# Effect of steady def lections on the aeroelastic stability of a turbine blade
B. S. Kallesøe
Wind Energy Department, Risø-DTU, Technical University of Denmark, DK-4000 Roskilde, Denmark
# ABSTRACT
This paper deals with effects of geometric non-linearities on the aeroelastic stability of a steady-state deflected blade. Today, wind turbine blades are long and slender structures that can have a considerable steady-state def lection which affects the dynamic behaviour of the blade. The f lapwise blade def lection causes the edgewise blade motion to couple to torsional blade motion and thereby to the aerodynamics through the angle of attack. The analysis shows that in the worst case for this particular blade, the edgewise damping can be decreased by half.
Copyright $\circled{\mathrm{C}}\ 2010$ John Wiley & Sons, Ltd.
# KEYWORDS
stability analysis; aeroelasticity
# Correspondence
B. S. Kallesøe, Wind Energy Division, Risø DTU, Frederiksborgvej 399, P.O. Box 49, DK-4000 Roskilde, Denmark. E-mail: bska@risoe.dtu.dk
Received 5 October 2009; Revised 17 May 2010; Accepted; 31 May 2010
# 1. INTRODUCTION
A second-order non-linear beam model is used for aeroelastic stability analysis of a wind turbine blade. The importance of including the effects of non-linear geometric couplings in the stability analysis is considered and the aeroelastic mechanisms driving the aeroelastic response are described in detail.
The effect of non-linear geometric couplings in a curved rotating blade on the stability has been investigated in the helicopter society for decades1–4 and state-of-the-art comprehensive helicopter stability codes of today, like Hodges et al.,5 include both material and geometric non-linearities. However, most aeroelastic stability tools for wind turbines are based on linear beam theory and do not include the non-linear geometric coupling caused by, for instance, steady-state blade def lection, pre-bend or swept blade.
In the late 1970s, the oil crisis stimulated many MW size turbine projects. In a review of research on aeroelastic stability Friedmann6 concluded that ‘Reliable aeroelastic stability analyses should be based on non-linear formulations which account for both moderately large deformations (i.e. finite slopes) and non-linear aerodynamic effects, such as stall’. All these MW size turbine projects however ended without any commercial success. Later, the wind turbine followed a development starting at small $30\;\mathrm{kW}$ units gradually growing to today’s MW size commercial turbine. During this period, wind turbines have been relatively stiff constructions with only limited geometric couplings. Chaviaropoulos7 examines the influence of non-linear effects on the aeroelastic stability of a $19\;\mathrm{m}$ blade. It was found that the most important effect to include is the unsteady aerodynamics and that the structural def lection is unimportant. Modern wind turbine blades are longer (up to $60\;\mathrm{m}$ ) and more slender, thus increasing the blade def lection under normal operation and thereby reintroducing stability issues concerning geometric couplings. Steady-state blade def lection will result in geometrically non-linear couplings between the different blade modes. For instance, a large flapwise blade def lection will enhance the coupling between edgewise and torsional blade motion and consequently affect the aerodynamics through the angle of attack. Therefore, it can be important to include the non-linear geometric coupling between for example edgewise and torsional motion of a flapwise deflected blade.
Research in utilizing sweep and pre-bend blades is ongoing. The European Union founded project UPWIND $^{8-10}$ deals, among other issues, with non-linear modelling of blades and the effects of including such non-linearities. Some stateof-the-art stability codes, such as TURBU,11 include the effect of geometric non-linearities. Riziotis et al.12 include these effects in a stability analysis of a turbine in closed-loop operation. There is also focus on utilizing the geometric couplings to reduce fatigue and/or ultimate loads, for instance Ashwill et al.,13 where a blade is swept to introduce a flapwise—torsion coupling.
Wind turbine stability can be analysed by a variety of different model types. The most detailed description of the turbine response is given by numerical non-linear time simulation tools.14–18 These tools show instabilities as well as non-linear effects limiting the response to for instance limit cycle oscillations. They can also be used to analyse the effect of, for instance, turbulence and wind shear’s effects on turbine stability. The referenced tools use different models and different model complexity. For instance, $\mathrm{FAST^{18}}$ is a modal-based code which on the one hand does not include a torsional degree of freedom of the blade and non-linear geometric couplings, but on the other hand is relatively computationally inexpensive. A code like $\mathrm{HAWC}2^{14,15}$ has a more complex model with a structural model based on a multi-body formulation where each body is a Timoshenko beam element including torsion. The drawback of these time-simulation tools is that they are computationally intensive and they can make it difficult to extract the important aeroelastic mechanisms from the large volume of results. Another approach is to use eigenvalue analysis of a linear (or linearized) model of the turbine.11,12,19–21 The HAWCStab code19,21 offers a platform for linearization of the undeflected turbine structure, while the code TURBU11 offers a platform for aeroservoelastic stability analysis based on linearization around the def lected/curved blade state. The structural model in TURBU is based on a simple co-rotational beam element approach. Each beam element consists of a rigid body with springs and dampers in its entry point; average strains in the springs and torque-free rotation offsets between the beam elements embody the average deflected/curved blade state. Riziotis et al.12 offers a multi-body platform which finds a reference state by time integration and linearizes the aeroservoelastic equations around this reference state to provide a stability tool including closed loop control. This type of tool can give both structural eigenfrequencies and eigenmodes that describe the basic structural dynamics of the turbine and aeroelastic frequencies, damping and modes of the aeroelastic motion. The aeroelastic damping reveals any stability problems for the turbine. However, since it is linear tools, they do not give any information concerning non-linear mechanism that limit the amplitude of a linear negative damped mode. The knowledge of structural and aeroelastic frequencies and mode shapes is very useful in the analysis and in the interpretation of results from aeroelastic time simulations. However, the modes of the aeroelastic response of the whole turbine can still be complex to analyse. To reduce the complexity, and thus make the results more transparent, a blade-only analysis is used.22 This allows a clear physical interpretation and insight into the mechanisms that govern the dynamic response of the blade and many basic characteristic of turbine stability can be extract from a blade-only analysis.
This paper uses a non-linear blade model23 which includes the effect of large blade def lections, pitch action and rotor speed variations. This blade model is strongly inspired by the work of Hodges and Dowell1 First, the structural model is combined with a steady-state aerodynamic model based on beam element momentum (BEM) theory and discritized by a f inite difference scheme. The resulting algebraic non-linear aeroelastic model is employed to compute steady-state blade def lections and induced velocities of a blade from the 5 MW Reference Wind Turbine (RWT) by National Renewable Energy Laboratory (NREL)24 at normal power production conditions. The steady-state def lections are compared with the results from HAWC2 simulations, showing good agreement. Throughout this paper, the 5 MW RWT by NREL is used as an example blade. The reference turbine is an artif icial turbine based on state-of-the-art turbines on the market. The blade is strongly inspired by the $61.5\mathrm{~m~LM}$ glasf iber blade (LM Wind Power, Kolding, Denmark). This blade belongs to the mid-region of f lexible designs of state-of-the-art blades, and hence, the geometric couplings can be more pronounced for other blade designs. The big advantage of this blade however is that all data is publicly available and it has been widely used in other research work and therefore a good reference with realistic f lexibility compared with most state-of-the-art blades. A non-linear structural blade model23 and an unsteady aerodynamic model25 are then linearized about the steadystate def lected blade, preserving the main effects of the geometric non-linearities. The linear model is discritized by the f inite difference scheme which along with boundary conditions form a differential eigenvalue problem. The solution to this eigenvalue problem gives the aeroelastic frequencies and damping, but also information concerning the fundamental aeroelastic behaviour of the blade. The analysis shows that the aeroelastic damping of the edgewise modes changes when the steady-state def lection is included. The aeroelastic motion is analysed in detail for three different operation conditions in which there is large differences in the damping when including or excluding steady-state blade def lections.
