Steady-state dynamics with nondiagonal damping using the AMS eigensolver

This problem contains basic test cases for one or more Abaqus elements and features.

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.

This page discusses:

One-element tests
Model with discrete material damping
Model with frequency-dependent material
Model with base motion
SIM-based steady-state analysis with multiple load cases

ProductsAbaqus/StandardAbaqus/AMS

One-element tests

Elements tested

CPE4

C3D8

Features tested

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.

Problem description

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.

Results and discussion

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.

Input files

ssd_ams_1cpe4.inp: Mode-based steady-state dynamic analysis using the AMS eigensolver (CPE4).
ssd_lnz_1cpe4.inp: Subspace-based steady-state dynamic analysis using the Lanczos eigensolver (CPE4).
ssd_lnz_1cpe4_sdamp.inp: Mode-based steady-state dynamic analysis using the Lanczos eigensolver (CPE4) including global damping and damping controls.
ssd_ams_1c3d8.inp: Mode-based steady-state dynamic analysis using the AMS eigensolver (C3D8). Global damping and damping controls tested.
ssd_lnz_1c3d8.inp: Subspace-based steady-state dynamic analysis using the Lanczos eigensolver (C3D8).

Model with discrete material damping

Elements tested

CONN3D2

SPRING1

DASHPOT1

MASS

T3D2

Features tested

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.

Problem description

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.

Results and discussion

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.

Input files

ssd_ams_1dof.inp: Mode-based steady-state dynamic analysis using the AMS eigensolver for a spring-mass-dashpot model with one degree of freedom.
ssd_lnz_1dof.inp: Subspace-based steady-state dynamic analysis using the Lanczos eigensolver for a spring-mass-dashpot model with one degree of freedom.
ssd_ams_conn3d.inp: 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.
ssd_lnz_conn3d.inp: 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.
t3d2_ssd_ams_sdamping.inp: Mode-based steady-state dynamic analysis using the AMS eigensolver tested with global damping and damping controls.

Model with frequency-dependent material

Elements tested

CPS4

Features tested

Mode-based steady-state dynamic step for a system with frequency-dependent viscoelastic material and property evaluation feature in the frequency extraction step.

Problem description

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.

Results and discussion

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

Input files

ssd_ams_viscoe_cps4.inp: Mode-based steady-state dynamic analysis using the AMS eigensolver for a two-dimensional model with frequency-domain viscoelasticity
ssd_lnz_viscoe_cps4.inp: Subspace-based steady-state dynamic analysis using the Lanczos eigensolver for a two-dimensional model with frequency-domain viscoelasticity

Model with base motion

Elements tested

B23

Features tested

Mode-based steady-state dynamic step with base motion, eigenmode selection, and beam general section along with material damping.

Problem description

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.

Results and discussion

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.

Input files

ssd_lnz_base_b23.inp: Two-dimensional model for a subspace-based steady-state dynamic analysis with base motion, selective eigenmodes, and Lanczos eigensolver.

SIM-based steady-state analysis with multiple load cases

Elements tested

B21

DASHPOTA

Features tested

SIM-based steady-state dynamic analysis with multiple load case definitions.

Problem description

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:

Table 1. Dashpot damping.
Frequency (Hz)	Damping value
0.0	0.01
100	0.001
200	0.0005

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.

Results and discussion

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:


Frequency (Hz)	Single load		Real load case		Imaginary load case
Frequency (Hz)	Magnitude	Phase	Magnitude	Phase	Magnitude	Phase
5.85466	2978.0	90.0	2978.0	90.0	2978.0	−180.0
35.4301	625.0	90.0	625.0	90.0	625.0	180.0
97.0515	474.8	90.0	474.8	90.0	474.8	180.0

Input files

cant_dash_ssds_mlc.inp: SIM-based steady-state dynamic analysis of the cantilever beam with dashpot, subspace, and multiple load cases. Units: mm, N, MPa.