Viscoelastic Structural Damper — Transient Analysis

The model studies the forced response of a typical viscoelastic damper. Damping elements involving layers of viscoelastic materials are often used for reduction of seismic and wind induced vibrations in buildings and other tall structures. The common feature is that the frequency of the forced vibrations is low.

Model Definition

The geometry of the viscoelastic damper is shown in Figure 1 (from Ref. 1). The damper consists of two layers of viscoelastic material confined between mounting elements made of steel.

Figure 1: Viscoelastic damping element.

The viscoelastic layers are modeled with the generalized Maxwell model available in COMSOL Multiphysics. The generalized Maxwell model represents the viscoelastic material as a series of branches, each with a spring-dashpot pair.

Eighteen viscoelastic branches guarantee the accurate representation of the material behavior for different excitation frequencies, when the damper is subjected to forced vibration. The values of the shear moduli and relaxation times for each branch are available in Ref. 1. They are summarized in the following table:

Table 1: Model Data for the Viscoelastic Damper Model.
Property	Value	Description
K	40 GPa	Bulk modulus
G	58.6 kPa	Long time shear modulus
ρ	1.06 g/cm3	Density
G1	13.3 MPa	Shear modulus, branch 1
τ1	10-7 s	Relaxation time, branch 1
G2	286 MPa	Shear modulus, branch 2
τ2	10-6 s	Relaxation time, branch 2
G3	291 MPa	Shear modulus, branch 3
τ3	3.16·10-6 s	Relaxation time, branch 3
G4	212 MPa	Shear modulus, branch 4
τ4	10-5 s	Relaxation time, branch 4
G5	112 MPa	Shear modulus, branch 5
τ5	3.16·10-5 s	Relaxation time, branch 5
G6	61.6 MPa	Shear modulus, branch 6
τ6	10-4 s	Relaxation time, branch 6
G7	29.8 MPa	Shear modulus, branch 7
τ7	3.16·10-4 s	Relaxation time, branch 7
G8	16.1 MPa	Shear modulus, branch 8
τ8	10-3 s	Relaxation time, branch 8
G9	7.83 MPa	Shear modulus, branch 9
τ9	3.16·10-3 s	Relaxation time, branch 9
G10	4.15 MPa	Shear modulus, branch 10
τ10	10-2 s	Relaxation time, branch 10
G11	2.03 MPa	Shear modulus, branch 11
τ11	3.16·10-2 s	Relaxation time, branch 11
G12	1.11 MPa	Shear modulus, branch 12
τ12	0.1 s	Relaxation time, branch 12
G13	0.491 MPa	Shear modulus, branch 13
τ13	0.316 s	Relaxation time, branch 13
G14	0.326 MPa	Shear modulus, branch 14
τ14	1 s	Relaxation time, branch 14
G15	8.25·10-2 MPa	Shear modulus, branch 15
τ15	3.16 s	Relaxation time, branch 15
G16	0.126 MPa	Shear modulus, branch 16
τ16	10 s	Relaxation time, branch 16
G17	3.73·10-2 MPa	Shear modulus, branch 17
τ17	100 s	Relaxation time, branch 17
G18	1.18·10-2 MPa	Shear modulus, branch 18
τ18	1000 s	Relaxation time, branch 18

One of the mounting elements is fixed; the other two are loaded with periodic forces with a frequency of 3 Hz.

Some viscoelastic branches may relax much faster or much slower than the time scale of the excitation period. In such cases the corresponding dashpots may be considered to be either fully relaxed or rigid. These branches may then be removed and instead grouped into equivalent branches. This reduction of the number of individual branches, or “pruning”, is beneficial because each branch adds auxiliary strain variables that need to be solved for in a time-dependent study. Therefore, including a large number of branches may increase the computational cost unnecessarily as some of the branches may have no significant effect on the results. Excluding branches which do not contribute in a relevant way given a range of the excitation frequency is described in more detail in the theory chapter of the user’s guide: Structural Mechanics Module>User’s Guide>Structural Mechanics Theory >Material Models>Linear Viscoelasticity.

Results and Discussion

Figure 2 shows the results of the transient computations after three seconds of forced vibrations.

Figure 2: Displacement field.

The typical viscoelastic hysteresis loops for a point within one of the mounting elements are shown in Figure 3. The hysteresis loops are plotted for the cases when including all 18 viscoelastic branches and when pruning some of the branches. In the latter case effectively only 6 branches are solved for. The case with pruning shows only a small difference when compared to the solution including all branches; however, the pruned case has a significantly lower computation time. In both time-dependent cases the solutions converge to the response obtained by a frequency-domain analysis.

Figure 3: Displacement vs. applied force.

Notes About the COMSOL Implementation

The damper is excited with a harmonic load at 3 Hz. Not all branches listed inTable 1 contribute in a significant way, and may therefore be “pruned” with a corresponding setting in the node of the model. The pruning is done automatically and only requires to enter cut-off frequencies; that is, frequencies deemed not relevant given the load frequency. To show the difference in results as well as in computation cost, two studies will be set up; one including all viscoelastic branches and the other with pruning enabled. The latter study has a lower computational cost as fewer auxiliary variables need to solved for.

References

1. S.W. Park “Analytical Modeling of Viscoelastic Dampers for Structural and Vibration Control,” Int. J. Solids and Structures, vol. 38, pp. 694–701, 2001.

2. K.L. Shen and T.T. Soong, “Modeling of Viscoelastic Dampers for Structural Applications,” J. Eng. Mech., vol. 121, pp. 694–701, 1995.

Structural_Mechanics_Module/Dynamics_and_Vibration/viscoelastic_damper_transient

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

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In the tree, select .

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In the tree, select .

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Geometry 1

Import the predefined geometry from a file.

In the toolbar, click and choose .

Browse to the model’s Application Libraries folder and double-click the file viscoelastic_damper_geom_sequence.mph.

The imported geometry should look similar to that shown in Figure 1.

Solid Mechanics (solid)

Linear Elastic Material 2

In the window, under right-click and choose .

In the window for , locate the section.

From the list, choose .

The local computations at Gauss points during assembly are expensive for the viscoelasticity model. Using a reduced integration scheme will reduce the overall simulation time.

Locate the section. Select the check box.

Select Domains 2 and 5 only.

Viscoelasticity (No Pruning)

In the toolbar, click

and choose .

In the window for , type Viscoelasticity (No Pruning) in the text field.

Since there are 18 branches in this material model, the data has been collected in a text file which you can load.

Locate the section. Click to select row number 1 in the table.

Click

Browse to the model’s Application Libraries folder and double-click the file viscoelastic_damper_viscoelastic_data.txt.

Viscoelasticity (With Pruning)

Right-click and choose .

In the window for , type Viscoelasticity (With Pruning) in the text field.

Locate the section. Select the check box.

In the flower text field, type 0.3.

In the fupper text field, type 30.

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select .

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Materials

Viscoelastic

In the window, under right-click and choose .

In the window for , type Viscoelastic in the text field.

Select Domains 2 and 5 only.

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Bulk modulus	K	4e8	N/m²	Bulk modulus and shear modulus
Shear modulus	G	5.86e4	N/m²	Bulk modulus and shear modulus
Density	rho	1060	kg/m³	Basic