The structural blade model described in Kallesøe23 is based on the work by Hodges and Dowell1 using second order Bernoulli–Euler beam theory to describe the blade motion by a non-linear partial integral-differential equation of motion
\overline{\bf M}\ddot{\overline{\bf u}}+\overline{{{\bf F}}}\left(\dot{\overline{\bf u}},\overline{{{\bf u}}}^{\prime\prime},\overline{{{\bf u}}}^{\prime},\overline{{{\bf u}}},\ddot{\beta},\dot{\beta},\beta,\ddot{\phi},\dot{\phi},\phi\right)\!=\overline{{{\bf f}}}\left({\bf f}_{a e r o},M_{a e r o},u^{\prime},\nu^{\prime}\right)
where $\bar{\bf M}$ is the mass matrix, $\bar{\mathbf{F}}$ is a non-linear function that includes stiffness, damping, gyroscopic terms together with centrifugal force-based integral terms. The state vector $\bar{\mathbf{u}}=[u\left(t,s\right),\nu\left(t,s\right),\theta\left(t,s\right)]$ holds edgewise, flapwise and torsional deformations, respectively.
Flapwise is defined as the direction normal to the rotor plane (positive downwind) and edgewise as in the rotor plane (positive towards leading edge) for a blade at zero pitch. When the blade pitches, the $(u,\,\nu)$ frame follows the blade. The position along the blades elastic axis is denoted $s$ , $t$ is the time, $\beta=\beta(t)$ is the global pitch of the blade, $\phi=\phi(t)$ is the azimuth angle of the rotor and the right hand side force function $\bar{\mathbf{f}}$ holds the effect of the aerodynamic forces $\mathbf{f}_{a e r o}$ and aerodynamic moment $M_{a e r o}$ on the blade. The dots denote time derivatives and the primes denote derivatives with respect to the longitudinal coordinate $s$ . As an example, the equation of motion for edgewise blade bending is given by
叶片挥舞方向定义为垂直于风轮平面的方向（顺风方向为正），摆振方向定义为在风轮平面内（前缘方向为正），当叶片变桨角度为零时。当叶片变桨时，$(u,\,\nu)$坐标系跟随叶片运动。叶片弹性轴上的位置用$s$表示，$t$表示时间，$\beta=\beta(t)$表示叶片的全局变桨角度，$\phi=\phi(t)$表示风轮的方位角，右侧力函数$\bar{\mathbf{f}}$包含作用于叶片的气动力$\mathbf{f}_{a e r o}$和气动力矩$M_{a e r o}$的影响。点表示时间导数，撇号表示对纵向坐标$s$的导数。例如，摆振叶片弯曲的运动方程为：
where the first term is the inertia forces, the second term $F_{u,1}$ describes the influence of pitch action, which will not be used in this work. The third term $\boldsymbol{F}_{u,2}$ describes centrifugal and Coriolis forces caused by the rotor speed. The fourth term $F_{u,3}$ describes the unsteady influence form gravity, which is neglected in this work. The fifth term describes the restoring forces
where the first term is the bending stiffness in the $x$ -direction, the second term is the coupling to the $y$ -direction and the last term in equation (3) is the coupling to the twist. The sixth term in equation (2) describes the influence of rotor speed variations, which is assumed constant in this work, so the term is not active. The right hand side holds the external forces, which in this case will be aerodynamic forces. Longitudinal forces on and in the blade, for example the centrifugal force, lead to integral terms in the equations of motion. A detailed description of all terms are found in Kallesøe.23
The boundary conditions employed in this paper are for simplification derived for blades without pre-curvature. The boundary conditions for the root of the blade are given by the geometric constraints
where $s=R$ is the tip of the blade, $m=m(s)$ is the mass per length of the blade, $l_{c g}=l_{c g}(s)$ is the offset of centre of gravity from the elastic axis, $E=E(s)$ is the Young’s modulus, $I=I\left(s\right)$ and $I_{\eta}=I_{\eta}(s)$ is the principle moments of inertia, $w=$ $w(s,t)$ is the radius to the position $s$ on the elastic axis, $g$ denotes gravity, $\tilde{\theta}=\tilde{\theta}\left(s\right)$ is the angle between chord and principle axis of elasticity and $\tilde{\theta}=\tilde{\theta}\left(s\right)$ is the angle between the chord and a line between elastic centre and centre of gravity along which $l_{c g}$ is measured. In the case that $l_{c g}(R)\neq0$ the boundary conditions for the tip are functions of the rotor speed $\dot{\phi}$ and the azimuth angle of the rotor $\phi$ and therefore time varying. This is because an offset of the centre of gravity from the elastic axis at the blade tip leads to a bending moment at the tip caused by gravity and centrifugal force. Most modern wind turbine blades are tapered at the tip, whereby $l_{c g}(s)\longrightarrow{\cal0}$ and $E I_{\xi}I_{\eta}\longrightarrow0$ . Hence, it depends on the individual blade design if this azimuth angle-dependent boundary conditions can be neglected or not. In this work, the blade is constructed such that $l_{c g}(R)=0$ and $E I_{\xi}I_{\eta}|_{s=R}\neq0$ , thus making the boundary conditions azimuth angel independent and hence all right hand sides of equation (5) become zero.
To determine the steady-state def lection for the blade, a non-linear steady-state aeroelastic model i.s derived. Steady-state conditions are def ined as uniform inf low, zero gravity, constant rotor speed and pitch an.gle $\ddot{\phi}=\dot{\beta}=0$ whereby all time derivatives in the structural equations of motion (1) become zero $\ddot{u}=\ddot{\nu}=\ddot{\theta}=\dot{u}=\dot{\nu}=\dot{\theta}=0$ . These uniform conditions remove the periodicity of the system. The steady-state aerodynamic model is based on blade element momentum (BEM) theory including Prendtl’s tip loss correction.26 The BEM theory computes a balance between the forces on the blade and the momentum change in the wind. The aerodynamic model is coupled to the structural model through the local wind speed and angle of attack and the structural model is coupled to the aerodynamic model through the aerodynamic forces acting on the blade.
