ROTATION-ACCELEROMETER

Description

Figure 1. Connection type ROTATION-ACCELEROMETER.

The ROTATION-ACCELEROMETER connection does not impose kinematic constraints. It defines three local directions at node a and three local directions at node b. The ROTATION-ACCELEROMETER connection's formulation is similar to that for the ROTATION connection. The ROTATION-ACCELEROMETER connection measures the finite rotation that takes the local directions at node a into the local directions at node b and parameterizes that finite rotation by the rotation vector. Let $\varphi$ be the rotation vector that positions local directions $\left\{{\mathbf{e}}_1^b,{\mathbf{e}}_2^b,{\mathbf{e}}_3^b\right\}$ relative to $\left\{{\mathbf{e}}_1^a,{\mathbf{e}}_2^a,{\mathbf{e}}_3^a\right\}$; that is,

for all $i=1,2,3$, where $\widehat{\varphi }$ is the skew-symmetric matrix with axial vector $\varphi$. See Rotation variables for a discussion of finite rotations. The connection measures the change in the rotation vector components in the local directions rotating with the body at node b. The rotation vector components are calculated as

There are no available components of relative motion for the ROTATION-ACCELEROMETER connection. The connector rotation is

where ${\varphi }_0$ is the initial rotation vector and $n\ge 0$ is an integer accounting for rotations with magnitude greater than $2\pi$.

The ROTATION-ACCELEROMETER connection differs from the ROTATION connection in the way angular velocity and acceleration are calculated. The ROTATION-ACCELEROMETER connection measures velocity and acceleration from the nodes as

where ${\omega }_a$, ${\omega }_b$, ${\alpha }_a$, and ${\alpha }_b$ are the nodal angular velocities and accelerations at nodes a and b, respectively.

In two-dimensional and axisymmetric analyses $u{r}_1=u{r}_2=0$.

Summary