Structural dynamics of wind turbines

A modern wind turbine is a complex engineering system. At first sight it is primarily a machine for converting kinetic energy into a usable electric form; usually with the wind creating lift forces when flowing along an aerodynamically shaped blade. This is the same principle as used in an airplane. But instead of lifting a plane into the air, on a wind turbine these forces translate into a mechanical torque on a shaft, which is driving a generator. This simple idea is surprisingly efficient. As explained elsewhere [other chapter], the blades are rotating at a wind-speed dependent optimal speed to magnify the resulting forces. In some sense, the rotor of a wind turbine can be considered a giant funnel that captures the kinetic energy from a large area, by rotating through it at a specific speed, which is then concentrated into a higher density mechanical energy in the shaft. It is clear that such a rotating machine, with large and flexible blades and a drivetrain that transmits mega-watts, has to be designed carefully. With a typical design lifetime of 20-30 years, the rotor and drivetrain have to withstand 100 million or more rotations, while experiencing a large variety of wind conditions.

On the other hand, a wind turbine is not only a power plant, but might also be considered a building. The large sweeping rotor has to be mounted high enough to avoid a hazard for passing people, cars or ships, and wind conditions at height are usually better and allow for higher power conversion factors. However, the structural requirements for supporting a large wind turbine rotor and heavy drivetrain at large height are not very favorable, so the support structure (tower and foundations) contribute significantly to the cost of a wind turbine - without contributing anything to the amount of power generated. This means that the support structure has to be designed efficiently.

As the basis for accurate and efficient structural design is an accurate understanding of the dynamics of the system and its components, it should be clear that understanding the structural dynamics of a wind turbine is very important, especially when the system is so dynamic.

The remainder of this short chapter therefore aims to provide a general overview of the approaches toward resolving the structural dynamics of wind turbines (and their main structural components), to mention and discuss the major challenges in this field, and to point the reader to relevant and more advanced literature where more detailed information can be found.

A wind turbine as a structural system

Before discussing structural dynamics in more detail, we should remember the goals of structural design. The most important concerns here are to guarantee the safety of the structural system and its fitness for purpose, while at the same time trying to be cost efficient. The requirements and approaches used are typically dependent on which component is analyzed.

The focus in this chapter will be on:

• The wind turbine blade, as the primary load-bearing structure that is interacting with the flow.
• The rotor and drivetrain, as the main rotating components of the wind turbine
• The support structure (consisting of tower and foundations), as the main supporting component [* The mooring system?]

The wind turbine blade

A wind turbine blade can be thought of in some distinct ways. Primarily it is a complex airfoil that interacts with the air flow, creating distributed pressure forces. However, a blade is also a structural load-bearing and -transmitting element. In fact, from this structural point of view we usually distinguish two levels of fidelty.

In its load-generating and -bearing capacity, a wind turbine blade is very complex. The aerodynamic forces are complex enough (e.g. history dependent in the case of stall) that computational fluid dynamics are usually needed to assess them accurately enough. Individual layers of the composite used in the blade hull are defined with their individual mechanical properties and rotations, and the deflections and stresses experienced by them are simulated in nonlinear finite element simulations, to assess the risk of failure. An additional complication is that the rotation of the blades causes centripetal forces that need to be accounted for in these simulations.

As a load-transmitting element, a wind turbine blade is often simplified and considered a one-dimensional beam that is subject to distributed aerodynamic lift and drag forces and an aerodynamic moment (the latter of which is often neglected). These forces are usually represented by lift, drag and moment coefficients that depend on the shape of the airfoil, the flow angle of attack, and the Reynolds number. Such coefficients can be obtained though simulations that simplify the involved physics, often relying on potential-flow and boundary-layer approximations [XFOIL…], or simplified computational fluid dynamics [vortex methods] - or, of course, through actual experiments in wind tunnels. Structurally, the complex internal design of a wind turbine blade is simplified to representative stiffness and mass [NREL software?] values.

[Illustration: composite layup, internal design, representative beam?]