The equations of motion (equation (1)) are discretized on an equidistant grid along the elastic axis with step size $h$ and $N$ computation points. The spatial derivatives of the partial differential equation of motion (1) are approximated by the f inite difference scheme given in Table I. The derivatives of parameters (such as mass, stiffness, etc.) are approximated by the same f inite difference scheme. The integral terms in the equation of motion are approximated by sums using the trapezoid rule.
The boundary conditions for the f inite difference formulation are derived by inserting the f inite difference approximations into the boundary conditions (equations (4) and (5)). It is assumed that the offset of the centre of gravity is zero at the blade tip, thus making the boundary condition independent of rotor position.
The discretized version of the partial differential equations of motion implemented on the $N$ discretization points forms a set of non-linear algebraic equations:
where $\mathbf{F}_{s t}\;(\mathbf{u}_{0},\;\dot{\phi}_{0},\;\beta_{0})$ holds the terms from the discretization of the structural equation and $\mathbf{u}_{0}=[u_{0,1},\,\nu_{0,1},\,\theta_{0,1},\,\dots\,,\,u_{0},$ $u_{0,N},$ $\nu_{0,N}$ , ${\theta_{0,N}}\mathrm{]}^{\mathrm{T}}$ holds the steady-state deformation at each discretization point. The f irst subscript 0 denotes that it is the steadystate solution (zero order) and the second subscript denotes the discretization point, counting from the root of the blade. The right hand side $\mathbf{f}_{0}$ holds the steady-state aerodynamic forces computed at each discretization point using BEM theory.
The finite difference discretized steady-state equation (equation (6)) has 3N unknown blade def lections (flapwise, edgewise and torsional def lections of the $_\mathrm{N}$ discretization points) and 2N unknown induction factors (longitudinal and tangential induction factor at the $_\mathrm{N}$ discretization points). This system of non-linear equations is solved using the following iterative scheme: i) Operational conditions are chosen: steady-state wind speed $\left(U_{0}\right)$ , the corresponding rotor speed $\dot{\phi}_{0}=\dot{\phi}_{0}(U_{0}))$ and pitch setting $\begin{array}{r}{\beta_{0}=\beta_{0}(U_{0}))}\end{array}$ ; ii) apparent wind speed and angle of attack based on inf low conditions, blade def lections and induction factors are computed; iii) the aerodynamic forces using BEM theory are computed; iv) equation (6) is solved for the deformations $\mathbf{u}_{0}$ ; v) new induction factors are computed; and vi) if no convergence return to step 2. This gives the steady-state deformations $\mathbf{u}_{0}=\mathbf{u}_{0}\left(U_{0},\,\dot{\phi}_{0},\,\beta_{0}\right)$ and the induction factors for the given operational condition.
Figure 1. Edgewise and f lapwise def lection and angle of attack at $55.5~\mathsf{m}$ radius $(88\%)$ vs. wind speed for the present second-order Bernoulli–Euler blade model (equation (6)) and the non-linear aeroelastic time simulation code HAWC2.
The steady-state model (equation (6)) is used to compute steady-state blade def lection and induction factors for the NREL 5 MW RWT24 blade at normal power production operation. The results are compared with results from the non-linear aeroelastic time simulation code HAWC2.14,15 The HAWC2 code is a multi-body formulation where each body is a linear Timoshenko beam element with a torsional degree of freedom. The geometric non-linearities are captured by the multibody formulation, in which the blades for example are modelled by 10 bodies each. If only one body per blade is used the code will become as a linear code since the beam model in each body is linear, whereas a convergence study has shown that with 10 bodies the geometric non-linearities are captured. In the present model, only one blade is considered and modelled as a f lexible beam. For f irst and second modes of blade motions, as considered in this paper, the rotary and shear effects are negligible, so the Bernoulli–Euler beam model in the present mode is comparable with the Timoshenko beam model in HAWC2. As for higher order modes of motion and other turbine components, the rotary and shear effects are of higher relevance. Figure 1 shows the blade flapwise and edgewise def lections and angle of attack at radius $55.5~\mathrm{m}$ $88\%$ blade length) at different wind speeds. The angle of attack indicates how well the torsional deformation from the two models agrees. It is seen that there is good agreement between the present second-order Bernoulli–Euler blade model and HAWC2 for all operational conditions. The kink at rated wind speed $(\approx11{\mathrm{~m~s~}^{-1}})$ at the blade tip def lection curve is caused by the shift from variable speed, constant pitch to constant speed, variable pitch operation.
In this section, the aeroelastic modes of blade motion are analysed with particular emphasis on effects of steady-state flapwise blade def lection. The stability of a specif ic blade at normal operation will be analysed in detail and differences including and excluding geometric couplings will be discussed. The effect of pre-bend is similar to the effects of steadystate blade def lection which is investigated in this analysis. The effect of sweep (edgewise curved blades) is different since it couples flapwise and torsional motion instead of edgewise and torsion as characterized by the flapwise deflection.
The non-linear partial differential equations of motion is linearized by inserting ${\mathbf{u}}(s,\,t)={\mathbf{u}}_{0}(s)+\varepsilon{\mathbf{u}}_{1}(s,\,t)$ into equation (1), where ${\bf u}_{0}(s)$ is the steady-state deflected blade position including any pre-bend and sweep, ${\mathbf{u}}_{1}(s,t)$ is time-dependent variations around this position and $\varepsilon$ is a bookkeeping parameter denoting smallness of terms. The external inf luences, such as wind speed, pitch setting, etc. are split into a steady part and a time-varying part (denoted by the subscript 0 and 1, respectively) with the bookkeeping parameter $\varepsilon$ . The equation of motion (equation (1)) is Taylor expanded assuming $\varepsilon<<1$.Balancingtermsoforder$\varepsilon^{\mathrm{l}}$givethelinearapproximationaroundthedeflectedbladeposition$\mathbf{u}_{0}$.Bylinearizingtheequationsofmotionaboutthedeflectedbladethemaineffectsforthegeometricnon-linearitiesarepreserved.Forexample,thenon-linearstiffnesstermintheedgewiseequation
when linearized about the deflected blade (using $\theta=\theta_{0}+\theta_{1}$ and $\nu=\nu_{0}+\nu_{1}$ ), whereby the important coupling between edgewise and torsional blade motion of a flapwise deflected blade is preserved. The subscript 1 denotes the linear variation around the linearization point $\mathbf{u}_{0}$ . Likewise the non-linear term in the torsional equation
when linearized about the def lected blade. The major effect of the important geometric coupling in the stiffness terms (equations (7) and (9)) between edgewise and torsional motion of a f lapwise def lected blade is preserved when linearized about the steady-state def lected blade (equations (8) and (10)).
