305 lines
81 KiB
Markdown
305 lines
81 KiB
Markdown
RESEARCH ARTICLE
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# Effect of steady def lections on the aeroelastic stability of a turbine blade
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B. S. Kallesøe
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Wind Energy Department, Risø-DTU, Technical University of Denmark, DK-4000 Roskilde, Denmark
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# ABSTRACT
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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.
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本文研究了几何非线性对稳态变桨叶片气弹振稳性的影响。如今，风电机组叶片是长而细的结构，可能存在相当大的稳态变形，这会影响叶片的动力学行为。挥舞方向的叶片变形会导致摆振方向的叶片运动与扭转叶片运动耦合，进而通过攻角影响气动特性。分析表明，对于特定叶片的最坏情况下，摆振阻尼可能会减小一半。
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Copyright $\circled{\mathrm{C}}\ 2010$ John Wiley & Sons, Ltd.
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# KEYWORDS
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stability analysis; aeroelasticity
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# Correspondence
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B. S. Kallesøe, Wind Energy Division, Risø DTU, Frederiksborgvej 399, P.O. Box 49, DK-4000 Roskilde, Denmark. E-mail: bska@risoe.dtu.dk
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Received 5 October 2009; Revised 17 May 2010; Accepted; 31 May 2010
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# 1. INTRODUCTION
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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.
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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.
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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.
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为了风电机组叶片的气动弹性稳定性分析，采用二阶非线性梁模型。考虑在稳定性分析中包含非线性几何耦合效应的重要性，并详细描述驱动空载气动响应的气动弹性机制。
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在直升机领域，人们已经研究了几十年关于弯曲旋转叶片中非线性几何耦合对稳定性的影响<sup>1–4</sup>，如今最先进的直升机稳定性综合计算代码，如 Hodges 等人<sup>5</sup>，都包括了材料和几何非线性。然而，大多数风电机组的气动弹性稳定性工具仍然基于线性梁理论，并未包含由例如稳态叶片变形、预弯或掠角等引起的非线性几何耦合。
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在20世纪70年代末，石油危机刺激了许多兆瓦级风力发电机项目。在对气动弹性稳定性研究的回顾中，Friedmann<sup>6</sup> 结论是：“可靠的气动弹性稳定性分析应基于能够考虑中等幅度的变形（即有限斜率）和非线性气动效应（如失速）的非线性公式。” 然而，这些兆瓦级风力发电机项目最终都未能获得商业成功。 之后，风力发电机的发展始于小型30 kW机组，逐渐发展到如今的兆瓦级商业风电机组。在此期间，风力发电机结构相对刚性，几何耦合效应有限。 Chaviaropoulos<sup>7</sup> 考察了非线性效应对19 m叶片气动弹性稳定性的影响。研究发现，最重要的是包含非稳态气动效应，而结构变形的影响不重要。 现代风力发电机叶片更长（高达60 m），更细长，从而在正常运行期间增加了叶片变形，重新引入了关于几何耦合的稳定性问题。 稳态叶片变形会导致不同叶片模态之间的几何非线性耦合。 例如，较大的挥舞叶片变形会增强摆振和扭角叶片运动之间的耦合，从而通过迎角影响气动特性。 因此，在例如摆振和扭角运动之间包含非线性几何耦合，对于挥舞变形的叶片来说可能很重要。
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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.
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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.
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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.
