General viscous damping

The general viscous damping property implements a viscous damping between the source and target interface.

Definition

Source and target interfaces can be chosen arbitrarily.

Parameters

General link couplings can be parametrised in two modes: simple and advanced, which are described in the following.

Advanced mode

In the advanced mode, the user is prompted to supply four damping matrices, \(\mathbf{D_{SS}}\), \(\mathbf{D_{ST}}\), \(\mathbf{D_{TS}}\), and \(\mathbf{D_{TT}}\). Following equation describes how the stiffness matrices define a coupling between the target and source interface, denoted by subscript \({T}\) and \({S}\), respectively.

\[\begin{split}\begin{bmatrix}\mathbf{F_T} \\ \mathbf{F_S} \end{bmatrix} = \begin{bmatrix}\mathbf{D_{TT}} & \mathbf{D_{TS}} \\ \mathbf{D_{ST}} & \mathbf{D_{SS}}\end{bmatrix} \, \begin{bmatrix} \dot{\mathbf{y}}_\mathbf{T} \\ \dot{\mathbf{y}}_\mathbf{S} \end{bmatrix}\end{split}\]

A list of symbols is shown in following table.

Symbol

Dimension

Meaning

\(\mathbf{F_T}\)

\(\in\mathbb{R}^{6\times 1}\)

Load vector to target interface

\(\mathbf{F_S}\)

\(\in\mathbb{R}^{6\times 1}\)

Load vector to target interface

\(\dot{\mathbf{y}}_\mathbf{T}\)

\(\in\mathbb{R}^{6\times 1}\)

Displacement vector of target interface

\(\dot{\mathbf{y}}_\mathbf{S}\)

\(\in\mathbb{R}^{6\times 1}\)

Displacement vector of source interface

\(\mathbf{D_{TT}}\)

\(\in\mathbb{R}^{6\times 6}\)

Damping matrix coupling target displacement and target load

\(\mathbf{D_{ST}}\)

\(\in\mathbb{R}^{6\times 6}\)

Damping matrix coupling target displacement and source load

\(\mathbf{D_{TS}}\)

\(\in\mathbb{R}^{6\times 6}\)

Damping matrix coupling source displacement and target load

\(\mathbf{D_{SS}}\)

\(\in\mathbb{R}^{6\times 6}\)

Damping matrix coupling source displacement and source load

Simple mode

In the simple mode, a symmetric behaviour between target and source interface is implied. The four damping matrices are replaced by a single damping matrix \(\mathbf{D}\) and applied to the coupling equation as follows:

\[\begin{split}\begin{bmatrix}\mathbf{F_T} \\ \mathbf{F_S} \end{bmatrix} = \begin{bmatrix}\mathbf{D} & -\mathbf{D} \\ -\mathbf{D} & \mathbf{D}\end{bmatrix} \, \begin{bmatrix} \dot{\mathbf{y}}_\mathbf{T} \\ \dot{\mathbf{y}}_\mathbf{S} \end{bmatrix}\end{split}\]