The linearized equations of motion are combined with a linearized Beddoes–Leishman27 type of unsteady aerodynamic model.25 The unsteady aerodynamic model is formulated in a state space formulation with four states; two states are second-order approximations to Thoedorsen’s function28 and two states describe the dynamics of the trailing edge separation point. Periodic effects, such as gravity, can be included in the linear model by considering $\sin(\phi_{1}\;+\;t\dot{\phi}_{0})$ and $\cos(\phi_{\mathrm{l}}\;+\;t\dot{\phi}_{\mathrm{0}})$ as independent variables, which subsequently can be obtained by a non-linear transformation, but are neglected in this work. The linear partial differential equation and the unsteady aerodynamic model are given by
where $\mathbf{u}=\mathbf{u}(s,\,t)=[u_{1}(s,\,t)$ , $\nu_{1}(s,\,t),\,\theta_{1}(s,\,t)]$ are the linear def lections around the linearization point $\mathbf{u}_{0}$ , $\begin{array}{r}{\tilde{\mathbf{M}}{=}\tilde{\mathbf{M}}(\mathbf{u}_{0},\,\dot{\phi}_{0},\,\beta_{0},}\end{array}$ $U_{n,0})$ , $\tilde{\mathbf{D}}=\tilde{\mathbf{D}}$ $(\mathbf{u}_{0},\,\dot{\phi}_{0},\,\beta_{0},\,U_{n,0})$ , $\tilde{\mathbf{K}}_{s s}=\tilde{\mathbf{K}}_{s s}\;(\mathbf{u}_{0},\;\dot{\phi}_{0},\;\beta_{0}),\;\tilde{\mathbf{K}}_{s}=\tilde{\mathbf{K}}_{s}\;(\dot{\phi}_{0},\;\beta_{0}),\;\tilde{\mathbf{K}}=\tilde{\mathbf{K}}\;(\mathbf{u}_{0},\;\dot{\phi}_{0},\;\beta_{0},\;U_{0})$ are collections of the linear coeff icients, where $U_{0}$ is the mean wind speed, $\tilde{\mathbf{C}}=\tilde{\mathbf{C}}\left(\mathbf{u}_{0},\;\dot{\phi}_{0},\;\beta_{0}\right)$ is the unsteady aerodynamic’s effect on the structure, where $U_{1}$ is the variation of the wind speed. The coupling to external inf luences such as pitch action and wind speed variations is described on t.he right hand side, where $\tilde{\mathbf{F}_{s}}=\tilde{\mathbf{F}}_{s}\left(\mathbf{u}_{0},\,\dot{\phi}_{0},\,\beta_{0},\,U_{n,0}\right)$ holds the linear gains on the external inf luences given by f˜ $=[\beta(t),\ \dot{\beta}(t),\ \ddot{\beta}(t),\ \sin(\phi_{1}(t)+t\dot{\phi}_{0}),\cos(\phi_{1}(t)+t\dot{\phi}_{0}),\ \dot{\phi}_{1}(t),\ \ddot{\phi}_{1}(t),\ U_{1}(s,\ t),\ \dot{U}_{1}(s,\ t)]^{\mathrm{T}}$ . The four aerodynamic states in $\mathbf{z}$ are modelled by steady-state wind speed-dependent time constants $\tilde{\mathbf{T}}$ and affected by the linear blade def lection, speed and acceleration through time-varying angle of attack and local wind speed described by the matrices $\tilde{\bf G}$ , H˜ and $\tilde{\mathbf{J}}$ .25 The linear gains on external inf luences are given by $\tilde{\mathbf{F}}_{a}$ .
# 4.2. Aeroelastic modes of motion
The spatial derivatives in the linear equations of aeroelastic motion (equation (11)) are approximated by the f inite difference scheme (Table I) with N discretization points. The f inite difference implementation includes the spatial boundary conditions (equations (4) and (5)). The second-order differential equation is then rewritten into f irst-order form by introducing the f irst-order time derivatives as states and combining it with the unsteady aerodynamic model. The spatial discretized f irst-order equation of aeroelastic motion becomes
where $\dot{\mathbf{x}}$ includes the linear deformation around the linearization point, speed and the aerodynamic states for each discretization point, giving $3N+3N+4N=10N$ degrees of freedom, A is the linear coeff icients, $\mathbf{B}$ is the linear gains on the external inf luences and f is the linear variation of the external inf luences. The unforced version of equation (12) forms a differential eigenvalue problem.29 The differential eigenvalue problem is casted into an algebraic eigenvalue problem by assuming a complex exponential solution. The eigenvalues and corresponding eigenvectors can be grouped into two sets: real valued and complex valued eigenvalues. Generally, the real valued eigenvalues are related to the aerodynamic states and correspond to the aerodynamic time lags. However, overdamped aeroelastic modes will also have real valued eigenvalues. The complex valued eigenvalues are related to the aeroelastic states and give the aeroelastic frequencies and damping. The corresponding eigenvectors give the aeroelastic mode shapes of the particular mode.
It is noted that since aerodynamic forces are included, the eigenvalue problem12 is not self-adjoint, and therefore, the eigenvectors are not orthogonal.
# 4.3. Frequency and damping of a blade at normal power production conditions
The model described above is used to analyse the effect of geometric non-linearities caused by steady-state blade def lections under normal operational conditions. The aeroelastic frequencies, damping and mode shapes of the NREL RWT blade are computed for different wind speeds in the power production region. The aeroelastic results are computed in two versions: one in which the model is linearized about the steady-state def lected blade, and another in which it is linearized about the undef lected blade, hereby including and excluding the effect of the geometric non-linearities, respectively.
The results from the present model are compared with the results from the non-linear aeroelastic time simulation code HAWC2. Since each body in this code is a linear beam model and the non-linearities are only included by the multi-body formulation, this model will produce linear results if only one body per blade is used and non-linear results if more bodies are used. Hence, a one body per blade model will correspond to the present model without geometric couplings and a model with more bodies will correspond to the present model with geometric couplings. Two versions of the HAWC2 model are used in this work: one with one body in the blade and one with 10 bodies in the blade. In both models, only the blade is considered as a f lexible beam. The frequencies and damping from the time simulation code are estimated by f itting the frequency, phase and damping of a number of exponentially decaying sinusoidal functions to the decay of the blade motion after an initial excitation at the expected aeroelastic frequency. In the simulations, the pitch angle is set to a prescribed value dependent on wind speed only.
Figure 2 shows the aeroelastic frequencies and damping of the two f irst f lapwise blade-bending modes. In the variable speed operation range (5 to $12\;\mathrm{m}\;\mathrm{s}^{-1}$ ), the aeroelastic frequency increases because of increased centrifugal stiffness. The disagreement between the undef lected and def lected blade case in aeroelastic frequency of the f irst f lapwise mode around $20\ \mathrm{m}\ \mathrm{s}^{-1}$ is caused by the increased steady-state blade twist, which changes the angle of attack and thereby the aerodynamic stiffness. The damping of the f irst f lapwise bending mode is almost the same for the undef lected and def lected blade case, there are only some minor differences at the same wind speeds that are also caused by the small change in steady angle of attack. For the second f lapwise bending mode, neither the frequency nor the damping are changed by the inclusion of the geometric non-linearities. The results for the second f lapwise mode from HAWC2 are seen to follow the same trend as the results from the present model. Because of the high damping of this mode, the decay of initial excited oscillations is very fast and the noise from other lower damped modes becomes relatively large, resulting in a large uncertainty on the f titing of damping to this short decay time. The geometric non-linearities do not have a large effect on the f lapwise bending modes since the edgewise steady-state def lection is relatively small, giving only a weak coupling from f lapwise motion to the other directions.
