CONSTANT VELOCITY

Description

Figure 1. Connection type CONSTANT VELOCITY.

The shaft direction at node a is ${\mathbf{e}}_3^a$, and the shaft direction at node b is ${\mathbf{e}}_3^b$. The constant velocity constraint is stated as follows. In any configuration there are two unit length orthogonal vectors ${\mathbf{b}}_1$ and ${\mathbf{b}}_2$ in the plane perpendicular to the shaft at node b. These vectors can be written

The angle $\beta$ is chosen such that

The constant velocity constraint requires that the angle $\beta$ is constant at all times. The constant velocity constraint is equivalent to constraining the torsion angle to be constant in a FLEXION-TORSION connection.

The name “constant velocity” for this connection type derives from the following property. If the angular velocities of the two shafts, ${\mathbf{w}}_a$ and ${\mathbf{w}}_b$, have components only along each shaft, respectively, and in the direction of the normal to the plane containing the two shafts (that is, along the ${\mathbf{e}}_3^b\times {\mathbf{e}}_3^a$ direction), the components of angular velocity along the respective shaft directions are equal:

Hence, the “spinning” angular velocity component is the same about each shaft.

The constraint moment imposing the constant velocity constraint has a single component about the average shaft direction ${\mathbf{e}}_3^a+{\mathbf{e}}_3^b$ and is written

Summary