Kinematic coupling

Each possible combination of selected constraints on a secondary node rotation results in rotation relationships that are unique but analogous to existing MPC types: zero constrained rotational degrees of freedom results in a pin constraint; one, results in a universal constraint; two, results in a revolute constraint; and three, results in a beam constraint. Each of these combinations is treated according to the appropriate theory employed in Abaqus/Standard. To implement these constraints, an additional node is created internally for each secondary node. For example, for revolute and universal constraints this additional node is used in a similar manner to the nodes required in the specification of the analog MPC types.

The additional internal node, described above, also reflects the motion of the secondary node, relative to that of a fully constrained secondary node in the following manner. Let ${\mathbf{X}}^m$ be the position of the main node and ${\mathbf{X}}^s$ be the position of the secondary node in the reference configuration. The reference configuration position of the secondary node with respect to the main node is then

Then, in the current configuration

where ${\widehat{\mathbf{x}}}^s$ is the fully constrained secondary node position and

where $\mathbf{C}\left({\mathit{\varphi }}^m\right)$ is the rotation matrix associated with the main node rotation, ${\mathit{\varphi }}^m$ . The selectively constrained secondary node position can be described as

where $y_i$ are the translation degrees of freedom at the additional node and ${\mathbf{e}}_i$ are the current configuration base vectors, which rotate from the reference global Cartesian base vectors ${\mathbf{e}}_i$ according to

This base vector rotation is made regardless of the choice of rotation constraint at the secondary node. Constraint or release of secondary node translation degree of freedom i can now be described as the constraint ( $y_i=0$ ) or release of translation degrees of freedom on the additional node. With these constraints on $y_i$ , Equation 1 can be used to define the constraint equations.

The linearized form is readily obtained as

where $\mathbf{r}={\mathbf{x}}^s-{\mathbf{x}}^m$.

The initial stress stiffness terms can be obtained from