Figure 3(a) shows the aeroelastic frequencies and damping for the f irst edgewise blade-bending mode. There is an offset of the frequency of the two different models (HAWC2 and the present model). The reason for this offset is that the present model only includes the blade whereas the HAWC2 model includes the whole turbine. The turbine’s effect on the blade dynamics is minimized by making all other turbine components very stiff in the HAWC2 computations, but nonetheless there will always be a small effect. This effect is more pronounced for the edgewise mode since the coupling is more direct through the drive train and the other blades than it is for the f lapwise mode. The change in frequency caused by the blade def lection is also seen to have a minor difference in offset for the two models. This is due to the fact that in the
Figure 2. Aeroelastic frequency and damping for the (a) f irst and (b) second f lapwise blade-bending modes. There are no HAWC2 results for f irst f lapwise mode because it is too highly damped for measuring the decay.
Figure 3. Aeroelastic frequency and damping ratio for (a) the f irst and (b) the second edgewise blade-bending modes. Damping ratio refers to the exponential damping rate.
HAWC2 model the aerodynamic forces are applied to the deformed blade position even if the blade is assumed linear whereas in the present model the forces are applied to the undef lected blade position. Regardless of these differences, the damping of the two models is qualitatively similar, and since the focus of this work is the qualitative effect of geometric couplings on the blade stability, the present model is well suited for this purpose. The aeroelastic damping around $14\;\mathrm{m}\;\mathrm{s}^{-1}$ decreases when the geometric non-linear couplings are included (def lected blade case). At the higher wind speeds, the damping of the model including the geometric non-linearities increase and becomes the highest. The reason for these differences will be analysed in the next section. Figure 3(b) shows the aeroelastic frequency and damping of the second edgewise bending mode. The frequency and damping from the present model differ from the results from HAWC2 at a wind speed around 11 m s−1, where the f lapwise def lection is largest. This case will be analysed in the next section.
# 4.4. Aeroelastic analysis of specif ic cases
The aeroelastic damping of the edgewise mode is a caused by both f lapwise and edgewise aerodynamic force variations, which results from angle off attack variations due to edge-torsion coupling of the f lapwise def lected blade and from f lap and edgewise blade motion. On the one hand, modal aerodynamic force variation that occurs in counter phase with the blade speed enhances the damping. On the other, when it is in-phase with speed, the damping decreases or even becomes negative. When modal aerodynamic force variations are in counter phase with the blade def lection, aerodynamic stiffening occurs and vice versa. The following discussion is clarif ied through phase-space plots of f lapwise and edgewise def lections; these phase-space plots also include distinct values of the belonging aerodynamic f lapping force variation through a scaled stem-like plot (vertical bars with an o-mark; sign from up/down orientation relative to trajectory). Furthermore, the elastic twist of the blade is included in a distinct number of points of the trajectory in the phase-space plot through a straight, mainly horizontally directed bar. The torsion will increase the angle of attack when the bar is decreasing from left to right and vice versa. The plots are included to clarify the aeroelastic damping mechanisms and to illustrate the difference for an undef lected and a def lected blade. The three cases where there are large differences between the def lected and undef lected blade cases are analysed in detail; the f irst edgewise bending mode at 14 and $25\ \mathrm{m\s^{-1}}$ and the second edgewise bending mode at $11~\mathrm{m~s}^{-1}$ . Summary: In the f irst cases (f irst edgewise bending mode at $14~\mathrm{m~s}^{-1}$ ), the damping of the def lected blade is lower than the damping of the undef lected blade. The damping decreases because the inclusion of geometric non-linearities reduces the f lapwise motion and the phase between f lapwise motion and f lapwise forces is changed. In the second case (f irst edgewise bending mode at $14~\mathrm{m~s}^{-1}$ ), the damping of the def lected blade is highest. The increased damping is due to the fact that the geometric non-linearities increase the torsional motion, and thereby the changes in angle of attack and thus the aerodynamic forces. The change in phase and amplitude of the aerodynamic forces relative to the edgewise motion increase the negative aerodynamic work, increasing damping. In the last case (second edgewise bending mode at $11~\mathrm{m~s^{-1}}$ ) relative large increase in damping is seen when the def lections are included. The increase is caused by an increased amount of torsional motion and negative aerodynamic work on the torsional motion.
Figure 4. Traces of cross-sectional blade motion at $90\%$ radius in the f irst structural edgewise eigenmode at $14\;\mathrm{m}\;\mathrm{s}^{-1}$ for the blade with exaggeration of the torsional component. Arrows denote the direction of motion and the bars denote the torsional component. (a) Steady-state blade def lection terms are excluded and (b) steady-state blade def lection terms are included. Note that the relative wind comes from right to left in the displayed cross-sectional coordinate system.
The f irst case to be analysed is the aeroelastic response of the f irst edgewise blade-bending mode at $14~\mathrm{m~s^{-1}}$ , where the def lected blade cases are less damped than the undef lected blade case (Figure 3(a)). First, looking at the case without steady-state blade def lections, which for this blade without pre-bend and sweep will mean a straight blade removing geometric non-linearities: Figure 4 shows the normalized cross-sectional blade def lection at $90\%$ radius for the f irst edgewise bending structural eigenmodes for the undef lected blade and the steady-state def lected blade at $14~\mathrm{m~s}^{-1}$ . When the blade moves forward (left to right) in the structural eigenmode, the local wind speed increases, consequently increasing the aerodynamic forces, and vice versa when the blade moves backwards. The extremes of this variation of aerodynamic forces appear at the points with the largest blade speed, i.e. the midpoint of the edgewise blade motion. The f lapwise motion in the structural eigenmode also affects the aerodynamic force, increasing the angle of attack when the blade moves downwards and thereby increasing the aerodynamic force. Since the edgewise and f lapwise motion are in counter phase (blade moves forward and downwards) in this structural eigenmode, both effects described above give the highest aerodynamic forces when the blade moves forward and lowest when the blade moves backwards. In this case, without steady-state deformations, there is only a very limited and insignif icant torsional motion. The variations in aerodynamic f lapwise forces affect the f lapwise motion in the aeroelastic mode of motion. The frequency of the f irst edgewise mode $(1.1\ \mathrm{Hz})$ is higher than the resonance frequency of f irst f lapwise bending mode $(0.79\ \mathrm{Hz})$ . Hence, the f lapwise def lection lags approximately 180 degrees after the f lapwise force according to basic dynamic considerations. The f lapwise force is highest at the midpoint of the forward edgewise motion, increasing the f lapwise def lection around the midpoint of the backward edgewise motion. This increased f lapwise motion at the midpoint of the edgewise motion will increase the f lapwise speed at the edgewise turning points, affecting the angle of attack and thereby the aerodynamic force. The increased f lapwise forces will increase the f lapwise def lection ${\approx}180$ degrees later, which is the other edgewise turning point. Summing up, the f lapwise motion in the f irst edgewise aeroelastic bending mode is an equilibrium between the f lapwise motion caused by the structural coupling (eigenmode motion) and the variations in f lapwise aerodynamic force caused by the structural eigenmode and the f lapwise motion itself. Figure 5(b) shows the unsteady aerodynamic f lapwise force for the cross-sectional motion of f irst edgewise aeroelastic bending mode. The resulting aerodynamic f lapwise force variation is seen to be largest around the edgewise turning points, indicating that it is dominated by the force variation caused by the f lapwise motion itself. The black dot denotes the point with the largest f lapwise force.