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正在进行关于利用翼梢和预弯曲叶片的研发。欧盟资助的UPWIND项目<sup>8-10</sup>，涉及叶片的非线性建模以及包含此类非线性的影响等问题。一些先进的稳定性代码，例如TURBU<sup>11</sup>，考虑了几何非线性效应。Riziotis等<sup>12</sup>在闭环操作的涡轮机稳定性分析中，将这些效应纳入考虑。此外，还着重利用几何耦合来降低疲劳和/或极限载荷，例如Ashwill等<sup>13</sup>，他们通过翼梢弯曲来引入翼叶摆振-扭转耦合。
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风电机组稳定性可以通过多种不同类型的模型进行分析。对涡轮机响应最详细的描述是由数值非线性时间模拟工具提供的<sup>14–18</sup>。这些工具显示了不稳定性以及非线性效应，限制了响应，例如极限周期振荡。它们还可以用于分析例如湍流和风切对涡轮机稳定性的影响。这些参考工具使用不同的模型和不同的模型复杂度。例如，FAST<sup>18</sup>是一种基于模态的代码，一方面不包括叶片的扭转自由度以及几何非线性耦合，但另一方面计算成本相对较低。像HAWC2<sup>14,15</sup>的代码具有更复杂的模型，该模型基于多体公式，其中每个体是包含扭转的Timoshenko梁单元。这些时间模拟工具的缺点是它们计算量大，并且难以从中提取重要的气动弹性机制，从而获得大量结果。**另一种方法是使用线性（或线性化）涡轮机模型进行特征值分析<sup>11,12,19–21</sup>**。HAWCStab代码<sup>19,21</sup>提供了一个平台，用于线性化未变形的涡轮机结构，而TURBU代码<sup>11</sup>提供了一个平台，用于基于线性化于变形/弯曲叶片状态进行气动弹性稳定性分析。TURBU中的结构模型基于简单的共旋转梁单元方法。每个梁单元由一个刚体和在其入口点上的弹簧和阻尼器组成；弹簧的平均应力和梁单元之间的无扭转旋转偏移量体现了平均变形/弯曲叶片状态。Riziotis等<sup>12</sup>提供了一个多体平台，通过时间积分找到参考状态，并在线性化气动弹性方程周围找到该参考状态，以提供一个包括闭环控制的稳定性工具。这类工具可以给出结构固有频率和描述涡轮机基本结构动力学的固有模态，以及气动弹性运动的频率、阻尼和模态。气动弹性阻尼揭示了涡轮机的任何稳定性问题。然而，由于它是线性工具，因此无法提供任何关于限制线性负阻尼模态幅度的非线性机制的信息。结构和气动弹性频率和模态形状的知识对于气动弹性时间模拟的分析和结果解释非常有用。然而，整个涡轮机的气动弹性响应的模态仍然难以分析。为了减少复杂性，从而使结果更清晰，可以使用仅叶片分析<sup>22</sup>。这允许对物理进行清晰的解释和对控制叶片动态响应的机制的见解，并且许多涡轮机基本稳定性的特征可以从仅叶片分析中提取。
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本文使用了一种非线性叶片模型<sup>23</sup>，该模型考虑了大角度叶片变形、变桨角度和转子速度变化的影响。该叶片模型深受Hodges和Dowell<sup>1</sup>的工作启发。首先，**将结构模型与基于梁单元动量（BEM）理论的稳态气动模型相结合，并通过有限差分方案离散化。将由此产生的代数非线性气动弹性模型用于计算在额定功率生产条件下，由美国国家可再生能源实验室（NREL）的5 MW参考风电机组（RWT）<sup>24</sup>的稳态叶片变形和诱导速度**。将稳态变形与HAWC2模拟结果进行比较，结果吻合良好。在本文中，使用NREL的5 MW RWT作为示例叶片。参考涡轮机是基于市场上先进涡轮机的虚拟涡轮机。叶片深受61.5 m LM风纤维叶片（LM Wind Power，Kolding，丹麦）的启发。该叶片属于最先进叶片设计的柔性设计的中部区域，因此其他叶片设计的几何耦合可能更明显。然而，该叶片的优点是所有数据均可公开获取，并且已被广泛用于其他研究工作，因此与大多数最先进叶片相比，它具有良好的参考和现实的柔性。然后，关于稳态变形叶片，对非线性结构叶片模型<sup>23</sup>和非稳态气动模型<sup>25</sup>进行线性化，同时保留几何非线性效应的主要影响。通过有限差分方案离散化线性模型，与边界条件一起形成微分特征值问题。对该特征值问题的解给出气动弹性频率和阻尼，同时也给出关于叶片基本气动弹性行为的信息。分析表明，当包含稳态变形时，翼叶摆振模态的气动弹性阻尼会发生变化。对三种不同的运行条件下进行详细的气动弹性运动分析，在包含或排除稳态叶片变形时，阻尼存在较大差异。
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# 2. STRUCTURAL BLADE MODEL
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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
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Kallesøe23 中描述的叶片结构模型，基于 Hodges 和 Dowell1 的工作，采用二阶 Bernoulli–Euler 梁理论，用一个非线性积分微分运动方程来描述叶片（叶片）的运动。
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$$
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\overline{\bf M}\ddot{\overline{\bf u}}+\overline{{{\bf F}}}\left(\dot{\overline{\bf u}},\overline{{{\bf u}}}^{\prime\prime},\overline{{{\bf u}}}^{\prime},\overline{{{\bf u}}},\ddot{\beta},\dot{\beta},\beta,\ddot{\phi},\dot{\phi},\phi\right)\!=\overline{{{\bf f}}}\left({\bf f}_{a e r o},M_{a e r o},u^{\prime},\nu^{\prime}\right)
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$$
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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.