Major challenges for understanding the structural dynamics of blades are:

• the nonlinear aerodynamic loads and their complex interaction with the structure,
• the effects and pseudo-forces caused by the rotation of the blades,
• and the size and flexibility of the blades, which lead to large, non-linear deflections.

The rotor

The rotor is the assembly of all wind turbine blades in a common hub. The main reason for considering the rotor as a separate structural component is that its dynamics can be studied independently of the rest of the turbine. In fact, the number and specific arrangement of blades can lead to structural challenges - but also to simplifications (see section XX).

The main reason that horizontal axis wind turbines have three blades is that periodically changing gravity forces from three or more blades in a symmetric configuration cancel each other [ref - proof?]. In contrast, turbines with single blades need not only a counteracting weight to avoid large drivetrain bending loads, but also experience much larger periodic excitations. And turbines with two blades experience such unfavorable periodic excitations that most such turbines employ a special mechanisms that allows the rotor to “teeter” around the hub in a controlled fashion [ref?].

The drivetrain

The drivetrain consists of one (or more) shafts that are rotating, constrained by one or more bearings, and a number of additional components that interact with it, such as the rotor, the generator and/or gearbox, and a shaft brake or possibly a coupling. The requirements for wind turbine gearboxes and generators are relatively high, so typically detailed finite element simulations (e.g. of tooth-tooth interactions) are performed when designing these components. In fact, gearboxes and the electrical components have some of the highest failure rates in wind turbines, which indicates how important their design is. That said, the relative stiffness of shafts means that drive trains vibrations are usually approximated by simple zero- or one-dimensional models (see section XX).

The support structure

The support structure consists of the tower and the foundations, possibly with a load-transmitting substructure in between. Various choices and different conceptual solutions exist for the support structure, depending on specific applications and environmental factors. For small (< 100 kW), land-based wind turbines the most economical solution is often a guyed pole [ref. to Small Wind book]. Slightly larger turbines were often built on multi-member lattice towers (similar to electricity towers) [ref. Muskulus?], before circular towers emerged that not only seem visually more pleasing, but which also offer protection of service personnel and equipment against the environment. In offshore applications such circular towers were often connected to circular monopiles, but depending on water depth gravity-base structures, multi-member jackets, or floating concepts are more economical [ref?].

Support structures are usually manufactured from either structural steel or concrete. The dynamics of such structures is usually assessed in the elastic range, where the structure is behaving in a linear way. In other words, the dynamics of support structure is, in principle, very simple and can be efficiently resolved. Unfortunately, the interaction with both the supporting soil, surrounding sea or ice, and the supported rotor are usually nonlinear. For very large or more flexible wind turbines also geometric nonlinearities will be present.

Structural dynamics modeling

As seen above, a wind turbine is a complex engineering system. Considered as a whole, its dynamics is characterized by large deflections and finite rotations, excited by nonlinear loads and other nonlinear interactions with the environment. In this section we will discuss the main approaches to simplify and simulate this dynamics, starting with the simplest ones and
ending with the general case.

Obviously the dynamics of a wind turbine follows Newton’s laws, so at its core we have to solve a $n$-dimensional second-order mechanical equation such as,

$$ M\ddot{x} + C\dot{x} + Kx = F(x,\dot{x},\ddot{x},t).$$

A solution of this equation is a vector-valued function $x(t)$ that describes the motion of all $n$ degrees of freedoms involved, where $n$ is the dimension of the coordinate, or displacement, vector $x$. The force term $F$ mainly represents external forces due to interaction with the enviroment. These forces are typically time-dependent and can depend on the displacements $x$, their velocities $\dot{x}$ or accelerations $\ddot{x}$ in a non-trivial, nonlinear manner. In general, also the mass matrix $M$, the damping matrix $C$, and the stiffness matrix $K$ will also depend on displacements, velocities or accelerations. This is not indicated in Eq.X, as the resulting terms can be moved to the right-hand side.

