Normal and Attachment Modes
The selection and calculation of mode shape functions follows the idea that was originally suggested by Craig, 2000 as a modification of the widely used Craig-Bampton method from Craig, 1968. For both methods the stations are subdivided into boundary stations that may couple to other components and interior stations that do not couple. The boundary stations also represent the component nodes that may link to nodes of other components. In particular the station representing the proximal node is constrained completely in order to exclude rigid body displacement modes.
With the applied method the modes are generally selected as the union between attachment modes that may couple to other components and normal modes that may be considered as internal vibration modes.
Attachment Modes
The attachment modes are calculated from the component stiffness matrix by a static equilibrium, where the component is fixed at the proximal node and point loads are applied in turn at the distal nodes, as seen in Figure 1.This method yields an attachment mode for every degree of freedom of the distal node of the component (fixed-free boundary condition).
For the attachment modes of the support structure, the calculation of the mode frequencies include the mass of the rotor and nacelle assembly, and would usually be of lower frequencies than the corresponding normal modes.
Normal Modes
The normal modes are determined directly from the fully assembled finite element mass and stiffness matrices using a generalised eigenvalue problem. This calculation is performed with the component fixed at the proximal node (fixed-free) or at both the proximal and distal nodes (fixed-fixed), as illustrated in Figure 1. The type of normal mode (fixed-free vs. fixed-fixed) depends solely on the presence of distal nodes.
For instance, in the case of a tower, there is always a distal node at the tower top, whereas single-part blades do not have distal nodes. Multi-part blades have both a proximal node and a distal node for all parts except the last part, which only has a proximal node. Consequently, the last part produces fixed-free normal modes, while the rest produce fixed-fixed normal modes. There might be multiple distal nodes present, such as in dynamic moorings, and in such cases, all the distal nodes are fixed.
This will produce one normal mode for every degree of freedom in the model, where the number of degrees of freedom is dependent on the boundary condition.
Frequencies and Structural Damping
The frequencies of the attachment modes are calculated by Rayleigh’s method (Clough and Penzien, 1993), while the frequencies of the normal modes result from the eigenvalue problem. These frequencies are solely used for describing damping.
Structural damping is modelled as modal damping (Clough and Penzien, 1993) in terms of a set of damping coefficients (damping ratio) that relate to the mode shape functions. These coefficients are defined as input parameters for the model and may usually be measured directly, for example by exiting a mode and measure the decay of the succeeding oscillation.