General linear coupling

General link couplings implement a general coupling between the source and target interface with linear elasticity and viscous damping properties.

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 stiffness matrices, \(\mathbf{K_{SS}}\), \(\mathbf{K_{ST}}\), \(\mathbf{K_{TS}}\), and \(\mathbf{K_{TT}}\). Following equation describes how the stiffness matrices define a coupling between the target and source interface, denoted by subscript \({T}\) and \({S}\), respectively.

\[\begin{split}\begin{bmatrix}\mathbf{F_T} \\ \mathbf{F_S} \end{bmatrix} = \begin{bmatrix}\mathbf{K_{TT}} & \mathbf{K_{TS}} \\ \mathbf{K_{ST}} & \mathbf{K_{SS}}\end{bmatrix} \, \begin{bmatrix} \mathbf{y_T} \\ \mathbf{y_S} \end{bmatrix}\end{split}\]

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
\(\mathbf{y_T}\)	\(\in\mathbb{R}^{6\times 1}\)	Displacement vector of target interface
\(\mathbf{y_S}\)	\(\in\mathbb{R}^{6\times 1}\)	Displacement vector of source interface
\(\mathbf{K_{TT}}\)	\(\in\mathbb{R}^{6\times 6}\)	Stiffness matrix coupling target displacement and target load
\(\mathbf{K_{ST}}\)	\(\in\mathbb{R}^{6\times 6}\)	Stiffness matrix coupling target displacement and source load
\(\mathbf{K_{TS}}\)	\(\in\mathbb{R}^{6\times 6}\)	Stiffness matrix coupling source displacement and target load
\(\mathbf{K_{SS}}\)	\(\in\mathbb{R}^{6\times 6}\)	Stiffness matrix coupling source displacement and source load

The damping matrices are to be defined accordingly by substituting the displacement vectors with velocity vectors and by substituting the stiffness matrices by damping matrices, i.e.:

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

Simple mode

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

\[\begin{split}\begin{bmatrix}\mathbf{F_T} \\ \mathbf{F_S} \end{bmatrix} = \begin{bmatrix}\mathbf{K} & -\mathbf{K} \\ -\mathbf{K} & \mathbf{K}\end{bmatrix} \, \begin{bmatrix} \mathbf{y_T} \\ \mathbf{y_S} \end{bmatrix}\end{split}\]