Line theory: Line end orientation

At line ends, we usually need to define not only the axial direction of the end fitting but also the twist orientation about that axial direction. This is done by first specifying the azimuth and declination of the axial direction and then specifying the twist orientation by giving a third angle called gamma.

Together, the three angles azimuth, declination and gamma fully determine the rotational orientation of the end fitting. To define how this is done we need a frame of reference $\Exyz$ for the end fitting, where

• $E$ is at the connection point
• $E\urm{z}$ is the axial direction of the fitting, using the "A to B" convention – i.e. $E\urm{z}$ is into the line at end A, but out of the line at end B
• $E\urm{x}$ and $E\urm{y}$ are perpendicular to $E\urm{z}$.

The azimuth, declination and gamma angles then define the orientation of $\Exyz$ relative to the local axes $\Lxyz$ of the object to which the end is connected, as follows:

• start with $\Exyz$ aligned with $\Lxyz$
• rotate $\Exyz$ by azimuth degrees about $E\urm{z}$ ($= L\urm{z}$ at this stage)
• rotate by declination degrees about the resulting $E\urm{y}$ direction
• finally, rotate by gamma degrees about the resulting (and final) $E\urm{z}$ direction.

For all these rotations, a positive angle means rotation clockwise about the positive direction along the axis of rotation, and a negative angle means anti-clockwise.

If the line end is not connected to another object, then it must be either fixed, anchored or free. In all of these cases, the three angles define $\Exyz$ relative to the global axes $\GXYZ$.

Three-dimensional rotations are notoriously difficult to describe and visualise. When setting the azimuth, declination and gamma, it can be helpful to check that the resulting $\Exyz$ directions are correct by drawing the local axes on the 3D view.

Here are some examples of the effect of various values of (azimuth, declination, gamma) for a fixed end. For ends connected to other objects, replace $\GXYZ$ by $\Lxyz$ in these examples.

• (0,0,0) sets $\Exyz$ to be aligned with $\GXYZ$
• (0,30,0) sets $E\urm{x}$ at 30° below $G\urm{X}$ (in the $G\urm{XZ}$ plane), $E\urm{y}$ along $G\urm{Y}$, $E\urm{z}$ at 30° to $G\urm{Z}$, towards $G\urm{X}$
• (0,180,0) sets $E\urm{x}$ along $-G\urm{X}$, $E\urm{y}$ along $G\urm{Y}$, $E\urm{z}$ along $-G\urm{Z}$.
• (90,90,0) sets $E\urm{x}$ along $-G\urm{Z}$, $E\urm{y}$ along $-G\urm{X}$, $E\urm{z}$ along $G\urm{Y}$
• (90,90,90) sets $E\urm{x}$ along $-G\urm{X}$, $E\urm{y}$ along $G\urm{Z}$, $E\urm{z}$ along $G\urm{Y}$.

End direction results

If the end orientation $\Exyz$ is defined, then OrcaFlex offers various results with respect to those axes. For a given vector $\vec{v}$ (such as the end-force, for instance), these include the components of $\vec{v}$ with respect to $\Exyz$, and the angles that $\vec{v}$ makes with the various axes of $\Exyz$. The angles offered are

• Ez angle, between $\vec{v}$ and the $E\urm{z}$ (axial) direction. This measures how far $\vec{v}$ is away from the end fitting axial direction.
• Ezx angle, from $E\urm{z}$ to the projection of $\vec{v}$ onto the $E\urm{zx}$ plane (measured positive from $E\urm{z}$ towards $E\urm{x}$). This is the angle $\vec{v}$ makes with $E\urm{z}$ when viewing the $zx$-plane.
• Ezy angle, from $E\urm{z}$ to the projection of $\vec{v}$ onto the $E\urm{zy}$ plane (measured positive from $E\urm{z}$ towards $E\urm{y}$). This is the angle $\vec{v}$ makes with $E\urm{z}$ when viewing the $zy$-plane.
• Exy angle, between the $E\urm{x}$ direction and the projection of $\vec{v}$ onto the $E\urm{xy}$ plane (measured positive from $E\urm{x}$ towards $E\urm{y}$). This is the angle $\vec{v}$ makes with $E\urm{x}$ when viewing the $xy$-plane.