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其中，$\bar{\bf M}$ 为质量矩阵，$\bar{\mathbf{F}}$ 为包含刚度、阻尼、陀螺惯性力以及基于离心力积分项的非线性函数。状态向量 $\bar{\mathbf{u}}=[u\left(t,s\right),\nu\left(t,s\right),\theta\left(t,s\right)]$ 分别表示摆振（edgewise）、挥舞（flapwise）和扭角（torsional）变形。
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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
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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：
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$$
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\begin{array}{r l}&{m\big(\ddot{u}-\ddot{\theta}l_{c g}\sin\big(\overline{{\theta}}\big)\big)+F_{u,1}\big(\ddot{\beta},\dot{\beta},\dot{\phi},\dot{\nu},\dot{\theta},u^{\prime},u,\theta,\beta\big)+F_{u,2}\big(\dot{\phi},\dot{u},\dot{\nu},u^{\prime},\nu,\theta,\beta\big)}\\ &{\quad+\,F_{u,3}(\phi,\beta,\theta,u^{\prime},\nu^{\prime})+F_{u,4}(u^{\prime\prime},\nu^{\prime\prime},\theta)+F_{u,5}\big(\ddot{\phi},\beta\big)}\\ &{\quad=f_{u}+\Big((u^{\prime}+l_{p i}^{\prime})\int_{s}^{R}f_{w}\mathrm{d}\rho\Big)^{\prime}}\end{array}
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$$
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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
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其中第一项为惯性力，第二项 $F_{u,1}$ 描述了变桨角度的作用，本研究中将不使用该项。第三项 $\boldsymbol{F}_{u,2}$ 描述了由风轮转速引起的离心力和科里奥利力。第四项 $F_{u,3}$ 描述了不稳定的重力影响，本研究中忽略该项。第五项描述了恢复力。
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$$
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\begin{array}{r l}&{F_{u,4}\!=\!\big(E\big(I_{\xi}\!\cos^{2}\!(\tilde{\theta})\!+I_{\eta}\sin^{2}\!(\tilde{\theta})\big)u^{\prime\prime}\big)^{\prime\prime}+\!\big(E(I_{\xi}\!-I_{\eta})\!\cos\!(\tilde{\theta})\!\sin\!(\tilde{\theta})\nu^{\prime\prime}\big)^{\prime\prime}}\\ &{\qquad-\big(E(I_{\xi}\!-I_{\eta})\theta\big(u^{\prime\prime}\!\sin\!\big(2\tilde{\theta}\big)\!-\nu^{\prime\prime}\!\cos\!\big(2\tilde{\theta}\big)\!+I_{p i}^{\prime\prime}\sin\!\big(\tilde{\theta}\big)\!\cos\!(\tilde{\theta})\!\big)\big)^{\prime\prime}}\end{array}
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$$
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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
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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
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其中第一项为 $x$ 方向的弯曲刚度，第二项为与 $y$ 方向的耦合，方程 (3) 中的最后一项为扭转耦合。方程 (2) 中的第六项描述了风轮转速变化的影响，本研究假设转速恒定，因此该项不活跃。右侧包含外部力，在本例中为气动力。叶片上的纵向力，例如离心力，会导致运动方程中的积分项。所有项的详细描述见 Kallesøe.23
