# 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.

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

$$\mathbf{F_T}$$

$$\in\mathbb{R}^{6\times 1}$$

$$\mathbf{F_S}$$

$$\in\mathbb{R}^{6\times 1}$$

$$\dot{\mathbf{y}}_\mathbf{T}$$

$$\in\mathbb{R}^{6\times 1}$$

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:

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