Figure 6(b) shows the change in cross-sectional motion caused by the aerodynamic forces. It is seen that the largest f lapwise def lection caused by the aerodynamic forces is ${\approx}180$ degree offset from the largest f lapwise force.
When the steady-state def lections are included in the model, the geometric non-linear couplings between edgewise and torsional motion of a f lapwise def lected blade (equations (8) and (10)) become active and increase the torsional motion in the f irst f lapwise structural eigenmode (Figure 4(a)). The torsional motion is seen to decrease the angle of attack, and thereby the aerodynamic force, at the forward position of the edgewise motion so this torsional motion counteracts the angle of attack changes caused by the f lapwise speed at the edgewise turning points. The reduced effect of the f lapwise motion on the aerodynamic forces changes the phase between f lapwise and edgewise motion. The f lapwise motion relative to the local wind becomes smaller but only looking at the change in f lapwise motion caused by aerodynamic forces (Figure 6) the def lections are similar, so it is mainly the phase between f lapwise and edgewise motion that has changed.
Figure 5. Traces of cross-sectional blade motion at $90\%$ radius in the f irst aeroelastic edgewise mode at $14~\mathsf{m}~\mathsf{S}^{-1}$ for the blade with exaggeration of the torsional component. Arrows denote the direction of motion, the bars denote the torsional component and the vertical lines the unsteady f lapwise aerodynamic force with the black dot at the point with highest force. (a) Steady-state blade def lection terms are excluded and (b) steady-state blade def lection terms are included. Note that the relative wind comes from right to left in the displayed cross-sectional coordinate system.
Figure 6. Change in cross-sectional blade motion at $90\%$ radius of the f irst aeroelastic edgewise mode at $14\;\mathrm{m}\;\mathrm{s}^{-1}$ caused by aerodynamic forces. The f igure shows the difference between the structural eigenmode (Figure 4) and the aeroelastic mode (Figure 5) showing that the maximum f lapwise def lection caused by aerodynamic forces are 90 degrees phase shifted from the maximum force. Arrows denote the direction of motion and the vertical lines the unsteady f lapwise aerodynamic force with the black dot at the point with highest force. (a) Steady-state blade def lection terms are excluded and (b) steady-state blade def lection terms are included. Note that the relative wind comes from right to left in the displayed cross-sectional coordinate system.
Table II shows the aerodynamic sectional work for the two cases in Figure 5. Both the f lapwise and edgewise aerodynamic works are seen to be negative, thus extracting energy from the motion (adding damping to the mode). For the undef lected blade case, the total work is dominated by the f lapwise work. The relatively high f lapwise work is due to the fact that the f lapwise force is ${\approx}90\$ degrees phase shifted from the f lapwise motion, so for this reason the largest forces counteract at the highest velocities. The f lapwise force is mainly caused by the f lapwise component of the lift force on the airfoil. This lift force will also have an edgewise component pointing forward (the component driving the wind turbine) so the point with the highest f lapwise force also has a relatively large edgewise force component pointing forward. For the undef lected blade case (Figure 5(b)), the blade moves forward at the point with the highest forces. Consequently at this point, the edgewise component of the lift will add energy to the system, reducing the damping. This is the reason for the low damping value for the edgewise motion of the undef lected blade (Table II). In the def lected blade case, two effects reduce the f lapwise damping: f irst, the reduced f lapwise motion relative to the local wind, reduces the amount of work. Second, the f lapwise force and motion are almost in counter phase, so the maximum forces act at a low f lapwise velocity, extracting less energy from the system. The edgewise work is increased since the point of maximum force is moved towards the edgewise turning point compared with the undef lected blade case, which reduces the amount of energy that the lift force component on the edgewise motion adds to the system, leading to a higher edgewise damping contribution (Table II).
Table II. Aerodynamic sectional work for the sectional motion shown in Figure 5. The work is normalized with respect to the total work of the undef lected blade, except for the sign.
Figure 7. Traces of cross-sectional blade motion at $90\%$ radius in the f irst structural edgewise eigenmode at $25~\mathsf{m}~\mathsf{s}^{-1}$ for the blade with exaggeration of the torsional component. Arrows denote the direction of motion and the bars denote the torsional component. (a) Steady-state blade def lection terms are excluded and (b) steady-state blade def lection terms are included. Note that the relative wind comes from right to left in the displayed cross-sectional coordinate system.
The next case to be analysed is the f irst edgewise blade-bending mode at $25\ \mathrm{m\s^{-1}}$ where the damping of the def lected blade is higher than the damping of the undef lected blade case (Figure 3(a)). At this higher wind speed, the f lapwise tip def lection shifts sign (Figure 1) changing the sign of the coupling between edgewise and torsional motion for the f lapwise def lected blade (equations (8) and (10)). Figure 7 shows how the torsional def lection in the f irst edgewise structural eigenmode at $25\mathrm{~m~s~}^{-1}$ has changed sign compared with the results for $14~\mathrm{m~s^{-1}}$ (Figure 4). Figure 8 shows the crosssectional def lection of the f irst edgewise aeroelastic mode and the unsteady aerodynamic f lapwise forces at $25\mathrm{~m~s~}^{-1}$ . At this wind speed, the average angle of attack at the shown cross-section is ${\approx}{-4}$ degrees. At this negative angle of attack, the lift force is negative, so the effect of edgewise vibration change, since the forward motion, which gives larger local wind speed, increases the absolute value of the negative lift force. Hence, the forward motion decreases the lift and the backward motion increases the lift, opposite the case at $14~\mathrm{m~s^{-1}}$ . The effect of f lapwise motion is the same as before since this affects the angle of attack. So the two effects counteract each other, resulting in smaller unsteady aerodynamic forces in this mode at $25\ \mathrm{m\s^{-1}}$ than at $14~\mathrm{m~s}^{-1}$ (Figure 8). The phase between the f lapwise and edgewise motion determines how well the forces from the two effects cancel each other out and thereby also where the highest force appears. Because of the reduced aerodynamic forces, the aeroelastic mode is less affected by the aerodynamic forces and the direction of motion is similar to the structural eigenmode when compared with the previous case at $14~\mathrm{m~s}^{-1}$ . The edgewise force is mainly caused by the lift force on the blade, and since the angle of attack in this $25\ \mathrm{m}\ \mathrm{s}^{-1}$ case is negative $\approx\!-4$ degrees), a lift force giving a positive f lapwise force will give a negative edgewise force. Thus, for the f irst ${\approx}2/3$ for the forward and backward edgewise motion, the aerodynamics will contribute with negative work (Figure 8(b)). For the f lapwise motion, the f lapwise force is almost constantly in the opposite direction than the f lapwise motion, extracting energy from the motion. Table III shows that the f lapwise and edgewise works contribute equally to the damping of the undef lected blade case at $25\ \mathrm{m\s^{-1}}$ . The changes in blade twist, and thereby angle of attack, caused by the geometric nonlinearities increase the aerodynamic force at the forward edgewise position of the blade and decreases the forces at the backward position. Adding this extra effect to the effects of f lapwise and edgewise motion moves the point of highest f lapwise force towards the forward position and places it almost at the midpoint for both the f lapwise and edgewise motion. Having the highest f lapwise force (indicating high negative edgewise force at this negative angle of attack) close to the highest f lapwise and edgewise speed, results in high damping even though the force level is relatively low.