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本文采用的边界条件为简化推导，适用于无预弯叶片。叶片根部的边界条件由几何约束给出。
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$$
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u(0,t)=u^{\prime}(0,t)=\nu(0,t)=\nu^{\prime}(0,t)=\theta(0,t)=0
|
||
$$
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||
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because the frame used to describe the blade follows the root of the blade. The boundary conditions for the tip of the blade are23
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||
因为用于描述叶片的框架遵循叶片根部。叶片尖部的边界条件是23。
|
||
|
||
|
||
|
||
$$
|
||
\begin{array}{l}{{{u^{\prime\prime}}(R,t)=\nu^{\prime\prime}(R,t)\,{=}\,\theta^{\prime}(R,t)=0}}\\ {{u^{\prime\prime\prime}(R,t)\,{=}\,\displaystyle\frac{m l_{c g}}{E I_{\xi}I_{\eta}}\,\big(\dot{\phi}^{2}w-g\cos{(\phi)}\big)\big(I_{\eta}\sin{\big(\tilde{\theta}-\overline{{\theta}}\big)}\sin{\big(\tilde{\theta}\big)}+I_{\xi}\cos{\big(\tilde{\theta}-\overline{{\theta}}\big)}\cos{\big(\tilde{\theta}\big)}\big)}}\\ {{{\nu^{\prime\prime\prime}}(R,t)\,{=}\,\displaystyle\frac{m l_{c g}}{E I_{\xi}I_{\eta}}\big(\dot{\phi}^{2}w-g\cos{(\phi)}\big)\big(I_{\eta}\cos{\big(\tilde{\theta}-\overline{{\theta}}\big)}\sin{\big(\tilde{\theta}\big)}+I_{\xi}\sin{\big(\tilde{\theta}-\overline{{\theta}}\big)}\cos{\big(\tilde{\theta}\big)}\big)}}\end{array}
|
||
$$
|
||
|
||
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.
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其中，$s=R$ 为叶片末端，$m=m(s)$ 为叶片单位长度的质量，$l_{c g}=l_{c g}(s)$ 为重心偏离弹性轴的偏移量，$E=E(s)$ 为杨氏模量，$I=I\left(s\right)$ 和 $I_{\eta}=I_{\eta}(s)$ 为惯性主矩，$w=$ $w(s,t)$ 为弹性轴上到位置 $s$ 的半径，$g$ 表示重力，$\tilde{\theta}=\tilde{\theta}\left(s\right)$ 为弦线与弹性主轴之间的夹角，且 $\tilde{\theta}=\tilde{\theta}\left(s\right)$ 为弦线与从弹性中心到重心之间的连线夹角，沿该连线测量 $l_{c g}$。当 $l_{c g}(R)\neq0$ 时，叶片末端的边界条件是风轮转速 $\dot{\phi}$ 和方位角 $\phi$ 的函数，因此随时间变化。这是因为叶片末端重心偏离弹性轴会导致重力和离心力引起的弯矩。大多数现代风电机组叶片在末端呈锥形，其中 $l_{c g}(s)\longrightarrow{\cal0}$ 且 $E I_{\xi}I_{\eta}\longrightarrow0$ 。因此，是否可以忽略这些方位角相关的边界条件取决于具体的叶片设计。在本工作中，叶片被构造成 $l_{c g}(R)=0$ 且 $E I_{\xi}I_{\eta}|_{s=R}\neq0$，从而使边界条件与方位角无关，并使方程 (5) 的所有右侧项变为零。
|
||
# 3. STEADY–STATE AEROELASTIC MODEL
|
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|
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To determine the steady-state def lection for the blade, a non-linear steady-state aeroelastic model i.s derived. Steady-state conditions are defined as uniform inf low, zero gravity, constant rotor speed and pitch angle $\ddot{\phi}=\dot{\beta}=0$ 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.
