Turbine theory: Blades

Blade structural model

Blades are represented by a structural model similar in outline to that used by lines, with massless segments connecting nodes at which inertia is lumped, ordered from end A to end B.

The blade extends from the root at end A to the tip at end B. The root frame of reference of the blade is coincident with that of its associated fitting frame, except that it may be rotated about its $z$-axis by the blade pitch angle.

Unlike lines, the blade DOFs can be fixed, in which case the nodes are rigidly constrained to each other and the blade forms a rigid body system (the blade can still be pitched relative to its fitting frame in this mode of operation).

If the blade has free DOFs, the nodes can rotate and translate relative to each other, allowing blade flexibility to be modelled and aeroelastic coupling effects to be captured. The structural model here is analogous to a line with torsion included: each node has six calculated degrees of freedom (three translational and three rotational), bending stiffness is represented by linear elastic rotational springs between the node $z$-axis and the segment $z$-axis, and axial and torsional stiffness by linear elastic springs at the mid-segment.

A and B frames

Each blade segment is split, at the mid-segment arc length, into two half-segments which we call the A-half and the B-half. The A-half is the half-segment nearest the blade root (end A), the B-half that nearest the blade tip (end B). Each half-segment is connected, at its outer end, to a frame carried by its adjacent node; the A-half is connected to an A-frame carried by its root-side node and the B-half is connected to a B-frame carried by its tip-side node. The other ends of the pair half-segments are connected together, to form the whole segment, by the mid-segment frame – we will come to this below.

Thus a node, which (other than an end node) connects two segments, carries two frames; the segment on the node's root-side connects to the B-frame and the segment on the node's tip-side connects to the A-frame. At the node, these two frames are rigidly constrained together and share a common origin (the node's position) and common $z$-axis direction (the blade's axial direction at this arc length). However, their respective $x$-axes and $y$-axes are allowed to be rotated away from each other, by a constant angle, about their shared $z$-axis. This angle is the difference in the user-specified structural twist of the two adjacent segments at their mid-segment arc lengths.

At the ends of the blade, the B-frame carried by the first node is coincident with the blade root frame and the A-frame carried by the last node is coincident with the tip frame.

Segment frames

As is the case for a line segment, each blade half-segment carries a frame of reference. The orientation of the segment A-half frame is obtained by rotating the A-frame carried by its root-side node such that its $z$-axis is parallel to the segment $z$-axis. Similarly, the orientation of the segment B-half frame is obtained by rotating the B-frame carried by its tip-side node such that its $z$-axis is parallel to the segment's $z$-axis. If the blade DOFs are free, the half-segment frames can twist away from each other, about the segment's $z$-axis, at the mid-segment arc length. The rotational stiffness between the half-segment frames is determined by the user-specified torsional stiffness GJ. The blade segment also carries the mid-segment frame, positioned at the mid-segment arc length and connecting together the two half-segments on either side. The mid-segment frame and half-segment frames all have a common $z$-axis direction, while the mid-segment frame's $x$-axis and $y$-axis are oriented to be half way between those of the two half-segments, i.e. it is oriented to have the average segment twist. These segment frames, and the A and B frames carried by the nodes, are collectively known as the blade structural frames. If the blade DOFs are fixed, then all the structural frames associated with a segment, from the A-frame carried by its root-side node through to the B-frame carried by its tip-side node, have identical orientation.

Structural frames are the frames of reference of the finite element model. They define the axes with respect to which structural data is defined. For example, the non-isotropic bend stiffness components EI define the stiffness of rotational springs between the $x$ and $y$ axes of half-segment frames and those carried by the nodes. The blade profile specifies the relationship between these frames and the data frames, with respect to which other data is defined.

For lines, external loading is always applied to the nodes. For blades, however, external aerodynamic loading is applied to the mid-segment frame at the aerodynamic centre.

Node structural frames

Used for drawing and reporting, each node also carries a node structural frame. This frame accounts for the user-specified structural twist at the node arc length. In the finite element model, it is found by rotating the node's B-frame by the structural twist at the node arc length relative to that of the previous mid-segment arc length. At the end A node, it's known as the root node structural frame and is simply found by rotating the node's B-frame (i.e. the root frame) by the structural twist at zero arc length.