Figure 8. Traces of cross-sectional blade motion at $90\%$ radius in the f irst aeroelastic edgewise mode at $25~\mathsf{m}~\mathsf{s}^{-1}$ for the blade with exaggeration of the torsional component. Arrows denote the direction of motion, the bars denote the torsional component and the vertical lines the unsteady f lapwise aerodynamic force with the black dot at the point with highest force. (a) Steady-state blade def lection terms are excluded and (b) steady-state blade def lection terms are included. Note that the relative wind comes from right to left in the displayed cross-sectional coordinate system.
Table III. Aerodynamic sectional work for the sectional motion shown on Figure 8. The work is normalized with respect to the total work of the undef lected blade, except for the sign.
The last case to be analysed is the second edgewise blade-bending mode at $11~\mathrm{m~s}^{-1}$ , where the damping of the def lected blade case is much higher than the damping of the undef lected blade case (Figure 3(b)). On a pitch-regulated wind turbine, as the present one, the f lapwise tip def lection is largest around rated wind speed since the pitch regulation of the turbine relieves the aerodynamic loads at higher wind speeds. The large f lapwise steady-state def lection (indicating large curvature $\nu_{0}^{\prime\prime}\propto\nu_{0})$ together with the relatively large edgewise curvature $u_{1}^{\prime\prime}$ gives a large torsional component in the second edgewise bending mode (equation (10)). Figure 9 shows the content of f lapwise, edgewise and torsional motion in the second edgewise bending mode and it is seen how the inclusion of the non-linearities increases the torsional motion. Figure 10 shows the distribution of aerodynamic work done by the edgewise, f lapwise and torsional aerodynamic forces along the blade. It is on the outer $10\%$ of the blade, beyond the node of the second bending mode, that the majority of the aerodynamic work is done and the difference between the two blade def lection cases arises. The main differences in aerodynamic work between undef lected and def lected blade cases are in the torsional motions, which increase when the geometric non-linearities are included. Figure 11 shows the cross-sectional motion for the second aeroelastic edgewise bending mode for the undef lected and the def lected blade at $95\%$ blade radius. The modal aeroelastic cross-sectional motion of the undef lected blade is very similar to the structural eigenmode: this is because the unsteady aerodynamic forces are smaller relative to the higher inertia and structural restoring forces in this higher bending mode compared with the f irst edgewise mode. Figure 10 shows that the edgewise motion is slightly negatively damped for the outer part of the blade. This is because the edgewise component of the unsteady lift force acts in the direction of edgewise motion adding energy to the system. This results in minor negative damping because the drag force on the edgewise motion always adds damping. The f lapwise motion is positively damped since the unsteady f lapwise force works against the direction of f lapwise motion.
Figure 9. Edgewise, f lapwise and torsional components of the second edgewise aeroelastic bending mode at $11\ m\ s^{-1}$ for the undef lected and the def lected blade.
Figure 10. Edgewise, f lapwise and torsional cross-sectional work in the second edgewise aeroelastic vibration mode at $11\ m\ s^{-1}$ for the undef lected and the def lected blade. Negative aerodynamic work corresponds to positive aeroelastic damping.
When the steady-state def lections are included, the large torsional component caused by the geometric non-linearities (equation (10)) has a large effect on the unsteady aerodynamic forces. Note that the direction of the loop has changed compared with the undef lected blade case. The edgewise force adds energy to the system, since the force acts in the same direction and the motion for the f irst ${\approx}2/3$ of the edgewise motion. The f lapwise forces in the def lected blade case add energy to the system (Figure 10) since they act in the same direction as the f lapwise motion. The amount of work is relatively small because the f lapwise amplitude normal to the local wind direction is relatively small. The large increase in aeroelastic damping of the def lected blade case compared with the undef lected blade case is caused by negative aerodynamic work of the torsional motion (Figure 10). The aerodynamic lift force acts at the aerodynamic centre, which is located in front of the elastic centre, where the blade twists. Thus, an increased lift results in an increased rotational moment on the cross-section. The cross-sectional motion of the undef lected blade (Figure 11(b)) has almost no torsional motion, resulting in small aerodynamic work (Figure 10). The def lected blade case, on the other hand, has much more torsional motion (Figure 11(b)). The cross-section has a nose down motion on its way forward to the lift force and thereby also the torsional moment is high and a nose up motion on its way back where the lift is low, resulting in negative aerodynamic work, increasing the damping.
Figure 11. Traces of cross-sectional blade motion at $95\%$ radius in the second aeroelastic edgewise mode at $11m\ s^{-1}$ for the blade with exaggeration of the torsional component. Arrows denote the direction of motion, the bars denote the torsional component and the vertical lines the unsteady f lapwise aerodynamic force with the black dot at the point with highest force. The dotted line shows the structural eigenmode. (a) Steady-state blade def lection is excluded and (b) steady-state blade def lection is included. Note that the relative wind comes from right to left in the displayed cross-sectional coordinate system.
# 5. CONCLUSION
In this paper, a second-order non-linear beam model is used for aeroelastic stability analysis of a turbine blade. The aeroelastic mechanisms of the different modes and the difference between including and excluding non-linear geometric couplings caused by steady-state def lection at normal operation are discussed in detail. The methodology can also be used to analyse the effects of pre-bend or swept blades.
The analysis is based on the non-linear structural blade model from Kallesøe,23 which in this work is extended to include an aerodynamic model. The resulting non-linear aeroelastic blade model is linearized about a curved blade position, caused by e.g. sweep, pre-bend or steady-state def lections. The linearized model is used to perform stability analysis of a steady-state def lected blade and to examine the effects of the linearized geometric non-linearities.
