Appendix G: Sign convention for internal loads

1. General

The RIFLEX forces stored by SIMA are those internal to an element, directed along/about the element’s local axes.

If the stored force/moment is positive, it means the force/moment is directed as indicated by green arrows in Figure 1.

These values are not necessarily positive along/about the element’s positive local axes, this depends on whether values are extracted at node 1 or 2 of the element.

internal2local
Figure 1. Notation for internal loads in an element local system

It should be noted that when extracting loads from a given element, only stiffness based loads for that element are included, inertial and damping loads of the particular element are not included. For submerged elements, or elements with internal contents, the effective tension is reported.

2. Transforming internal loads to the local coordinate system

If internal loads are to be represented as loads with signs consistent with the element’s local coordinate system (i.e. positive force directed along positive axis, positive moment directed based on right hand rule), the following sign transformations should be made:

Force/Moment Node 1 Node 2

\(F_x\)

\(-V_x\)

\(V_x\)

\(F_y\)

\(V_y\)

\(-V_y\)

\(F_z\)

\(V_z\)

\(-V_z\)

\(M_x\)

\(-Q_x\)

\(Q_x\)

\(M_y\)

\(Q_y\)

\(-Q_y\)

\(M_z\)

\(-Q_z\)

\(Q_z\)

3. Example

Consider one of the examples included in SIMA, accessed via Help::Examples::Wind Turbine::INO WINDMOOR semi 12 MW, see Figure 2.

INO WindMOOR semi 12MW
Figure 2. INO WindMOOR semi 12MW

In Model::Calculation Parameters::Dynamic Calculation::Storage, modify:

  • Displacements to ensure that displacements are stored for:

    • shaft3 (at least Segment 2, node 1)

    • bl<i>foil1 (at least Segment 1, node 1), for i = 1, 2, 3

  • Force response to ensure that Store Transformation Matrices is selected, and that internal forces are stored for:

    • shaft3 (at least Segment 2, element 1)

    • bl<i>foil1 (at least Segment 1, element 1), for i = 1, 2, 3

    • optionally: bl<i>ecc1 (at least Segment 1, element 1), for i = 1, 2, 3

Consider the schematic of the shaft, blade eccentricity element, and first blade foil element, see Figure 3. sqwqheqowheqwhe

blade loads at shaft
Figure 3. Schematic layout of elements and reference systems relevant for expressing blade loads at the shaft

It is desired to extract loads at node 1 of the 1st foil element, convert them to the fixed shaft coordinate system, and compare to wind turbine aerodynamic loads, expressed in the same reference system. The procedure is outlined as follows:

  1. For each blade \(i\), extract:

    1. The internal loads at node 1:

      \(V_{i}^b, \, Q_{i}^b\)

    2. Convert from internal loads to loads in the local coordinate system (see the table shown earlier): \(V_{i}^b \rightarrow F_{i}^b, \, Q_{i}^b \rightarrow M_{i}^b\)

      In Figure 3 the loads extracted at node 1, are the internal loads required to keep the element in equilibrium, i.e. those applied by the rest of the system to the element. We are interested in the opposite, i.e. the loads applied by the element to the rest of the system. Thus \(F_{i}^b = -F_{i}^b, \, M_{i}^b = -M_{i}^b\). If loads were e.g. extracted at node 2 of the eccentricity element bl<i>ecc1, they are already the desired loads, and no further modification is required

    3. The position of node 1 (in global system): \(p_{i}^g\)

    4. Transformation matrix from local to global for element 1: \(T_{b,i}^g\)

  2. For the fixed shaft element (Segment 2 of the shaft), extract:

    1. The position of node 1(in global system): \(p_{s}^g\)

    2. Transformation matrix from local to global for element 1: \(T_{s}^g\)

  3. For each blade \(i\):

    1. Define the offset from node 1 of the fixed shaft, to node 1 of the 1st foil element on each blade:

      \(r_{i}^g = p_{i}^g - p_s^g\)

    2. Convert forces and moments from local to global system:

      \(F_{i}^g = T_{b,i}^g F_{i}^b\)

      \(M_{i}^g = T_{b,i}^g M_{i}^b\)

  4. Accumulate forces and moments from all blades at the fixed shaft node (expressed in global system):

    \(F^g = \sum _{i} F_{i}^g\)

    \(M^g = \sum _{i} M_{i}^g + r_{i}^g \times F_{i}^g\)

  5. Convert forces and moments from the global to the fixed shaft coordinate system:

    \(F^s = {T_{s}^g}^T F^g\)

    \(M^s = {T_{s}^g}^T M^g\)

The results are are shown in Figure 4 and Figure 5, and can be compared to forces and moments stored with the Wind Turbine object.

Forces
Figure 4. Forces at node 1 of the 1st foil element
Moments
Figure 5. Moments at node 1 of the 1st foil element

The difference in torque (moment about the fixed shaft \(x\)-axis) calculated from Riflex loads at the first blade nodes (Riflex: \(\vec{M} + \vec{r} \times \vec{F}\)) and generator torque at the low speed shaft (Gen. Torq. LSS) can be attributed to inertial loads of the first blade elements that are not included in the loads extracted from Riflex. Extracting loads at node 2 of the blade eccentricity elements instead, largely eliminates the observed differences, see Figure 6 and Figure 7.

Forces
Figure 6. Forces at node 2 of the blade eccentricity elements
Moments
Figure 7. Moments at node 2 of the blade eccentricity elements

The remaining large offsets between Riflex and aerodynamic loads visible in \(F_x\), \(F_z\) and \(M_y\) can be attributed to the mass of the rotor, this can be confirmed through some simple calculations:

Consider the mass of the rotor, \(m_{r} = 189397.0 \, \text{kg}\), and a shaft tilt of \(\theta = 6.0^\circ\).

In the shaft coordinate system, this implies

\[F_x^s = m_r g \sin(\theta) = 194.2 \, \text{kN}\]

and

\[F_z^s = -m_r g \cos(\theta) = -1847.8 \, \text{kN}.\]

These numbers match up closely with the initial offsets seen in the figure Forces in the fixed shaft system. Lastly, the rotor centre of mass can be calculated [1] to be located approximately

\[r_x^s = -3.17 \, \text{m}\]

(i.e. upwind) of node 1 on the fixed shaft element, implying

\[M_y^s = -r_x^s F_z^s = -5857.5 \, \text{kNm}\]

also closely matching the initial offset in the figure Moments in fixed shaft system.

1. Sima 4.6.3 introduced a feature to calculate mass and inertia properties for selected lines during static analysis, that can be used to calculate the rotor mass and centre of mass. This is selected in Calculation Parameters::Static Calculation::Mass summary.