# Improved Modal Dynamics of Wind Turbines to Avoid Stall-induced Vibrations M. H. Hansen∗ Wind Energy Department, Risø National Laboratory, PO Box 49, DK-4000 Roskilde, Denmark Key words: horizontal axis turbines; mechanical vibrations; modal analysis; optimization Stall-induced edgewise blade vibrations have occasionally been observed on three-bladed wind turbines over the last decade. Experiments and numerical simulations have shown that these blade vibrations are related to certain vibration modes of the turbines. A recent experiment with a 600 kW turbine has shown that a backward whirling mode associated with edgewise blade vibrations is less aerodynamically damped than the corresponding forward whirling mode.Inthisarticlethemodeshapesoftheparticularturbineareanalysed, based on a simplified turbine model described in a multi-blade formulation. It is shown that the vibrations of the blades for the backward and forward edgewise whirling modes are different, which can explain the measured difference in aerodynamic damping. The modal dynamics of the entire turbine is important for stability assessments; blade-only analysis can be misleading. In some cases the modal dynamics may even be improved to avoid stall-induced vibrations. 过去十年中，在三叶风轮上偶有观察到失速诱发的摆振叶片振动。实验和数值模拟表明，这些叶片振动与机组的某些振动模态相关。最近对一台 600 千瓦风轮的实验表明，与摆振叶片振动相关的向后旋转模态的空气动力学阻尼小于相应的向前旋转模态。本文分析了特定风轮的模态形状，基于多叶片公式描述的简化风轮模型。结果表明，向后和向前摆振叶片振动的振动情况不同，这可以解释空气动力学阻尼的测量差异。整个风轮的模态动力学对于稳定性评估至关重要；仅对叶片进行分析可能会产生误导。在某些情况下，模态动力学甚至可以得到改善，以避免失速诱发的振动。 Copyright $\circledcirc$ 2003 John Wiley & Sons, Ltd. # Introduction A theoretical modal analysis of a three-bladed wind turbine is presented, with the aim of explaining observed differences in aerodynamic damping of two apparently similar modes.1 Similar results have previously been published,2 but without the present details on the method and analysis. 一篇关于三叶风轮的理论模态分析被呈现，旨在解释两种看似相似的模态在气动阻尼上的观测差异。类似的结果已发表过，但缺乏本研究方法和分析的详细信息。 Stall-induced vibrations of wind turbine blades have become an issue in the optimized design of stallregulated wind turbines with increasing rotor diameters. Until the 1990s, no problems with such vibrations had been observed, although work performed in the 1980s indicated that these problems could arise.3When one of the first examples of stall-induced vibrations was reported on a $500~\mathrm{kW}$ wind turbine,4 it led to further investigations which have provided a detailed understanding of the problem.1,5–8 From the studies of stall-induced vibrations, different types of solutions have been developed. The common solution for a wind turbine in operation is to change the stall characteristics of the blade by placing stall strips on the leading edge of the blade.8,9 When designing a new blade, this aerodynamic issue must of course be considered in the choice of aerofoil profiles. However, the structural dynamics of the blade is also important for the risk of stall-induced vibrations. The aerodynamic damping of a blade section operating in stall is highly dependent on the direction of blade vibration (relative to the rotor plane) associated with the primary bending modes.5,7,8 For most aerofoil profiles at high angles of attack it can be shown, using quasi-steady aerodynamics (see Appendix), that vibrations in the rotor plane (edgewise) are less damped by the air flow than those out of the rotor plane (flapwise). Another important role of the structural dynamics is the structural damping of the blade. For example, if the first edgewise mode of a blade has a negative aerodynamic damping at some wind speeds, then it may be possible to avoid an instability by adding sufficient structural damping to the blade, or a distinct vibration damper.10 Thus, besides