387e-03 |
| 69.69 Hz | 0.4381 | –3.2477e-02 | 0.4381 | –3.1929e-02 |
| 79.80 Hz | 0.3575 | –0.1051 | 0.3575 | –0.1033 |
| 91.92 Hz | 0.3248 | –0.2611 | 0.3248 | –0.2566 |
| 100.00 Hz | 0.3514 | –0.6925 | 0.3507 | –0,6808 |
Mode-based steady-state dynamic analysis using the AMS eigensolver for a two-dimensional model with frequency-domain viscoelasticity
Subspace-based steady-state dynamic analysis using the Lanczos eigensolver for a two-dimensional model with frequency-domain viscoelasticity
Mode-based steady-state dynamic step with base motion, eigenmode selection, and beam general section along with material damping.
The model consists of 20 Euler-Bernoulli beams sequentially connected; each end of the beams is constrained to the ground. Primary base motion is prescribed with user-defined amplitude, and the first 25 modes are selected for mode-based steady-state dynamic analysis.
The results from both the mode-based steady-state dynamic analysis procedure and the subspace-based steady-state dynamic analysis procedure for this model are identical.
Two-dimensional model for a subspace-based steady-state dynamic analysis with base motion, selective eigenmodes, and Lanczos eigensolver.
Model:
A cantilever beam with a dashpot at the tip.
Material:
Young's modulus = 2.0 × 105, Poisson’s ratio = 0.3, density = 2.0 × 10–6. Dashpot damping is frequency dependent as follows:
The beam is fixed at one end and is free at the other. The dashpot is connected to the tip and grounded at the other end. A concentrated load of amplitude 1200 is applied at the tip of the cantilever beam. For the second load case the same load is applied as an imaginary part of the load for comparison. The steady-state dynamic analysis is run from 0 to 100 Hz using subspace projection based on modes computed up to 200 Hz.
The results from the two load cases match in magnitude to the results from a single load case step. The results from the imaginary load case are off by 90° in phase as expected. The following table shows the peak response values:
SIM-based steady-state dynamic analysis of the cantilever beam with dashpot, subspace, and multiple load cases. Units: mm, N, MPa.