Rigid Constraints

For a rigid constraint there are a set of constraint equations which respect the geometry of the constraint. So for a single constrained node the constraint equations are

$$\begin{Bmatrix} u_{s} \\ v_{s} \\ w_{s} \\ \theta_{us} \\ \theta_{vs} \\ \theta_{ws} \\ \end{Bmatrix} = \begin{bmatrix} 1 & & & & z & - y \\ & 1 & & - z & & x \\ & & 1 & y & - x & \\ & & & \delta & & \\ & & & & \delta & \\ & & & & & \delta \\ \end{bmatrix}\begin{Bmatrix} u_{p} \\ v_{p} \\ w_{p} \\ \theta_{up} \\ \theta_{vp} \\ \theta_{wp} \\ \end{Bmatrix}$$

where the $\delta$ terms are $1$ for a fixed and $0$ for a pinned rigid constraint.

Different terms in the matrix are dropped for reduced constraint types. The two most common special types are plane and plate constraints with equations

$$\begin{Bmatrix} u_{s} \\ v_{s} \\ \theta_{ws} \\ \end{Bmatrix} = \begin{bmatrix} 1 & & - y \\ & 1 & x \\ & & \delta \\ \end{bmatrix}\begin{Bmatrix} u_{p} \\ v_{p} \\ \theta_{wp} \\ \end{Bmatrix}$$

for an $xy$ plane constraint. Similarly for $yz$ and $zx$ plane constraints.

$$\begin{Bmatrix} w_{s} \\ \theta_{us} \\ \theta_{vs} \\ \end{Bmatrix} = \begin{bmatrix} 1 & y & - x \\ & \delta & \\ & & \delta \\ \end{bmatrix}\begin{Bmatrix} w_{p} \\ \theta_{up} \\ \theta_{vp} \\ \end{Bmatrix}$$

for a $z$ plate constraint. Similarly for $x$ and $y$ plate constraints.