the aerodynamic issue, it is necessary to consider the mode shapes and their structural damping when designing a new blade. Another issue of stall-induced vibrations is the interaction of the blades with the remaining wind turbine. A numerical simulation of a turbine using the aeroelastic code HAWC11 showed that, by stiffening the support of the rotor (shaft and nacelle), undamped edgewise blade vibrations could be avoided.7 This observation may be related to an observation made by Thomsen et al.1 They have developed an experimental method for estimating the total damping of a wind turbine during operation. Figure 1 shows the results of their experiments with a stall-controlled Bonus $600~\mathrm{kW}$ wind turbine. The observation is that the forward rotor whirling mode associated with edgewise blade vibrations is more damped than the backward rotor whirling mode. The structural damping of these two modes is assumed to be the same, because their natural frequencies and mode shapes are almost identical; the difference in total damping is therefore a difference in aerodynamic damping. In this article it is suggested that the dependence of aerodynamic damping on whirl direction and rotor support stiffness can be explained by analysing the turbine mode shapes. It is shown how a theoretical modal analysis can be performed for a rotating turbine based on the so-called multi-blade co-ordinate transformation.12,13 For the particular $600~\mathrm{kW}$ turbine it is shown that the blades vibrate more out of the rotor plane in the forward edgewise whirling mode than in the backward edgewise whirling mode. This observation can explain the measured difference in aerodynamic damping for the two whirling modes, because the direction of blade vibration with respect to the rotor plane is important for the aerodynamic damping in stall. It is suggested that, using the modal analysis, the design of wind turbines can be tailored to increase the blade vibration out of the rotor plane for all modes, and thereby increase the aerodynamic damping of stall-regulated turbines. The next section describes the model and method for modal analysis of rotating turbines. The modal analysis of the $600~\mathrm{kW}$ turbine is then presented, followed by a section on the measured differences in aerodynamic damping and the optimization of its dynamical behaviour. 随着风轮直径不断增大，叶片失速诱导振动已成为优化失速调控风力发电机设计的难题。直到 1990 年代之前，人们并未观察到此类振动问题，尽管 1980 年代的研究表明这些问题可能会出现。3 当第一个失速诱导振动案例在 500 千瓦的风力发电机上被报告时，4 它引发了进一步的调查，这些调查提供了对该问题的详细理解。1,5–8 针对失速诱导振动，已经开发出多种解决方案。风力发电机运行时的常见解决方案是在叶片前缘放置失速条，从而改变叶片的失速特性。8,9 在设计新叶片时，必须在气翼剖面选择中考虑这一空气动力学问题。然而，叶片的结构动力学也对失速诱导振动的风险至关重要。叶片在失速状态下运行的叶片段的空气动力学阻尼高度依赖于叶片振动方向（相对于风轮平面）与主要弯曲模态之间的关系。5,7,8 对于大多数气翼剖面在较高攻角时，使用准稳态空气动力学（见附录）可以证明，风轮平面内的振动（摆振）比风轮平面外的振动（挥舞）受气流阻尼较小。结构动力学的另一个重要作用是叶片的结构阻尼。例如，如果叶片的第一个摆振模态在某些风速下具有负的空气动力学阻尼，那么可以通过向叶片添加足够的结构阻尼或一个独立的振动阻尼器来避免不稳定性。10 因此，除了空气动力学问题，在设计新叶片时，必须考虑模态形状及其结构阻尼。 失速诱导振动的另一个问题是叶片与剩余风力发电机之间的相互作用。使用气弹耦合代码 HAWC11 的涡轮数值模拟表明，通过加固风轮（轴和舱室）的支撑，可以避免未阻尼的摆振叶片振动。7 这一观察结果可能与 Thomsen 等人所做的观察相关。1 他们开发了一种实验方法来估计风力发电机在运行期间的总阻尼。图 1 显示了他们对失速调控 Bonus 600 千瓦风力发电机进行的实验结果。观察结果是，与摆振叶片振动相关的正向风轮回转模态比反向风轮回转模态更受阻尼。假设这两个模态的结构阻尼相同，因为它们的固有频率和模态形状几乎相同；因此，总阻尼的差异是空气动力学阻尼的差异。 本文建议，空气动力学阻尼对回转方向和风轮支撑刚度的依赖性可以通过分析涡轮模态形状来解释。展示了如何基于所谓的“多叶片坐标变换”12,13 对旋转涡轮进行理论模态分析。对于特定的 600 千瓦涡轮，表明在正向摆振回转模态中，叶片比在反向摆振回转模态中更倾向于风轮平面外振动。这一观察结果可以解释两种回转模态的空气动力学阻尼差异的测量结果，因为叶片相对于风轮平面的振动方向对失速状态下的空气动力学阻尼至关重要。建议使用模态分析，可以调整风力发电机设计，以增加所有模态的风轮平面外叶片振动，从而增加失速调控涡轮的空气动力学阻尼。 下一节描述了旋转风机模态分析的模型和方法。然后，对 600 千瓦涡轮进行模态分析，随后是关于测量到的空气动力学阻尼差异和其动力学行为优化的部分。 # Model and Method A linear model of a three-bladed wind turbine is needed for the modal analysis. It must qualitatively describe the dynamics of the turbine up to the second tilt/yaw modes in order to study the phenomena mentioned above. Blade torsion is not included in the analysis, because the corresponding natural frequencies are several times higher than the dynamics of interest. 为了模态分析，需要建立一个三叶风电机组的线性模型。为了研究上述现象，它必须定性描述风电机组直至第二倾覆/偏航模态的动力学。由于其固有频率远高于感兴趣的动力学，因此未在分析中包含叶片扭转。 ![](146a343862b1487203627349c8c528f86914d6bbb025069461ed2230d451d999.jpg) Figure 1. Total damping of the forward and backward edgewise whirling modes of the Bonus 600 kW estimated by Thomsen et al.1 ![](c54c4be6a0b3a29c3948930025463b595b9a6124393996423d58702f62526d82.jpg) Figure 2. Simplified structural model of a wind turbine # Equations of Motion A schema of the structural model is shown in Figure 2. The motion of the nacelle and tower is described by seven degrees of freedom (DOFs). The nacelle can translate in the two horizontal directions, described in the ground fixed frame $(X,Y,Z)$ by $u_{x}^{\mathrm{t}}$ (lateral) and $u_{y}^{\mathrm{t}}$ (longitudinal), and can tilt and yaw about the tower top, described by the angles $\theta_{x}^{\mathrm{t}}$ and $\theta_{z}^{\mathrm{t}}$ respectively. The rotor can rotate out of its plane owing to bending of the shaft, described by the angles $\theta_{x}^{\mathrm{s}}$ and $\theta_{z}^{\mathrm{s}}$ defined in the corotating frame of blade number 1. Finally, the torsion of the shaft and drive-train is described by the angle $\theta_{y}^{\mathrm{s}}$ . The motions of the blades are described by a modal expansion defined in their own co-rotating frames $(x,\,y,\,z)$ , where the $z_{i}$ -axis is the blade axis and the $y$ -axis at rest coincides with the $Y$ -axis. The in- and out-of-rotor-plane motions of blade number $i$ are given by 图2所示为结构模型的示意图。机舱和塔架的运动由七个自由度（DOF）描述。机舱可以在两个水平方向上平移，分别在地面固定坐标系 $(X,Y,Z)$ 中描述为 $u_{x}^{\mathrm{t}}$（X）和 $u_{y}^{\mathrm{t}}$（Y），并且可以绕塔顶倾覆和偏航，分别由角度 $\theta_{x}^{\mathrm{t}}$ 和 $\theta_{z}^{\mathrm{t}}$ 描述。风轮由于轴的弯曲而可以偏离其平面，分别由角度 $\theta_{x}^{\mathrm{s}}$ 和 $\theta_{z}^{\mathrm{s}}$ 在叶片编号1的共转坐标系中定义。最后，轴和驱动系的扭转由角度 $\theta_{y}^{\mathrm{s}}$ 描述。 叶片的运动由在其自身共转坐标系 $(x,\,y,\,z)$ 中定义的模态展开来描述，其中 $z_{i}$ 轴为叶片轴，静止时的 $y$ 轴与 $Y$ 轴重合。叶片编号 $i$ 的进出风轮平面运动由给出。 $$ u_{x}^{\mathrm{B}_{i}}(t,z)=\sum_{n=1}^{N}q_{i,n}(t)\Phi_{n}^{x}(z),\qquad u_{y}^{\mathrm{B}_{i}}(t,z)=\sum_{n=1}^{N}q_{i,n}(t)\Phi_{n}^{y}(z) $$ where $q_{i,n}$ are modal blade $c o$ -ordinates describing the contents of mode number $n$ in the motion of blade number $i$ , and $\Phi_{n}^{x}$ and $\Phi_{n}^{y}$ are the mode shape functions. These shape functions can be obtained numerically or experimentally. For the derivation of the inertia forces it is assumed that the centre of mass (CM) for the undeformed blade lies along the $z_{i}$ -axis. A vector in the blade frame $(x,\,y,z)$ from the rotor centre to the CM at radius $z$ is therefore given by 其中，$q_{i,n}$ 为模态叶片坐标，描述了第 $i$ 叶片运动中第 $n$ 模态的内容，而 $\Phi_{n}^{x}$ 和 $\Phi_{n}^{y}$ 为模态形函数。这些形函数可以通过数值计算或实验获得。 在推导惯性力时，假设未变形叶片的质量中心（CM）位于 $z_{i}$ 轴上。因此，从机组中心到半径为 $z$ 的质量中心的叶片坐标系 $(x,\,y,z)$ 中的向量表示为： $$ \mathbf{r}_{i}^{\mathrm{B}}=\sum_{n=1}^{N}\{\Phi_{n}^{x}(z),\,\Phi_{n}^{y}(z),z\}^{\mathrm{T}}q_{i,n}(t) $$ In the ground fixed frame $(X,Y,Z)$ the motion of the CM at radius $z$ on blade number $i$ becomes 在固定地面坐标系 $(X,Y,Z)$ 中，半径为 $z$，叶片编号为 $i$ 的质心运动变为： $$ \mathbf{r}_{i}=\mathbf{r}_{\mathrm{{t}}}+\mathbf{T}_{\mathrm{{t}}}[\mathbf{r}_{\mathrm{{ts}}}+\mathbf{T}_{\mathrm{{a}}}\mathbf{T}_{\mathrm{{s}}}(\mathbf{r}_{\mathrm{{sh}}}+\mathbf{T}_{\mathrm{{B}}_{i}}\mathbf{r}_{i}^{\mathrm{{B}}})] $$ where ${\bf r}_{\mathrm{t}}=\{u_{x}^{\mathrm{t}},u_{y}^{\mathrm{t}},0\}^{\mathrm{T}}$ is a vector from the origin to the tower top, ${\bf r}_{\mathrm{ts}}=\{0,-L_{\mathrm{n}},0\}^{\mathrm{T}}$ is a vector from the