|
||
为了确定叶片的稳态变形，推导了一个非线性稳态气弹耦合模型。稳态条件被定义为均匀来流、零重力、恒定风轮转速和变桨角度 $\ddot{\phi}=\dot{\beta}=0$，从而使运动方程(1)中的所有时间导数为零 $\ddot{u}=\ddot{\nu}=\ddot{\theta}=\dot{u}=\dot{\nu}=\dot{\theta}=0$。这些均匀条件消除了系统的周期性。稳态气动模型基于叶片元动量（BEM）理论，包括普兰德尔的梢流损失修正。BEM理论计算叶片上的力和风的动量变化之间的平衡。气动模型通过局部风速和迎角与结构模型耦合，结构模型通过作用在叶片上的气动力与气动模型耦合。
|
||
# 3.1. Discretization of structural model
|
||
|
||
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:
|
||
|
||
运动方程（方程（1））在弹性轴方向上以步长 $h$ 和 $N$ 个计算点进行离散化。偏微分运动方程（1）的空间导数采用表I中给定的有限差分方案进行近似。参数（如质量、刚度等）的导数也采用相同的有限差分方案进行近似。运动方程中的积分项采用梯形法则进行求和近似。
|
||
|
||
有限差分公式的边界条件是通过将有限差分近似代入边界条件（方程（4）和（5））得到的。假设叶片尖端重心偏移量为零，从而使边界条件与风轮位置无关。
|
||
|
||
在 $N$ 个离散化点上实现的偏微分运动方程的离散版本形成一组非线性代数方程：
|
||
|
||
$$
|
||
\mathbf{F}_{s t}\big(\mathbf{u}_{0},\dot{\phi}_{0},\beta_{0}\big)\!=\!\mathbf{f}_{0}
|
||
$$
|
||
|
||
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.
|
||
其中 $\mathbf{F}_{s t}\;(\mathbf{u}_{0},\;\dot{\phi}_{0},\;\beta_{0})$ 包含结构方程离散化项，而 $\mathbf{u}_{0}=[u_{0,1},\,\nu_{0,1},\,\theta_{0,1},\,\dots\,,\,u_{0},$ $u_{0,N},$ $\nu_{0,N}$ , ${\theta_{0,N}}\mathrm{]}^{\mathrm{T}}$ 表示每个离散化点的稳态变形。第一个下标 0 表示它是稳态解（零阶），第二个下标表示离散化点编号，从叶片（blade）的根部开始计数。右侧项 $\mathbf{f}_{0}$ 包含使用 BEM 理论计算出的每个离散化点的稳态气动力。
|
||
# 3.2. Solution scheme
|
||
|
||
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.
|
||
有限差分离散的稳态方程（方程（6））具有 3N 个未知叶片变形（$\mathrm{N}$ 离散点的挥舞、摆振和扭转变形）和 2N 个未知诱导系数（$_\mathrm{N}$ 离散点的纵向和切向诱导系数）。该非线性方程组采用以下迭代方案求解：i) 选择工况：稳态风速 $\left(U_{0}\right)$ ，对应的风轮转速 $\dot{\phi}_{0}=\dot{\phi}_{0}(U_{0}))$ 和变桨角度 $\begin{array}{r}{\beta_{0}=\beta_{0}(U_{0}))}\end{array}$；ii) 基于无穷小条件计算视风速和迎角，并计算叶片变形和诱导系数；iii) 使用 BEM 理论计算气动力；iv) 求解方程（6）以获得变形 $\mathbf{u}_{0}$；v) 计算新的诱导系数；vi) 如果没有收敛，返回步骤 2。这给出了稳态变形 $\mathbf{u}_{0}=\mathbf{u}_{0}\left(U_{0},\,\dot{\phi}_{0},\,\beta_{0}\right)$ 和给定工况下的诱导系数。
|
||
|
||
<html><body><table><tr><td colspan="2">Tablel.Second-orderfinitedifferenceformulationforuniformstepsize. fi+(t)-f-{(t)</td></tr><tr><td rowspan="2">f’(s, t) ds</td><td>af(s, t)</td></tr><tr><td>2h</td></tr><tr><td>f"(s, t) ²f(s, t)</td><td>fi+1(t)-2f(t)+f-(t)</td></tr><tr><td rowspan="2">ds2</td><td>h2</td></tr><tr><td>-f-2(t)+2f_(t)-2fi+1(t)+fi+2(t)</td></tr><tr><td>f"(s, t) a²f(s, t) ds3</td><td>2h3</td></tr><tr><td rowspan="3">f"(s, t)</td><td>a4f(s, t)</td><td></td></tr><tr><td>ds4</td><td>f-2(t)-4f-(t)+6f;(t)-4f+1(t)+f+2(t)</td></tr><tr><td></td><td>h4</td></tr></table></body></html>
|
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|
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|
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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.
