Blade modes

The motion of the tapered, twisted and very flexible rotor blades is among the most complex phenomena related to the structural dynamics of a wind turbine. In Bladed, a blade can be represented by one component or several rigidly connected components. Use of several components allows rigorous modelling of large deflections.

For the single-part blade model, only normal modes with fixed-free boundary conditions are used. This is the classical approach for selecting the vibration modes of a wind turbine blade. In the multi-part blade model, the inner parts use both normal modes with fixed-fixed boundary conditions and attachment modes with fixed-free boundary conditions, while the outermost part uses only normal modes with fixed-free boundary conditions. For more details see normal and attachment modes.

Each mode is defined in terms of the following parameters:

• Modal frequency, \(\bscalar{\omega_i}\)
• Modal damping coefficient, \(\bscalar{\xi_i}\)
• Mode shape represented by a vector of the displacement at the stations

where the subscript \(i\) indicates properties related to the \(i^{th}\) mode.

For blade with several parts, it is still desirable to calculate and review the coupled eigenmodes for the whole blade. To facilitate this, Bladed performs a subsequent eigen analysis of the blade parts to calculate the modes corresponding to the natural modes of the blade. This is useful both for physical interpretation of the blade mode shapes and for applying damping, as explained in the next section.

Specifying blade damping (whole blade damping)

The blade damping is specified for the natural modes of the whole blade. To allow this, Bladed must calculate the damping on each blade part mode (or generalised finite element freedom) based on the damping of the natural modes of the whole blade. This is done according to theory presented in Clough and Penzien, 1993, pp240-242.

Damping should be specified for the coupled modes which would be expected to contribute significantly to the dynamic response of the blade (typically the first ~10 modes). For subsequent higher frequency coupled modes, the damping is assumed to be proportional to modal stiffness, calculated as \(\bmatrix{C}=\bscalar{a_1}\bmatrix{K}\). This results in damping that is proportional to modal frequency, so that the responses of higher frequency modes are effectively eliminated by high damping ratios.

The coefficient \(\bscalar{a_1}\) is defined according to the highest mode for which damping is specified:

\[ \begin{equation} \bscalar{a_1} = \frac{2\bscalar{\xi_{c}}}{\bscalar{\omega_{c}}} \label{eq:acoefficient} \end{equation} \]

where

• the subscript \(c\) refers to highest mode with damping specified,
• \(\bscalar{\xi}\) refers to the coupled mode damping ratio.

Damping on these higher frequency coupled modes is given by

\[ \begin{equation} \bscalar{\xi_{n}} = \bscalar{\xi_{c}}\left( \frac{\bscalar{\omega_{n}}}{\bscalar{\omega_{c}}} \right). \label{eq:dampingratio} \end{equation} \]

Once the coupled mode damping values are calculated, the blade part modal frequencies are calculated in Bladed according to Clough and Penzien, 1993.