tower top to the bending point on the shaft in nacelle co-ordinates, and $\mathbf{r}_{\mathrm{sh}}=\{0,-L_{\mathrm{s}},0\}^{\mathrm{T}}$ is a vector from this point to the rotor centre in rotor co-ordinates. The transformation matrices ${\mathbf{T}}_{\mathrm{a}}$ and ${\bf{T}}_{{\bf{B}}_{i}}$ handling the azimuth rotations are written as 其中 ${\bf r}_{\mathrm{t}}=\{u_{x}^{\mathrm{t}},u_{y}^{\mathrm{t}},0\}^{\mathrm{T}}$ 是从原点到塔顶的矢量，${\bf r}_{\mathrm{ts}}=\{0,-L_{\mathrm{n}},0\}^{\mathrm{T}}$ 是在机舱坐标系中，从塔顶到曲柄轴弯曲点的矢量，而 $\mathbf{r}_{\mathrm{sh}}=\{0,-L_{\mathrm{s}},0\}^{\mathrm{T}}$ 是从该点到风轮中心的矢量，坐标系为风轮坐标系。处理方位角旋转的变换矩阵 ${\mathbf{T}}_{\mathrm{a}}$ 和 ${\bf{T}}_{{\bf{B}}_{i}}$ 如下： $$ \begin{array}{r l}&{\mathbf{T}_{\mathbf{B}_{i}}=\left[\begin{array}{c c c}{\cos\left[\frac{2}{3}\pi(i-1)\right]}&{0}&{\sin\left[\frac{2}{3}\pi(i-1)\right]}\\ {0}&{1}&{0}\\ {-\sin\left[\frac{2}{3}\pi(i-1)\right]}&{0}&{\cos\left[\frac{2}{3}\pi(i-1)\right]}\end{array}\right]}\\ &{\mathbf{T}_{\mathbf{a}}=\left[\begin{array}{c c c}{\cos(\Omega t+\theta_{\mathbf{y}}^{\mathrm{s}})}&{0}&{\sin(\Omega t+\theta_{\mathbf{y}}^{\mathrm{s}})}\\ {0}&{1}&{0}\\ {-\sin(\Omega t+\theta_{\mathbf{y}}^{\mathrm{s}})}&{0}&{\cos(\Omega t+\theta_{\mathbf{y}}^{\mathrm{s}})}\end{array}\right]}\end{array} $$ where $\Omega$ is the rotation speed. It is assumed that all rotations of the tower top and hub are small $(\theta_{x}^{\mathrm{t}},\theta_{z}^{\mathrm{t}},\theta_{x}^{\mathrm{s}},\theta_{z}^{\mathrm{s}}<<1)$ , whereby the remaining transformation matrices of equation (3) become 其中 $\Omega$ 为转速。假设塔顶和轮毂的所有旋转均较小 $(\theta_{x}^{\mathrm{t}},\theta_{z}^{\mathrm{t}},\theta_{x}^{\mathrm{s}},\theta_{z}^{\mathrm{s}}<<1)$ ，因此方程(3)中剩余的变换矩阵变为 $$ \begin{array}{r}{\mathbf{T}_{\mathrm{s}}=\left[\begin{array}{l l l}{1}&{-\theta_{z}^{\mathrm{s}}}&{0}\\ {\theta_{z}^{\mathrm{s}}}&{1}&{-\theta_{x}^{\mathrm{s}}}\\ {0}&{\theta_{x}^{\mathrm{s}}}&{1}\end{array}\right],\ \mathbf{T}_{\mathrm{t}}=\left[\begin{array}{l l l}{1}&{-\theta_{z}^{\mathrm{t}}}&{0}\\ {\theta_{z}^{\mathrm{t}}}&{1}&{-\theta_{x}^{\mathrm{t}}}\\ {0}&{\theta_{x}^{\mathrm{t}}}&{1}\end{array}\right]}\end{array} $$ where ${\bf{T}}_{\mathrm{{s}}}$ handles the transformation from rotor to nacelle co-ordinates due to shaft bending, and ${\bf T}_{\mathrm{t}}$ handles the transformation from nacelle co-ordinates to the ground fixed frame due to tilt and yaw. The total kinetic energy of the wind turbine in this model can now be derived as 其中 ${\bf{T}}_{\mathrm{{s}}}$ 处理由于主轴弯曲引起的风轮到机舱坐标系的变换，而 ${\bf T}_{\mathrm{t}}$ 处理由于倾覆和偏航引起的机舱坐标系到地面固定坐标系的变换。 在此模型中，风电机组的总体动能现在可以推导如下： $$ T=\frac{1}{2}\left(M(\dot{u}_{x}^{t^{2}}+\dot{u}_{y}^{t^{2}})+I_{x}\dot{\theta}_{x}^{t^{2}}+I_{z}\dot{\theta}_{z}^{t^{2}}+I_{y}\dot{\theta}_{y}^{s^{2}}+\sum_{i=1}^{3}\int_{0}^{R}m(z)|\dot{\bf r}_{i}|^{2}{\bf d}z\right) $$ where $\dot{\left(\,\right)}\equiv\partial/\partial t$ denotes the time derivative, $M$ is the total mass of the nacelle (and a part of the tower), $I_{x}$ and $I_{z}$ are the rotational inertias of the nacelle about the tilt and yaw axes respectively, $I_{y}$ is the rotational inertia of the shaft and drive-train, $R$ is the length of the blades including the hub and root extension, and $m(z)$ is the mass per unit length of the entire blade–root–hub assembly. It is assumed that there is an elastic stiffness coupling between the nacelle tilt $\theta_{x}^{\mathrm{t}}$ and the longitudinal tower motion $u_{y}^{\mathrm{t}}$ , whereby the potential energy is defined as 其中 $\dot{\left(\,\right)}\equiv\partial/\partial t$ 表示时间导数，$M$ 为机舱（及塔架的一部分）的总质量，$I_{x}$ 和 $I_{z}$ 分别为机舱绕tilt轴和偏航轴的转动惯量，$I_{y}$ 为轴和传动系的转动惯量，$R$ 为叶片（包括轮毂和根部延伸）的长度，$m(z)$ 为整个叶片-根部-轮毂组件的单位长度质量。 