Modal analysis and decomposition

Although it is possible, and indeed often necessary, to integrate the equation of motion Eq.X numerically, under certain conditions the problem can be significantly simplified. Let us assume that the mass and stiffness matrix are constant and consider the case that no external forces are acting. If we neglect damping, we end up with the equation for the free response of a mechanical system,

$$ M\ddot{x} + Kx = 0.$$

Being a linear equation, its solutions $x$ span a linear vector space. Assuming that functions of the form $x(t) = e^{\lambda t}$ are solutions results in an eigenvalue problem

$$\lambda^2 M + K = 0,$$

which has a non-trivial solution for all values $\lambda$ such that $\det (\lambda^2 M + K) = 0$.

When the matrices $K$ and $M$ are symmetric semi-positive definite, the eigenvalues $\lambda_j$ are purely imaginary, $\lambda_j = \pm i \omega_j$, where $\omega_j$ are the natural frequencies of the system. The corresponding eigenvectors $\phi_j$ are solutions of

$$(K - \omega_j^2 M) \phi_j = 0,$$

and called the mode shapes of the system. The properties of $K$ and $M$ guarantee that these mode shapes are real. Obviously, mode shapes are only defined up to a constant. One often used convention in structural dynamics is to normalize $\phi_j$ such that its largest component is equal to a unit displacement, $||\phi_j||_\infty = 1$.

Under these assumption, there will be $n$ independent mode shapes that span the space of solutions, and the general solution $x(t)$ of the equation of motion Eq.X has the form

$$x(t) = \sum_{i=1}^r (A_i + B_i t) \phi_i + \sum_{i=r+1}^n (A_i \cos \omega_i t + B_i \sin \omega_i t) \phi_i,$$

with $2n$ constants $A_i, B_i$ that can be determined from the $2n$ initial conditions $x(0), \dot{x}(0)$. The first term represents rigid body motion, which corresponds to motion with natural frequencies that are zero, and the second term corresponds to vibration modes. For most turbines the number of rigid body modes $r$ will be zero. In the case of floating wind turbines it is the mooring system that provides stiffness terms that result in non-zero frequencies. Note that one then usually still talks, somewhat incorrectly, about rigid body modes of the floating support structure. We will assume $r=0$ in the remainder.

The vector of eigenmodes $\Phi =(\phi_1, \dotsc, \phi_n)$ is a $n$-by-$n$ matrix that allows to define modal coordinates $z$, defined by

$$x = \Phi z.$$

Consider now the full equation of motion Eq.X,

$$ M\ddot{x} + C\dot{x} + Kx = F(x,\dot{x},\ddot{x},t).$$

It can be shown that eigenmodes are orthogonal with respect to the mass and stiffness matrix. Regarding damping, one often assumes modal (or classical) damping, where it is assumed that the modal damping is diagonal,

$$ \Phi^T C \Phi = \mathrm{diag}(2 \zeta_i \mu_i \omega_i),$$

where $\mu_i$ is the modal mass and the $\zeta_i$ are the modal fraction of critical damping of the $i$-th mode. Substituting in the equation of motion, Equation Eq.X then decouples into a number of independent equations, one for each mode,

$$ \ddot{z}_i + 2 \zeta_i \omega_i \dot{z}_i + \omega_i^2 z_i = \frac{1}{\mu_i} \Phi_i^T F.$$

This modal decomposition can be used to simplify the dynamics of the system. Most excitations are usually characterized by exhibiting most energy at relatively low frequencies. It is therefore customary to only consider a small number $m \ll n$ of modes to be dynamically relevant. Applying such modal truncation, the corresponding dynamic respose in given by the first $m$ modes only,

$$x_d(t) = \sum_{i=1}^m \phi_i z_i(t).$$

However, it would be a mistake to simply neglect the remaining modes. Although not acting dynamically, these modes still contribute a quasi-static response,

$$x_s(t) = \sum_{i=m+1}^n \frac{\phi_i^T F(t)}{\mu_i \omega_i^2} \phi_i.$$

Dynamics in the frequency domain

The modal expansion Eq.X does not only allow for simplification through modal truncation. It also shows how the response often can be obtained without numerically integrating the equations over time. For a linear equation such as Eq.X it is an important (and easily established) fact that a harmonic excitation $f(t)=F e^{j \omega t}$ at a single frequency $\omega$ results in a harmonic response at the same frequency, $x(t) = X e^{j \omega t}$. The complex amplitudes $F$ and $X$ are related by

