Default Orientation of Section Planes

Section Planes

As is mentioned in blade axis systems, the blade is defined in a series of 2D planes. The user has full control over the orientation of these planes, and can specify an explicit ZAxis plane normal \((Z_{ref})\) and YAxis \((Y_{ref})\) to orient the reference coordinate system.

Figure 1: Blade section reference system.

In the case where both the ZAxis and YAxis properties are provided, first \(X_{ref}\) is calculated by the cross product of the two vectors, and then \(Y_{ref}\) is regenerated from \(X_{ref}\) and \(Z_{ref}\) to ensure all three axes are orthogonal to each other.

\[ \begin{equation} X_{ref} = \frac{\mathrm{YAxis} \times \mathrm{ZAxis}}{||\mathrm{YAxis} \times \mathrm{ZAxis}||} \end{equation} \]

\[ \begin{equation} Z_{ref} = \frac{\mathrm{ZAxis}}{||\mathrm{ZAxis}||} \end{equation} \]

\[ \begin{equation} Y_{ref} = Z_{ref} \times X_{ref} \end{equation} \]

Defaults for Reference Coordinate System

Although many customers may wish to make use of the full control they have of the section plane orientations, a simple default orientation of the reference coordinate system is calculated if the ZAxis (reference plane normal) or YAxis (the orientation of the axis system) is omitted.

Where the reference plane normal is left at default, the plane will cut approximately normal to the blade's local curvature, and this would be maintained if prebend was subsequently added to the blade (by moving the reference system origins in the x direction), or presweep was added (by moving the reference system origins in the y direction).

Presuming that the origins of the reference system for a straight blade lie upon a straight line such as the root axis, then the default section planes will all remain parallel with each other.

Figure 2: Default of section plane normal when sections are aligned with the root axis.

If prebend is added, then this will result in the planes - if left at their default - remaining approximately normal to the local curvature.

Figure 3: Default of section plane normal with prebend added by moving the reference x location.

Tip

The polyline created by joining the origins of the reference coordinate systems is used to determine the default plane normal. Since the reference system is independent of any physical characteristic of the blade (such as the neutral axis or leading edge), it means that the user can position their reference origins to best suit their modelling practice. If leaving the plane normals at their defaults, thought must be put into where to position the reference coordinate system and how this will affect the section planes.

Implementation of ZAxis Default

The plane normal used if no ZAxis property is provided is the average vector of the directions from the previous reference system's origin and to the next reference system's origin, as defined by:

\[ \begin{equation} dir_{previous} = \frac{O_{current}-O_{previous}}{||O_{current}-O_{previous}||} \end{equation} \]

\[ \begin{equation} dir_{next} = \frac{O_{next}-O_{current}}{||O_{next}-O_{current}||} \end{equation} \]

\[ \begin{equation} Z_{ref} = \frac{dir_{previous} + dir_{next}}{||dir_{previous} + dir_{next}||} \end{equation} \]

Where \(O\) is the origin of the reference coordinate system.

Figure 4: Averaging of direction vectors.

For the case of the first and last sections, only a single direction is used to calculate the plane normal, and no averaging is undertaken:

\[ \begin{equation} Z_{ref}^{first} = \frac{O_{next}-O_{current}}{||O_{next}-O_{current}||} \end{equation} \]

\[ \begin{equation} Z_{ref}^{last} = \frac{O_{current}-O_{previous}}{||O_{current}-O_{previous}||} \end{equation} \]

Implementation of YAxis Default

The reference Y-axis orientation if no YAxis property is provided is the intersection of the \(YZ_{root}\) plane with the section plane, as defined by the reference system's origin and z-axis as the point and normal respectively. This line is taken in the direction aligned with \(Y_{root}\) to form the \(Y_{ref}\) direction.

In mathematical notation, it is the cross-product of the two plane normals, \(X_{root}\) and \(Z_{ref}\):

\[ \begin{equation} Y_{ref} = \frac{Z_{ref} \times X_{root}}{||Z_{ref} \times X_{root}||} \end{equation} \]

The \(X_{ref}\) vector is then calculated thus:

\[ \begin{equation} X_{ref} = Y_{ref} \times Z_{ref} \end{equation} \]

Figure 5: Generation of default reference y-axis.

Warning

When providing the YAxis property, an effort should be made to ensure that it is close to being perpendicular to the ZAxis property, as the above process will rotate it to be exactly ninety degrees to the z-axis. This could lead to a reinterpretation of the y-axis, and therefore a misinterpretation of the scalar properties associated with it.

It would also be considered more robust if either both of the ZAxis and YAxis properties are provided, or just the ZAxis. This avoids the situation where the provided y-axis does not correlate perfectly with the default z-axis calculated by Bladed.