Equations of Motion

Because of the complexity of the coupling of the modal degrees of freedom of the rotating and non-rotating components, the algebraic manipulation involved in the derivation of the equations of motion for a wind turbine is relatively complicated, and the following only gives a brief description of the theory.

Degrees of freedom

Examples of possible degrees of freedom involved in the equations of motion for the structural dynamic model for Bladed are as follows:

• Blade deflection
• Nacelle yaw
• Tower fore-aft, side-side and torsional deflection (axisymmetric tower model)
• General tower deflection (multi-member tower model): a large number of modes is allowed, including the torsional and axial degrees of freedom.

In addition, the following dynamics can also be included as required:

• A sophisticated representation of the power train dynamics.
• A range of different representations of generator and power converter dynamics, including both mechanical models and electrical models which can include network interactions.
• A range of pitch actuator models, from simple passive models to detailed representations of electric servo drives and hydraulic actuators.
• Teeter restraints, passive blade vibration dampers and tower dampers, and yaw system dynamic response.
• Transducer dynamics for control signals.
• All controller dynamics.

Formulation of equations of motion

As described briefly in the multibody dynamics approach the equations of motions of the complete system have been derived using the multibody dynamics approach based on the principle of virtual work. It appears that the solution of the resulting equations is generally difficult to obtain as the augmented mass matrix including the constraint matrices are generally ill-conditioned. The system is therefore transformed into a system where the only unknown is the strain accelerations \(\ddot{\bvector{\epsilon}}\) using the so-called constraint elimination method (Géradin and Cardona, 2001) together with the transformation for the velocity given in the form \(\bmatrix{D}_{\mathrm{r}}^{\mathrm{c}} \bvector{v}+\bmatrix{D}_{\epsilon}^{\mathrm{c}} \dot{\bvector{\epsilon}}=\mathbf{0}\). The final result of this straightforward transformation can be written in the conventional form

\[ \begin{equation} \bmatrix{M} \dot{\bvector{\epsilon}}+\bmatrix{C} \dot{\bvector{\epsilon}}+\bmatrix{K} \bvector{\epsilon}=\upsigma. \label{eq:constrainteliminationmethod1} \end{equation} \]

In cases with no prescribed strains it is straightforward to show, that the three system matrices appearing on the left-hand side of the above equation become:

\[ \begin{equation} \bmatrix{M}=\bmatrix{M}_{\epsilon \epsilon}+\bmatrix{D}_{\epsilon \mathrm{r}}^T \bmatrix{M}_{\mathrm{rr}} \bmatrix{D}_{\epsilon \mathrm{r}}+\bmatrix{D}_{\epsilon \mathrm{r}}^T \bmatrix{M}_{\mathrm{r} \epsilon}+\bmatrix{M}_{\mathrm{r} \epsilon}^T \bmatrix{D}_{\epsilon \mathrm{r}}, \quad \bmatrix{C}=\bmatrix{C}_{\epsilon \epsilon}, \quad \bmatrix{K}=\bmatrix{K}_{\epsilon \epsilon} \label{eq:constrainteliminationmethod2} \end{equation} \]

where \(\bmatrix{D}_{\epsilon \mathrm{r}}=-\bmatrix{D}_{\mathrm{r}}^{-l} \bmatrix{D}_{\epsilon}\) is the time-dependent part of the reduction matrix. The right-hand side stress vector of the global system of equations becomes:

\[ \begin{equation} \bvector{\upsigma}=\bvector{\upsigma}_{\mathrm{a}}+\bvector{\upsigma}_0-\bvector{\upsigma}_{\mathrm{i}}-\bmatrix{M}_{\mathrm{r} \epsilon}^T \bvector{g}_2+\bmatrix{D}_{\epsilon \mathrm{r}}^T\left(\bvector{f}_{\mathrm{a}}-\bvector{f}_{\mathrm{i}}-\bmatrix{M}_{\mathrm{rr}} \bvector{g}_2\right), \label{eq:constrainteliminationmethod3} \end{equation} \]

where \(\bvector{g}_2=\bmatrix{D}_{\epsilon \mathrm{r}}^{-1} \bvector{a}_2\).

In general, the system mass matrix M is full, due to the coupling of the degrees of freedom, and it contains periodic coefficients because of the time-dependent interaction of the dynamics of the rotor and tower. The system damping and stiffness matrices C and K are generally diagonal and constant.

Because of their complexity, further details of the equations of motion are not presented here.

Solution of the equations of motion

Typically, the equations of motion are solved by time-marching numerical integration of the system of ordinary differential equations using a variable step size, fourth order Runge-Kutta integrator (Kreyszig, 2006). For so-called stiff systems with many high frequency modes (for example wind turbine models with multi-part blades), a fixed step Newmark-\(\beta\) integrator (Newmark, 1959) or the Generalised-\(\alpha\) integrator (Chung, 1993) integrator are recommended to improve simulation performance.

It is noted that all fixed step integrators in Bladed assume zero structural state accelerations when calculating the geometric stiffness loads. This assumption was done to enable fixed-step integrators to converge within reasonable time step size. In contrast, the implicit Newmark-\(\beta\) and Runge-Kutta integrators do not require this simplification in the formulation, which will provide a more accurate solution for dynamic response of long-flexible blades. Therefore, the implicit Newmark-\(\beta\) integrator or Runge-Kutta integrator is recommended over other integrators in Bladed.

Last updated 15-11-2024