假设机舱tilt角 $\theta_{x}^{\mathrm{t}}$ 和纵向塔架运动 $u_{y}^{\mathrm{t}}$ 之间存在弹性刚度耦合，势能定义如下： $$ \begin{array}{r l r}{{V=\frac{1}{2}\left[k_{x}u_{x}^{\mathrm{t}^{2}}+k_{y}u_{y}^{\mathrm{t}^{2}}+g_{x y}u_{y}^{\mathrm{t}}\theta_{x}^{\mathrm{t}}+G_{x}\theta_{x}^{\mathrm{t}^{2}}+G_{z}\theta_{z}^{\mathrm{t}^{2}}\right.}}\\{+K_{s}(\theta_{x}^{s^{2}}+\theta_{z}^{s^{2}})+G_{y}\theta_{y}^{s^{2}}+\sum_{i=1}^{3}\left(\sum_{n=1}^{N}k_{n}q_{i,n}^{2}+V_{\mathrm{c},i}\right)]}\end{array} $$ where $k_{x}$ and $k_{y}$ are the translational stiffnesses of the tower at its top, $g_{x y}$ is the coupling stiffness between nacelle tilt and longitudinal tower motion, $G_{x}$ and $G_{z}$ are the rotational stiffnesses of the tower at its top, $K_{\mathrm{s}}$ is the bending stiffness of the shaft, and $G_{y}$ is the torsional stiffness of the shaft and drive-train. The modal stiffness $k_{n}$ is defined as $k_{n}\equiv\omega_{n}^{2}M_{n}$ , where $\omega_{n}$ is the natural frequency of blade mode number $n$ , and $M_{n}$ is the modal mass, $\begin{array}{r}{M_{n}\equiv\int_{0}^{R}m(z)(\Phi_{n}^{x^{2}}+\Phi_{n}^{y^{2}})\mathbf{d}z}\end{array}$ . The potential energy term $V_{\mathrm{c},i}$ is added for modelling of the centrifugal stiffening effect. The tensile force in the blade due to centrifugal body forces is $\begin{array}{r}{\Omega^{2}\int_{z}^{R}m(\zeta)\zeta\mathrm{d}\zeta}\end{array}$ , and the additional potential energy due to the centrifugal body forces can be approximated by 其中，$k_{x}$ 和 $k_{y}$ 分别为塔顶的平动刚度，$g_{x y}$ 为机舱倾斜与纵向塔架运动之间的耦合刚度，$G_{x}$ 和 $G_{z}$ 分别为塔顶的转动刚度，$K_{\mathrm{s}}$ 为轴的弯曲刚度，$G_{y}$ 为轴和驱动系统的扭转刚度。模态刚度 $k_{n}$ 定义为 $k_{n}\equiv\omega_{n}^{2}M_{n}$ ，其中 $\omega_{n}$ 为叶片模态数 $n$ 的固有频率， $M_{n}$ 为模态质量，$\begin{array}{r}{M_{n}\equiv\int_{0}^{R}m(z)(\Phi_{n}^{x^{2}}+\Phi_{n}^{y^{2}})\mathbf{d}z}\end{array}$ 。 为了模拟离心加固效应，添加了势能项 $V_{\mathrm{c},i}$。由于离心惯性力，叶片中的拉力为 $\begin{array}{r}{\Omega^{2}\int_{z}^{R}m(\zeta)\zeta\mathrm{d}\zeta}\end{array}$ ，由于离心惯性力产生的附加势能可以近似为 $$ V_{\mathrm{c},i}=\frac{1}{2}\Omega^{2}\int_{0}^{R}\left[\left(\sum_{n=1}^{N}q_{i,n}\frac{\mathrm{d}\Phi_{n}^{x}}{\mathrm{d}z}\right)^{2}+\left(\sum_{n=1}^{N}q_{i,n}\frac{\mathrm{d}\Phi_{n}^{y}}{\mathrm{d}z}\right)^{2}\right]\int_{z}^{R}m(\zeta)\zeta\mathrm{d}\zeta\mathrm{d}z $$ where it is assumed that rotations of the blade cross-sections due to bending are small. The blade and tower/nacelle co-ordinates are collected in a state vector 其中假设因弯曲引起的叶片横截面旋转较小。 叶片和塔架/机舱坐标被收集到一个状态向量中。 $$ \mathbf{x}=\{q_{1,1},\dotsc,q_{1,N},q_{2,1},\dotsc,q_{2,N},q_{3,1},\dotsc,q_{3,N},\theta_{x}^{\mathrm{s}},\theta_{z}^{\mathrm{s}},u_{x}^{\mathrm{t}},u_{y}^{\mathrm{t}},\theta_{x}^{\mathrm{t}},\theta_{z}^{\mathrm{s}},\theta_{y}^{\mathrm{s}}\}^{\mathrm{T}} $$ and Lagrange’s equations are derived from the Lagrangian $L=T-V$ . After linearization about the undeformed state by assuming small amplitudes of the co-ordinates, $x_{j}<<1$ , the equations of motion can be written in the form 并且拉格朗日方程可由拉格朗日量 $L=T-V$ 推导得出。在假设坐标变量 $x_{j}$ 具有小幅值，即 $x_{j}<<1$，并对未变形状态进行线性化后，运动方程可以写成如下形式： $$ \mathbf{M}(t){\ddot{\mathbf{x}}}+\mathbf{C}(t){\dot{\mathbf{x}}}+\mathbf{K}(t)\mathbf{x}=\mathbf{0} $$ Where M, C and $\mathbf{K}$ are the mass, gyroscopic and stiffness matrices respectively. All three matrices are time-dependent, containing harmonic terms with period $T=2\pi/\Omega$ . Note that structural damping is neglected in this simplified wind turbine model. The lower-order modes of interest to this study are low damped, thus structural damping has practically no effect on their mode shapes. For studies of resonance phenomena and stability it would be necessary to include the structural damping.14 The equations of motion (10) have a form that is general for linear structural models of wind turbines. The periodic terms arising from the azimuthal rotation prevent a direct derivation of the natural frequencies and mode shapes from an eigenvalue problem. Still, the mathematical approach of using Floquet theory should be avoided, because there is a more physically comprehensive way to set up an eigenvalue problem. 