$$X = \left(-\omega^2 M + j \omega C + K \right)^{-1} F = G(\omega) F,$$

where

$$G(\omega) = \sum_{i=1}^n \frac{\phi_i \phi_i^T}{\mu_i \omega_i^2} \cdot \frac{1}{1-\frac{\omega^2}{\omega_i^2} + 2 j \zeta_i \frac{\omega}{\omega_i}}$$

is called the dynamic flexibility matrix [ref. to Preumont section 2.5]. It is the generalization to multiple degrees of freedom of the transfer (or frequency response) function $H(\omega) = X/F$.

In fact, the Fourier transform $X(\omega)$ of the response is related to the Fourier transform $X(\omega)$ of a general excitation by

$$X(\omega) = G(\omega) F(\omega).$$

Since Fourier transform can be calculated efficiently with the FFT algorithm, this approach allows to determine the response of a linear structural system in a highly efficient manner. [This has been used in QuLa/QuLaF, for example]

Moreover, it is possible to study the behavior of the structure to random excitations by replacing the Fourier transform $F(\omega)$ of the excitation with a spectrum $S_F(\omega)$ (i.e., effectively neglecting the phases in Fourier transform). The corresponding response spectrum is given by

$$S_X(\omega) = G(\omega) S_F(\omega) G(\omega)^* = |G(\omega)|^2 S_F(\omega).$$

Multi-blade coordinate transformation

A major complication in the structural analysis of wind turbines is that some components are rotating. This raises the question if and how we can define eigemodes then. A first try might be to perform modal analysis with the rotor in a specific configuration, for example with one blade at zero azimuth (pointing exactly upwards). However, the rotation usually results in changes in the (apparent) structural properties (i.e., stiffnesses and masses), due to centrifugal stiffening, gyroscopic forces, and other nonlinearities. One strategy has been to average these system matrices over one rotational period before performing an eigenanalysis. This will, however, eliminate periodic terms and can therefore lead to erroneous results.

The standard way to deal with these problems is to use a multi-blade coordinate transformation [ref. Bir?] (also called Coleman transformation or Fourier coordinate transformation). We assume that the rotor consists of $N$ blades that are symmetrically distributed along the rotation axis, with an instantaneous azimuth angle $\psi_m = \psi_1 + \frac{2 \pi}{N}m$, for $m = 1, \dotsc, N$. Consider a (generalized) degree of freedom $q^{(m)}$ for each blade, for example edgewise bending (in the rotor plane) or flapwise bending (out of the rotor plane). We can transform this set of rotating freedoms from the rotating frame into a set of non-rotating degrees of freedom by defining

$$ \begin{align} q_0 & = \frac{1}{N} \sum_{m=1}^N q^{(m)} \ q_{nc} & = \frac{1}{N} \sum_{m=1}^N q^{(m)} \cos n \psi_m \ q_{ns} & = \frac{1}{N} \sum_{m=1}^N q^{(m)} \sin n \psi_m \ q_{N/2} & = \frac{1}{N} \sum_{m=1}^N q^{(m)} (-1)^m
\end{align} $$

The first of these is called the collective degree of freedom, the next two lines represent cyclic degrees of freedom, and the last freedom is only present when the number of blades $N$ is even. The index $n$ runs from $1$ to $(N-1)/2$ (if $N$ is odd) or $(N-2)/2$ (if $N$ is even), such that we end up with $N$ degrees of freedom again. The (cyclic) degrees of freedom with $n > 1$ are only present for rotors with five or more blades and describe mainly internal rotor motions. These are therefore called reactionless freedoms. The collective and first cyclic freedoms, on the other hand, are the dominant coupling with the drivetrain [ref. Johnson]. The (reactionless) freedom $q_{N/2}$ is only present if the number of blades is even, and usually results in some special dynamics. Moreover, for a two-bladed turbine $q_1 = \frac{1}{2}(q^{(2)} - q^{(1)})$ replaces the usual cyclic modes. The dynamics of two-bladed rotors is therefore fundamentally different from other rotor configurations.

