Blade Coordinate Systems

The blades in Bladed are defined as a series of 2D sections, distributed along the blade's span. These 2D planes are defined using the blade root coordinate system and at each of the 2D sections, it is possible to define property systems that describe the structural and aerodynamic properties of the blade. The underlying theory that Bladed implements for coordinate transformations is outlined in the blade coordinate systems theory article.

This article summarises how the blade section planes are defined and how the reference coordinate systems are created.

The Blade Root Coordinate System

The blade root coordinate system is the cartesian coordinate system in which the blade sections are positioned, as illustrated in Figure 1 and summarised in Table 1.

Figure 1: Blade root coordinate system, illustrated for zero cone and pitch.

Table 1: Root axis' definitions.

Axis	Alignment
\(X_{root}\)	When the MountingPoints cone angle, Mounting cone angle, and the blade pitch angle are zero, the blade \(X_{root}\) axis is parallel to the rotor rotational axis.
\(Y_{root}\)	The \(Z_{root}\) axis follows from the [right-hand rule convention].
\(Z_{root}\)	The \(Z_{root}\) axis points along the blade towards the blade tip. This is equal to the pitch axis of the blade if there is no sweep or cone outboard of the pitch bearing.

The origin of the blade root system is the distal node of the hub. This is often also referred to as the body-fixed coordinate system of the blade . The orientation of the body-fixed coordinate system can be altered by a set of inboard and outboard mounting properties. As illustrated in Figure 2, the inboard properties change the pitch axis, whereas the outboard properties do not. The inboard mounting properties are defined by the MountingPoints properties of the hub, whereas the Mounting properties of the Blade component define the outboard properties.

Note

The \(z_{root}\) axis is not necessarily the same as the pitch axis as the Mounting cone and sweep is applied outboard of the pitch bearing.

The Mounting properties, applied outboard of the pitch bearing:

• ConingAngle: The blade cone angle in radians, has the effect of adding a contribution to the prebend deflection which increases linearly with distance along the blade. A positive cone angle means that the blade tip is coned away from the nacelle as shown in Figure 2.
• SweepAngle: The blade sweep angle in radians, a positive sweep angle means that the blade tip is swept back from the pitch axis, away from the direction of rotation.

An example of some arbitrary values for the Mounting property, note that these are not required:

"Mounting": {
    "ConingAngle": 0.05,
    "SweepAngle": 0.05
},

Figure 2: Left: positive cone angle inboard and outboard. Right: positive sweep angle outboard (inboard not supported)

Blade Sections

The properties of the turbine blade are specified at a series of 2D sections along the blade, as illustrated in Figure 3 and Figure 4.

Figure 3: Blade section planes.

Figure 4: Blade section reference system

Each of the section planes is defined in the blade root coordinate system and has an origin point \([X,Y,Z]_{\mathrm{ref}_0}\) (Origin), a \({Z}_{\mathrm{ref}}\) axis (ZAxis defining the section plane's normal), and a \({Y}_{\mathrm{ref}}\) axis (YAxis), which together define an orthogonal coordinate system referred to as the section's reference coordinate system (ReferenceCoordinateSystem).

All of the properties for the section (such as stiffnesses, masses, etc.) are then defined in specific coordinate systems which are defined in relation to this reference system (see Blade Section Properties Definition).

The origin of each reference system defines a reference location with respect to which other axis origins are defined. This can be assumed to represent any chosen location in the blade section, such as on the leading edge of the section, at the half-chord, or at the neutral axis. Because all of the section properties are defined in relation to the reference system, if the reference system is moved or rotated, all of the section property coordinate systems will move and rotate by the same amount.

Within the SectionDefinitions, the users can set the Origin, the ZAxis and the YAxis of the ReferenceCoordinateSystem. Instead of explicitly specifying the YAxis, users can alternatively choose to implicitly define it by setting RotationAboutReferenceZ, which is a rotation angle around the ZAxis.

The Origin point can also be defined by the use of DistanceAlongSpan, which is the cumulative 3D distance traced along the line made by the reference axis origins, as shown in Equation \(\eqref{eq:distance_along_span}\) and Figure 5.

\[ \begin{equation} \mathrm{DistanceAlongSpan}_n\ =\ \sum_{i=1}^{n-1}{\mathrm{\Delta}_{ref}}_{i+1,\ i}={\mathrm{\Delta}_{ref}}_{2,1}+\ldots+{\mathrm{\Delta}_{ref}}_{n,n-1} \label{eq:distance_along_span} \end{equation} \]

Where \({\mathrm{\Delta}_{ref}}_{i,j}\) is the distance between two section origins:

\[ \begin{equation} {\mathrm{\Delta}_{ref}}_{i,j}=\left|\begin{matrix}X_{0_i}-X_{0_j}\\Y_{0_i}-Y_{0_j}\\Z_{0_i}-Z_{0_j}\\\end{matrix}\right| \label{eq:delta_ref} \end{equation} \]

Figure 5: Illustration of the DistanceAlongSpan definition, and how it is using \({\mathrm{\Delta}_{ref}}_{i,j}\)

Note

Origin's Z coordinate or DistanceAlongSpan at the first sections must be set to 0.

Only the Origin's Z (\(Z_{ref_0}\)) coordinate or as an alternative theDistanceAlongSpan are an absolute requirement to define the section plane. All section reference planes in the same blade should be defined using the same options, whereas the code does not prevent the user from mixing both options, the outcomes might not be what the user expects. In the same way the sections will always be represented in the blade geometry as monotonically increasing in distance from the root.

Table 2: Reference axis' properties definitions.

Property	Units	Default Behaviour
Origin.X	meters	Defaults to \(0.0\)
Origin.Y	meters	Defaults to \(0.0\)
Origin.Z or DistanceAlongSpan	meters	Required user input
ZAxis	[ - ]	Defaults to the average direction from the last section's Origin to the next section's
YAxis1	[ - ]	[Defaults to the projection of the blade Y-axis axis onto the section plane]
RotationAboutReferenceZ2	radians	If not supplied YAxis or its default value will be used

The default orientation of the reference section plane's axis depend of the values set to the Origin point of that section and the location and orientation of the previous (closest on the root direction) defined section. For more details see the Default Orientation of Section Planes section.

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1. The reference system's \(X_{ref}\) axis is not an input, but is calculated by a cross product of the \(Y_{ref}\) and \(Z_{ref}\) vectors to produce a vector normal to both. The \(Z_{ref}\) and \(X_{ref}\) are then crossed with each other once more to produce a rectified \(Y_{ref}\) such that the axis set obeys the [right-hand rule convention].↩

2. The options RotationAboutReferenceZ and YAxis are mutually exclusive. If neither are specified Bladed will default the YAxis property. Details about how RotationAboutReferenceZ is implemented by Bladed can be found calculation of the reference frame article.↩