|
||
|
||
# 3.3. Steady–state blade deflection at power production conditions
|
||
|
||
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.
|
||
(6)式稳态模型被用于计算风电机组在正常功率生产运行条件下叶片的稳态变形和诱导因子。并将结果与非线性气弹振动时间模拟代码HAWC2.14,15的结果进行比较。HAWC2代码采用多体公式，其中每个体都是具有扭转自由度的线性Timoshenko梁单元。几何非线性通过多体公式捕捉，例如，每个叶片被建模为10个体。如果每个叶片仅使用一个体，则代码将变为线性代码，因为每个体的梁模型是线性的，而收敛性研究表明，使用10个体可以捕捉几何非线性。在本模型中，仅考虑一个叶片，并将其建模为柔性梁。对于本文考虑的叶片运动的第一和第二模态，旋转和剪切效应可以忽略不计，因此本模型中的Bernoulli–Euler梁模型与HAWC2中的Timoshenko梁模型相当。对于运动的更高阶模态和其他机组组件，旋转和剪切效应的相关性更高。图1显示了在不同风速下，半径为$55.5~\mathrm{m}$（占叶片长度的$88\%$）处的叶片挥舞和摆振变形以及攻角。攻角表明两个模型之间的扭角变形吻合程度。可以看出，对于所有运行条件，本模型的二阶Bernoulli–Euler叶片模型与HAWC2之间具有良好的吻合度。叶片尖端变形曲线在额定风速处$(\approx11{\mathrm{~m~s~}^{-1}})$出现的突变是由从变桨角度恒速运行模式切换到恒速变桨角度运行模式引起的。
|
||
# 4. AEROELASTIC MODES OF BLADE MOTION
|
||
|
||
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.
|
||
在本节中，将分析叶片挥舞模态的空气动力学弹性行为，特别关注稳态叶片挥舞变形的影响。将详细分析风轮在正常运行状态下的稳定性，并讨论包括和排除几何耦合时的差异。预弯的影响类似于本分析中研究的稳态叶片挥舞变形的影响。扫角（摆振弯曲叶片）的影响有所不同，因为它耦合了挥舞和扭转运动，而不是像挥舞变形所特有的摆振和扭转。
|
||
# 4.1. Linear aeroelastic model
|
||
|
||
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 $\varepsilon<<1$ . Balancing terms of order $\varepsilon^{\mathrm{l}}$ give the linear approximation around the def lected blade position $\mathbf{u}_{0}$ . By linearizing the equations of motion about the def lected blade the main effects for the geometric non-linearities are preserved. For example, the non-linear stiffness term in the edgewise equation
|
||
|
||
通过将 ${\mathbf{u}}(s,\,t)={\mathbf{u}}_{0}(s)+\varepsilon{\mathbf{u}}_{1}(s,\,t)$ 代入方程 (1) 来线性化运动的非线性偏微分方程，其中 ${\bf u}_{0}(s)$ 为包括任何预弯和展向的稳态变形叶片位置，${\mathbf{u}}_{1}(s,t)$ 为相对于该位置的时间相关变化，$\varepsilon$ 为表示项的微小程度的记账参数。外部影响，例如风速、变桨角度等，被分解为稳态部分和随时间变化的部（分别用下标 0 和 1 表示），并使用记账参数 $\varepsilon$ 。假设 $\varepsilon<<1$，对运动方程（方程 (1)）进行泰勒展开。平衡 $\varepsilon^{\mathrm{l}}$ 阶项得到关于变形叶片位置 $\mathbf{u}_{0}$ 的线性近似。通过对运动方程进行线性化，保留了几何非线性效应的主要部分。例如，摆振方程中的非线性刚度项。
|
||
|
||
$$
|
||
\left(\left(E I_{\xi}-E I_{\eta}\right)\cos\left(\tilde{\theta}+\theta\right)\sin\left(\tilde{\theta}+\theta\right)\nu^{\prime\prime}\right)^{\prime\prime}
|
||
$$
|
||
|
||
becomes
|
||
|
||
$$
|
||
\left(\left(E I_{\xi}-E I_{\eta}\right)\cos\left(2\left(\tilde{\theta}+\theta_{0}\right)\right)\nu_{0}^{\prime\prime}\theta_{1}\right)^{\prime\prime}+\dots
|
||
$$
|
||
|
||
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
|
||
|
||
$$
|
||
(E I_{\xi}-E I_{\eta})\cos(2\big(\tilde{\theta}+\theta\big)\big)u^{\prime\prime}\nu^{\prime\prime}
|
||
$$
|
||
|
||
becomes
|
||
|
||
$$
|
||
\begin{array}{r}{\big(E I_{\xi}-E I_{\eta}\big)\mathrm{cos}\big(2\big(\widetilde{\theta}+\theta_{0}\big)\big)u_{0}^{\prime\prime}u_{1}^{\prime\prime}+...\,.}\end{array}
|
||
$$
|
||
|
||
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
|
||
|
||
$$
|
||
\begin{array}{r}{\tilde{\mathbf{M}}\ddot{\mathbf{u}}+\tilde{\mathbf{D}}\dot{\mathbf{u}}+\left(\tilde{\mathbf{K}}_{s s}\mathbf{u}^{\prime\prime}\right)^{\prime\prime}+\left(\tilde{\mathbf{K}}_{s}\mathbf{u}^{\prime}\right)^{\prime}+\tilde{\mathbf{K}}\mathbf{u}+\tilde{\mathbf{C}}\mathbf{z}=\tilde{\mathbf{F}}_{s}\tilde{\mathbf{f}}\ }\\ {\dot{\mathbf{z}}+\tilde{\mathbf{T}}\mathbf{z}+\tilde{\mathbf{G}}\ddot{\mathbf{u}}+\tilde{\mathbf{H}}\dot{\mathbf{u}}+\tilde{\mathbf{J}}\mathbf{u}=\tilde{\mathbf{F}}_{a}\tilde{\mathbf{f}}}\end{array}
|
||
$$
|
||
|
||
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
|
||
|
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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
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# $\dot{\mathbf{x}}=\mathbf{A}\mathbf{x}+\mathbf{B}\mathbf{f}$
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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.
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It is noted that since aerodynamic forces are included, the eigenvalue problem12 is not self-adjoint, and therefore, the eigenvectors are not orthogonal.
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# 4.3. Frequency and damping of a blade at normal power production conditions
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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.
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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.
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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.
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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
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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.
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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.
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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.
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# 4.4. Aeroelastic analysis of specif ic cases
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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).
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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.
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<html><body><table><tr><td></td><td>Edgewise</td><td>Flapwise</td><td>Total</td></tr><tr><td>Undeflectedblade</td><td>-0.03</td><td>-0.97</td><td>-1.00</td></tr><tr><td>Deflectedblade</td><td>-0.25</td><td>-0.27</td><td>-0.52</td></tr></table></body></html>
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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.
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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.
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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.
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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.
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<html><body><table><tr><td></td><td>Edgewise</td><td>Flapwise</td><td>Total</td></tr><tr><td>Undeflectedblade</td><td>-0.56</td><td>-0.44</td><td>-1.00</td></tr><tr><td>Deflectedblade</td><td>-1.13</td><td>-0.43</td><td>-1.57</td></tr></table></body></html>
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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.
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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.
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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.
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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.
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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.
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# 5. CONCLUSION
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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.
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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.
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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.
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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.
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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.
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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.
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