其中 M、C 和 $\mathbf{K}$ 分别为质量、陀螺惯性力和刚度矩阵。这三个矩阵均随时间变化，包含周期为 $T=2\pi/\Omega$ 的谐波项。 在简化的风电机组模型中，忽略了结构阻尼。本研究感兴趣的低阶模态阻尼较小，因此结构阻尼实际上对它们的模态形状没有影响。若要研究共振现象和稳定性，则需要包含结构阻尼。14 运动方程 (10) 具有适用于风电机组线性结构模型的通用形式。由于方位角旋转产生的周期项，无法从特征值问题直接推导出固有频率和模态。然而，仍应避免使用Floquet理论进行数学处理，因为存在一种更具物理意义的方式来设置特征值问题。 # Multi-blade Co-ordinate Transformation The use of multi-blade co-ordinates, or Fourier co-ordinates, for bladed rotors12,13 is a method to describe the motions of individual blades in the same co-ordinate system as the structure supporting the rotor, whereby the periodic terms in the governing equations are eliminated. The fundamental assumption of this method is that the rotor is isotropic, i.e. all blades are identical, identically pitched and symmetrically mounted on the hub. In the wind turbine model given by equation (10), the motion of the tower/nacelle is described in the ground fixed frame, while the shaft bending is described in rotating shaft co-ordinates, and the motions of the three blades are described in their own co-rotating frames. The shaft bending can easily be described in the ground fixed frame by a co-ordinate transformation with respect to the azimuth angle. The transformation of the blade co-ordinates into the ground fixed frame can be done by the multi-blade co-ordinate transformation, which is defined by 使用多叶片坐标，或傅里叶坐标，来描述叶片风轮的运动12,13是一种方法，它将单个叶片的运动描述在支撑风轮的结构相同的坐标系下，从而消除了控制方程中的周期项。该方法的根本假设是风轮是各向同性的，即所有叶片都相同，变桨角度相同，并且对称地安装在轮毂上。 在由公式(10)给出的风电机组模型中，塔架/机舱的运动在地面固定坐标系中描述，主轴弯曲在旋转主轴坐标系中描述，而三个叶片的运动则在各自的共旋转坐标系中描述。通过相对于方位角的坐标变换，主轴弯曲可以很容易地在地面固定坐标系中描述。叶片坐标变换到地面固定坐标系可以通过多叶片坐标变换来实现，该变换由以下定义： $$ q_{i,n}=a_{0,n}+a_{1,n}\cos\psi_{i}+b_{1,n}\sin\psi_{i} $$ where $\begin{array}{r}{\psi_{i}=\Omega t+\frac{2}{3}\pi(i-1)}\end{array}$ is the azimuth angle to blade number $i$ (without shaft torsion) measured from the $Z$ -axis. 其中 $\begin{array}{r}{\psi_{i}=\Omega t+\frac{2}{3}\pi(i-1)}\end{array}$ 是以 $Z$ 轴为基准，测得的叶片编号 $i$ 的方位角（无主轴扭转）。 The three multi-blade co-ordinates $a_{0,n},\,a_{1,n}$ and $b_{1,n}$ replace the three blade co-ordinates $q_{1,n},\;q_{2,n}$ and $q_{3,n}$ . To see that the multi-blade co-ordinates describe the rotor motion in the ground fixed frame, assume that blade mode $n$ is the first flapwise bending mode. Then $a_{0,n}$ describes a simultaneous flapwise deflection of all three blades, while $a_{1,n}$ and $b_{1,n}$ describe tilt and yaw motions respectively. 三片叶片坐标 $a_{0,n},\,a_{1,n}$ 和 $b_{1,n}$ 取代了三片叶片坐标 $q_{1,n},\;q_{2,n}$ 和 $q_{3,n}$ 。为了说明多片叶片坐标描述了风轮在地面固定参考系中的运动，假设叶片模态 $n$ 是第一挥舞摆振模态。那么 $a_{0,n}$ 描述了所有三片叶片的同步挥舞变形，而 $a_{1,n}$ 和 $b_{1,n}$ 分别描述了倾覆和偏航运动。 The multi-blade transformation of the state vector $\mathbf{x}$ can be represented by $$ \mathbf{x}=\mathbf{B}(t)\mathbf{z} $$ where the state vector $\mathbf{z}$ containing the multi-blade co-ordinates is $$ {\mathbf z}=\{a_{0,1},\ldots,a_{0,N},a_{1,1},\ldots,a_{1,N},b_{1,1},\ldots,b_{1,N},\theta_{X}^{\mathrm{s}},\theta_{Z}^{\mathrm{s}},u_{x}^{\mathrm{t}},u_{y}^{\mathrm{t}},\theta_{x}^{\mathrm{t}},\theta_{z}^{\mathrm{s}},\theta_{y}^{\mathrm{s}}\}^{\mathrm{T}} $$ and the transformation matrix is $$ \mathbf{B}\equiv\left[\begin{array}{c c c c c c}{\mathbf{I}_{N}}&{\mathbf{I}_{N}\cos\psi_{1}}&{\mathbf{I}_{N}\sin\psi_{1}}&{\mathbf{0}}&{\mathbf{0}}&{\mathbf{0}}\\ {\mathbf{I}_{N}}&{\mathbf{I}_{N}\cos\psi_{2}}&{\mathbf{I}_{N}\sin\psi_{2}}&{\mathbf{0}}&{\mathbf{0}}&{\mathbf{0}}\\ {\mathbf{I}_{N}}&{\mathbf{I}_{N}\cos\psi_{3}}&{\mathbf{I}_{N}\sin\psi_{3}}&{\mathbf{0}}&{\mathbf{0}}&{\mathbf{0}}\\ {\mathbf{0}}&{\mathbf{0}}&{\mathbf{0}}&{\cos\Omega t}&{-\sin\Omega t}&{\mathbf{0}}\\ {\mathbf{0}}&{\mathbf{0}}&{\mathbf{0}}&{\sin\Omega t}&{\cos\Omega t}&{\mathbf{0}}\\ {\mathbf{0}}&{\mathbf{0}}&{\mathbf{0}}&{\mathbf{0}}&{\mathbf{0}}&{\mathbf{I}_{S}.