It should be noted that the MBC transformation does not necessarily eliminate all periodic terms, so the dynamics can still be time-variant after the transformation. However, for isotropic rotors - where all blades are equal and symmetrically distributed - the coupling to the support depends only on azimuth and not blade number, and the MBC transformation results in a time-invariant system that can then be subject to eigenvalue analysis and modal decomposition.

In general, the MBC transformation filters out all frequencies that are not integral multiples of $\Omega N$, where $\Omega$ is the rotor speed. It can be used for non-symmetrical rotors, but then other frequencies will remain after the transformation, and it is necessary to use more advanced techniques such as Floquet theory.

Campbell diagram

The inverse to Eq. X is

$$ q^{(k)} = q_0 + \sum_{n} (q_{nc} \cos n \psi_k + q_{ns} \sin n \psi_k) + q_{N/2}(-1)^k, $$

where the last term is only present when $N$ is even. For a three bladed rotor, in particular, the three resulting equations read

$$ q^{(k)} = q_0 + q_c \cos\left( \Omega t + \frac{2 \pi}{3}(k-1) \right) + q_s \sin \left( \Omega t + \frac{2 \pi}{3}(k-1) \right), \qquad k=1, 2, 3.$$

Eigenvalue problem… ???

mode shapes in blade coordinates becomes:

$$ q^{(k)} = A_0 \sin(\omega t + \phi_0) + A_{BW} \sin\left[ (\omega + \Omega)t + \frac{2 \pi}{3}(k-1) + \phi_{BW} \right] + A_{FW} \sin \left[ (\omega -\Omega) t - \frac{2 \pi}{3}(k-1) + \phi_{FW} \right], \qquad k=1, 2, 3,$$

where $A_{BW}, \phi_{BW}$ and $A_{FW}, \phi{FW}$ are new modal amplitudes and phases of a backward-whirling and a forward-whirling mode. [ref. Hansen 2007 - instability problems]

Floquet theory

Floquet theory is a general approach for studying the dynamics of periodic systems. In fact, the MBC transformation can be considered a special case of it [ref. Skjoldan & Hansen 2009].

Whirling modes?**

Backward and forward whirling modes… part of the fan generated by the Floquet analysis

Mode veering?!

Bottasso & Cacciola

Common system period $T$

Dynamics of a wind turbine

Dynamics of the rotor

Dynamics of the drivetrain

Dynamics of support structures

Notes

• Intro: basis for simulations, safety, etc.

• Complex: rotating parts, large deflections, etc.

• Approximations: some parts can be described by FEM or modes

• Scope: no historical overview, no in-depth treatment

• Basics: need to assess deformations (strains) and resulting internal forces (stresses)

• material usually assumed to be elastic

• global dynamics vs. component-models

• rotorloads transmitted through shaft, so basically coupling through (max.) 6 DOF?

Modeling technology

• Modal analysis
• Finite element analysis
• Multibody dynamics
• Flexible multibody modelling
• More options?

Modal analysis

• Steady conditions (constant RPM - the only requirement?!?)
• Computational efficiency

Finite element analysis

• Small deflections, infinitesimal (?) rotations assumed

Multibody dynamics

• Finite rotations, joints (constraints) needed

Flexible multibody modeling

Rotor dynamics

Multi-coordinate formalism

• isotropic
• shortcut that is equivalent to one of the Floquet solutions?

Linearization and stability

Drivetrain

Support structure

Foundations

Structural analysis

Frequency-domain analysis

Fatigue

• Cycle counting
• Locations
• Meta-modeling

Extreme loads

• Extrapolation

Accidental and service limit states

Load-structure-interaction

FSI, SSI, ISI

Monitoring and structural health

$$m \ddot{x} + k \dot{x} = 0$$