First, the derived non-linear aeroelastic model is used to compute steady-state blade def lections. The steady-state def lections are validated against results from a non-linear aeroelastic time simulation code, showing good agreement. Next, the non-linear aeroelastic model is linearized about the steady-state def lected blade. By linearizing about the def lected blade, the main effects of geometric non-linearities are preserved and the results show how the relative large f lapwise blade def lection introduces a coupling between edgewise and torsional blade motion.
Two versions of the linearized model are used to compute the aeroelastic stability of the blade: one linearized about the def lected blade, preserving the non-linearities and one linearized about an undef lected blade excluding the nonlinearities. The stability results from the two versions are compared and the differences discussed. It is found that the f lapwise modes are not as affected by the steady-state blade def lection as the edgewise modes. The damping of f irst edgewise bending mode of the steady-state def lected blade decreases around $14~\mathrm{m~s}^{-1}$ but increases around $25\ \mathrm{m\s^{-1}}$ compared with the undef lected blade. The reason for this change of the effect of the blade def lection on the aeroelastic damping is caused by the steady-state f lapwise def lection shifting sign around $20\ \mathrm{m}\ \mathrm{s}^{-1}$ . When the f lapwise def lection shifts sign, the coupling between the edgewise and torsional motion also shifts, and thereby changing the non-linear geometric couplings effect on the aeroelastic damping contribution. The damping of second edgewise bending mode is high around $11~\mathrm{m~s}^{-1}$ for the steady-state def lected blade compared with the undef lected blade. This is because the f lapwise steady-state def lection is largest around $11~\mathrm{m~s^{-1}}$ giving the largest effect of the geometric non-linear coupling between edgewise and torsional motion.
This work shows that the blade def lection under normal operation conditions affects the aeroelastic stability properties of the blades. In the worst case for this particular blade, the edgewise damping can be decreased by half.
# REFERENCES
1. Hodges DH, Dowell EH. Nonlinear equations of motion for the elastic bending and torsion of twisted nonuniform rotor blades. Technical Report TN D-7818, NASA, 1974.
2. Friedmann PP, Kottapalli SBR. Coupled f lap-lag-torsional dynamics of hingeless rotor blades in forward f light. Journal of the American Helicopter Society 1982; 4: 28–36.
3. Panda B, Chopra I. Flap-lag-torsion in forward f ilght helicopter. Journal of the American Helicopter Society 1985; 30: 30–39.
4. Crespo da Silva MRM, Hodges DH. Nonlinear f lexure and torsion of rotating beams, with application to helicopter rotor blades—II. Response and stability results. Vertica 1986; 10: 171–186.
5. Hodges DH, Saberi H, Ormiston RA. Development of nonlinear beam elements for rotorcraft comprehensive analyses. Journal of the American Helicopter Society 2007; 52: 36–48.
6. Friedmann PP. Aeroelastic stability and response analysis of large horizontal-axis wind turbines. Journal of Wind Engineering and Industrial Aerodynamics 1980; 5: 373–401.
7. Chaviaropoulos PK. Flap/lead-lag aeroelastic stability of wind turbine blades. Wind Energy—Bognor Regis 2001; 4: 183–200.
8. Politis E, Riziotis V. The importance of nonlinear effects identif ied by aerodynamic and aero-elastic simulations on the 5mw reference wind turbine. Technical Report, Center for Renewable Energy Sources, Pikermi, Greece, 2007.
9. Kallesøe BS, Hansen MH. Effects of large bending def lections on blade f lutter limits. Technical Report, Risøe DTU, Roskilde, Denmark, 2008.
10. Riziotis VA, Voutsinas SG, Politis ES, Chaviaropoulos PK, Hansen AM, Madsen HA, Rasmussen F. Identif ication of structural non-linearities due to large def lections on a 5mw wind turbine blade. Proceedings of the EWEC ’08, Scientif ic Track, Brussels, 2008.
11. van Engelen TG. Control design based on aero-hydro-servo-elastic linear models from turbu (ecn). Proceedings of European Wind Conference, Milan, 2007.
12. Riziotis VA, Politis ES, Voutsinas SG, Chaviaropoulos PK. Stability analysis of pitch-regulated, variable-speed wind turbines in closed loop operation using a linear eigenvalue approach. Wind Energy 2008; 11: 517–535.
13. Ashwill TD, Kanaby G, Jackson K, Zuteck M. Development of the swept twist adaptive rotor (star) blade. Proceedings of the 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, Orlando, Florida, 2010.
14. Larsen TJ, Hansen A, Buhl T. Aeroelastic effects of large blade def lections for wind turbines. Proceedings of the Special Topic Conference ‘The Science of making Torque from Wind’, Copenhagen, Denmark, 2004; 238–246.
15. Larsen TJ, Madsen HA, Hansen MA, Thomsen K. Investigations of stability effects of an offshore wind turbine using the new aeroelastic code hawc2. Proceedings of the Conference ‘Copenhagen Offshore Wind 2005’, Copenhagen, Denmark, 2005.
16. NWTC Design Codes (ADAMS2AD by LainoDJ, Jonkman J). [Online]. Available: http://wind.nrel.gov/designcodes/ simulators/adams2ad/ (Accessed 12 August 2005).
17. NWTC Design Codes (AeroDyn by Laino DJ). [Online]. Available: http://wind.nrel.gov/designcodes/simulators/ aerodyn/ (Accessed 5 July 2005).
18. NWTC Design Codes (FAST by Jonkman J). [Online]. Available: http://wind.nrel.gov/designcodes/simulators/fast/ (Accessed 12 August 2005).
19. Hansen MH. Improved modal dynamics of wind turbines to avoid stall-induced vibrations. Wind Energy 2003; 6: 179–195.
20. Riziotis VA, Voutsinas SG, Politis ES, Chaviaropoulos PK. Aeroelastic stability of wind turbines: the problem, the methods and the issues. Wind Energy 2004; 7: 373–392.
21. Hansen MH. Aeroelastic stability analysis of wind turbines using an eigenvalue approach. Wind Energy 2004; 7: 133–143.
22. Hansen MH. Aeroelastic instability problems for wind turbines. Wind Energy 2007; 10: 551–577.
23. Kallesøe BS. Equations of motion for a rotor blade, including gravity, pitch action and rotor speed variations. Wind Energy 2007; 10: 209–230.
25. Hansen MH, Gaunaa M, Madsen HA. A Beddoes-Leishman type dynamic stall model in state-space and indicial formulation. Technical Report Risø-R-1354(EN), Risø National Laboratory, Roskilde, 2004. [Online]. Available: http://www.risoe.dk (Accessed August 2004).
26. Hansen MOL. Aerodynamics of Wind Turbines. James & James: Earthscan, London, 2001.
27. Leishman JG, Beddoes TS. A semi-empirical model for dynamic stall. Journal of the American Helicopter Society 1989; 34: 3–17.
28. Thoedorsen T. General theory of aerodynamic instability and the mechanism of f lutter. NACA Report 496, 1935.
29. Thomsen JJ. Vibrations and Stability: Advanced Theory, Analysis, and Tools. Springer-Verlag: Berlin—Heidelberg— New York, 2003.