}\end{array}\right] $$ where ${\mathbf{I}}_{N}$ is the identity matrix of size $N\times N$ . The matrix $\mathbf{B}$ has the following properties: $$ \mathbf{B}^{-1}{\dot{\mathbf{B}}}={\left[\begin{array}{l l l l l l}{\mathbf{0}}&{\mathbf{0}}&{\mathbf{0}}&{\mathbf{0}}&{\mathbf{0}}&{\mathbf{0}}\\ {\mathbf{0}}&{\mathbf{0}}&{{\boldsymbol{\Omega}}\mathbf{I}_{N}}&{\mathbf{0}}&{\mathbf{0}}&{\mathbf{0}}\\ {\mathbf{0}}&{-{\boldsymbol{\Omega}}\mathbf{I}_{N}}&{\mathbf{0}}&{\mathbf{0}}&{\mathbf{0}}&{\mathbf{0}}\\ {\mathbf{0}}&{\mathbf{0}}&{\mathbf{0}}&{0}&{-{\boldsymbol{\Omega}}}&{\mathbf{0}}\\ {\mathbf{0}}&{\mathbf{0}}&{\mathbf{0}}&{{\boldsymbol{\Omega}}}&{0}&{\mathbf{0}}\\ {\mathbf{0}}&{\mathbf{0}}&{\mathbf{0}}&{\mathbf{0}}&{\mathbf{0}}&{\mathbf{0}}\end{array}\right]}\equiv\mathbf{R} $$ $$ {\bf B}^{-1}=\mu{\bf B}^{\mathrm{T}},\quad\pmb{\mu}=\left[\begin{array}{c c c c}{\frac{1}{3}{\bf I}_{N}}&{{\bf0}}&{{\bf0}}&{{\bf0}}\\ {{\bf0}}&{\frac{2}{3}{\bf I}_{N}}&{{\bf0}}&{{\bf0}}\\ {{\bf0}}&{{\bf0}}&{\frac{1}{3}{\bf I}_{N}}&{{\bf0}}\\ {{\bf0}}&{{\bf0}}&{{\bf0}}&{{\bf I}_{7}}\end{array}\right] $$ which shows that $\dot{\mathbf{B}}=\mathbf{B}\mathbf{R}$ and $\ddot{\mathbf{B}}=\mathbf{B}\mathbf{R}^{2}$ . A transformation of equation (10) into the multi-blade co-ordinates (12) yields $$ \mathbf{M}_{\mathrm{B}}\ddot{\mathbf{z}}+(2\mathbf{M}_{\mathrm{B}}\mathbf{R}+\mathbf{C}_{\mathrm{B}})\dot{\mathbf{z}}+(\mathbf{M}_{\mathrm{B}}\mathbf{R}^{2}+\mathbf{C}_{\mathrm{B}}\mathbf{R}+\mathbf{K}_{\mathrm{B}})\mathbf{z}=0 $$ where $\mathbf{M}_{\mathrm{B}}\equiv\mu\mathbf{B}^{\mathrm{T}}\mathbf{M}\mathbf{B}$ , $\mathbf{C}_{\mathrm{B}}\equiv\mu\mathbf{B}^{\mathrm{T}}\mathbf{C}\mathbf{B}$ and $\mathbf{K}_{\mathrm{B}}\equiv\mu\mathbf{B}^{\mathrm{T}}\mathbf{K}\mathbf{B}$ are the transformed system matrices. If the rotor is isotropic, these matrices are constant, i.e. the transformed equations of motion in the multi-blade co-ordinates (16) contain no periodic terms. It is therefore possible to define an eigenvalue problem that gives the natural frequencies and mode shapes of the wind turbine with a rotating rotor. 其中 $\mathbf{M}_{\mathrm{B}}\equiv\mu\mathbf{B}^{\mathrm{T}}\mathbf{M}\mathbf{B}$ , $\mathbf{C}_{\mathrm{B}}\equiv\mu\mathbf{B}^{\mathrm{T}}\mathbf{C}\mathbf{B}$ 和 $\mathbf{K}_{\mathrm{B}}\equiv\mu\mathbf{B}^{\mathrm{T}}\mathbf{K}\mathbf{B}$ 是变换后的系统矩阵。如果风轮是各向同性的，这些矩阵是常数，即在多叶片坐标系（16）中变换后的运动方程不包含周期项。因此，可以定义一个特征值问题，以获得具有旋转风轮的风电机组的固有频率和模态。 # Eigenvalue Problem and Solution A substitution of the solution $\mathbf{z}=\mathbf{z}_{k}\mathbf{e}^{\lambda_{k}t}$ into (16) yields an eigenvalue problem that determines the complex eigenvalues $\lambda_{k}\equiv\sigma_{k}+\mathrm{i}\omega_{k}$ $(\mathbf{i}\equiv{\sqrt{-1}})$ and complex eigenvectors $\mathbf{Z}_{k}$ of mode number $k$ . Inserting the solution $\mathbf{z}=\mathbf{z}_{k}\mathbf{e}^{\lambda_{k}t}$ into (12), the blade co-ordinate $q_{i,n}$ for the modal content of blade mode number $n$ in the motion of blade number $i$ becomes 将解 $\mathbf{z}=\mathbf{z}_{k}\mathbf{e}^{\lambda_{k}t}$ 代入 (16) 可得到一个特征值问题，该问题确定了模态数 $k$ 的复特征值 $\lambda_{k}\equiv\sigma_{k}+\mathrm{i}\omega_{k}$ $(\mathbf{i}\equiv{\sqrt{-1}})$ 和复特征向量 $\mathbf{Z}_{k}$。 