ROTATION

Description

The rotation connection does not impose kinematic constraints. The rotation connection is a finite rotation connection where the local directions at node b are parameterized relative to the local directions at node a by the rotation vector. Let $ϕ$ be the rotation vector that positions local directions ${e_{1}^{b}, e_{2}^{b}, e_{3}^{b}}$ relative to ${e_{1}^{a}, e_{2}^{a}, e_{3}^{a}}$ ; that is,

e_{i}^{b} = \exp [\hat{ϕ}] \cdot e_{i}^{a}

for all $i = 1, 2, 3$ , where $\hat{ϕ}$ is the skew-symmetric matrix with axial vector $ϕ$ . See Rotation variables for a discussion of finite rotations.

The available components of relative motion in the ROTATION connection are the change in the rotation vector components positioning the local directions at node b relative to the local directions at node a. Therefore,

u r_{i} = ϕ_{i} - {(ϕ_{0})}_{i} + 2 n π \frac{ϕ_{i}}{∥ ϕ ∥},

where $ϕ_{0}$ is the initial rotation vector, $n \geq 0$ is an integer accounting for rotations with magnitude greater than $2 π$ , all vector components are components relative to the local directions $e_{i}^{a}$ , and $i = 1, 2, 3$ . The connector constitutive rotations are

u r_{i}^{m a t} = ϕ_{i} - θ_{i}^{r e f} + 2 n π \frac{ϕ_{i}}{∥ ϕ ∥} .

The kinetic moment in a rotation connection is

m_{i} = m_{r o t a t i o n} \cdot e_{i}^{a}, i = 1, 2, 3.

In two-dimensional and axisymmetric analyses $u r_{1} = u r_{2} = 0$ and $m_{1} = m_{2} = 0$ .

Summary


ROTATION
Basic, assembled, or complex:	Basic
Kinematic constraints:	None
Constraint moment output:	None
Available components:	$u r_{1}, u r_{2}, u r_{3}$
Kinetic moment output:	$m_{1}, m_{2}, m_{3}$
Orientation at a:	Optional
Orientation at b:	Optional
Connector stops:	$θ_{i}^{m i n} \leq ϕ_{i} \leq θ_{i}^{m a x}$
Constitutive reference angles:	$θ_{i}^{r e f}$
Predefined friction parameters:	None
Contact force for predefined friction:	None