CARDAN

Description

Connection type CARDAN.

The CARDAN connection does not impose kinematic constraints. A CARDAN connection is a finite rotation connection where the local directions at node b are parameterized in terms of Cardan (or Bryant) angles relative to the local directions at node a. Local directions $\left\{{\mathbf{e}}_1^b,{\mathbf{e}}_2^b,{\mathbf{e}}_3^b\right\}$ are positioned relative to $\left\{{\mathbf{e}}_1^a,{\mathbf{e}}_2^a,{\mathbf{e}}_3^a\right\}$ by three successive finite rotations $\alpha$, $\beta$, and $\gamma$ as follows:

1. Rotate by $\alpha$ radians about axis ${\mathbf{e}}_1^a$;

2. Rotate by $\beta$ radians about the intermediate 2-axis, ${\mathbf{e}}_2=\cos \alpha {\mathbf{e}}_2^a+\sin \alpha {\mathbf{e}}_3^a$; and

3. Rotate by $\gamma$ radians about axis ${\mathbf{e}}_3^b$.

Rotation angle $\beta$ should be moderate (magnitude less than $\pi /2$), whereas $\alpha$ and $\gamma$ may be arbitrarily large (i.e., magnitude greater than $2\pi$). The Cardan angles are determined by the local directions as

Here, m and n are integers that account for rotations with a magnitude greater than $\pi$.

The three available components of relative motion in the CARDAN connection are the changes in the Cardan angles positioning the local directions at node b relative to the local directions at node a. Therefore,

where ${\alpha }_0$, ${\beta }_0$, and ${\gamma }_0$ are the initial Cardan angles. The connector constitutive rotations are

The kinetic moment in a CARDAN connection is determined from the three component relationships:

Summary