将解 $\mathbf{z}=\mathbf{z}_{k}\mathbf{e}^{\lambda_{k}t}$ 代入 (12)，风轮叶片（blade）的坐标 $q_{i,n}$，用于描述第 $i$ 个叶片的第 $n$ 个模态（mode）运动的模态内容，变为： $$ \begin{array}{r l}&{q_{i,n,k}=A_{n,k}^{0}\mathrm{e}^{\sigma_{k}t}\cos(\omega_{k}t+\phi_{\beta,k}^{0})}\\ &{\quad\quad\quad+\frac{1}{2}A_{n,k}^{\mathrm{BW}}\mathrm{e}^{\sigma_{k}t}\cos[(\omega_{k}+\Omega)t+\frac{2}{3}\pi(i-1)+\phi_{n,k}^{\mathrm{BW}}]}\\ &{\quad\quad\quad+\frac{1}{2}A_{n,k}^{\mathrm{FW}}\mathrm{e}^{\sigma_{k}t}\cos[(\omega_{k}-\Omega)t-\frac{2}{3}\pi(i-1)+\phi_{n,k}^{\mathrm{FW}}]}\end{array} $$ where the amplitudes and phases are determined by the complex eigenvectors 其中，幅值和相位由复特征向量决定。 $$ \begin{array}{r l}&{A_{n,k}^{0}=\frac{1}{2}\sqrt{\mathrm{Re}\{a_{0,n,k}\}^{2}+\mathrm{Im}\{a_{0,n,k}\}^{2}}}\\ &{A_{n,k}^{\mathrm{BW}}=\frac{1}{2}\sqrt{(\mathrm{Re}\{b_{1,n,k}\}-\mathrm{Im}\{a_{1,n,k}\})^{2}+(\mathrm{Im}\{b_{1,n,k}\}+\mathrm{Re}\{a_{1,n,k}\})^{2}}}\\ &{A_{n,k}^{\mathrm{FW}}=\frac{1}{2}\sqrt{(\mathrm{Re}\{b_{1,n,k}\}+\mathrm{Im}\{a_{1,n,k}\})^{2}+(\mathrm{Re}\{a_{1,n,k}\}-\mathrm{Im}\{b_{1,n,k}\})^{2}}}\end{array} $$ and $$ \begin{array}{r l}&{\phi_{n,k}^{0}=\tan^{-1}(\operatorname{Im}\{a_{0,n,k}\}/\operatorname{Re}\{a_{0,n,k}\})}\\ &{\phi_{n,k}^{\mathrm{BW}}=\tan^{-1}[(\operatorname{Im}\{a_{1,n,k}\}-\operatorname{Re}\{b_{1,n,k}\})/(\operatorname{Im}\{b_{1,n,k}\}+\operatorname{Re}\{a_{1,n,k}\})]}\\ &{\phi_{n,k}^{\mathrm{FW}}=\tan^{-1}[(\operatorname{Re}\{b_{1,n,k}\}+\operatorname{Im}\{a_{1,n,k}\})/(\operatorname{Re}\{a_{1,n,k}\}-\operatorname{Im}\{b_{1,n,k}\})]}\end{array} $$ Equation (17) shows that the motion of blade number $i$ for mode number $k$ consists of three components: a symmetric component where all blades deflect simultaneously in blade mode number $n$ with the amplitude $A_{n,k}^{0}$ , and two asymmetric components where the blades deflect with phase shifts of $T/3$ and the amplitudes $A_{n,k}^{\mathrm{BW}}$ and $A_{n,k}^{\mathrm{FW}}$ respectively. As indicated by the superscripts, the asymmetric components represent backward and forward rotor whirling. The direction of the whirl is determined by the sign of the blade-dependent phase shifts $\textstyle{\frac{2}{3}}\pi(i-1)$ . The frequency of the symmetric component is $\omega_{k}$ , while the frequencies of the whirling components are shifted by $\pm\Omega$ . The reason for these frequency shifts is that the natural frequency $\omega_{k}$ is given in the ground fixed frame. An observer on the tower top (in the ground fixed frame) will measure the natural frequency $\omega_{k}$ for all three modes. An observer on a blade will also measure the frequency $\omega_{k}$ for symmetric modes, but for backward and forward whirling modes the same observer will measure the frequencies $\omega_{k}+\Omega$ and $\omega_{k}-\Omega$ respectively. The damping of the turbine modes is described by the term $\mathrm{e}^{\sigma_{k}t}$ ; the modal damping factor $\sigma_{k}$ determines the exponential decay, or growth, of amplitude for a vibration in mode number $k$ . The computation of the damping factors can therefore be used for stability analyses, e.g. the flutter analysis of a wind turbine.15 公式(17)表明，对于模态数k，叶片数i的运动由三个分量组成：一个对称分量，所有叶片同时变形，变形模态数为n，振幅为$A_{n,k}^{0}$，以及两个非对称分量，叶片变形具有$T/3$的相位差，振幅分别为$A_{n,k}^{\mathrm{BW}}$和$A_{n,k}^{\mathrm{FW}}$。正如上标所示，非对称分量代表后向和前向风轮摆振。摆振方向由叶片相关的相位差$\textstyle{\frac{2}{3}}\pi(i-1)$的符号决定。 对称分量的频率为$\omega_{k}$，而摆振分量的频率则分别偏移了$\pm\Omega$。产生这些频率偏移的原因是，自然频率$\omega_{k}$是在地面固定参考系中给定的。塔顶观察者（在地面固定参考系中）将测量所有三个模态的自然频率$\omega_{k}$。叶片上的观察者也会测量对称模态的频率$\omega_{k}$，但对于后向和前向摆振模态，相同的观察者将分别测量频率$\omega_{k}+\Omega$和$\omega_{k}-\Omega$ 。 机组模态的阻尼由项$\mathrm{e}^{\sigma_{k}t}$描述；模态阻尼系数$\sigma_{k}$决定了模态数k的振动振幅的指数衰减或增长。因此，阻尼系数的计算可用于稳定性分析，例如风电机组的颤振分析。15 # Modal Analysis of the Experimental 600 kW Turbine This section contains derivations of the natural frequencies and mode shapes of the Bonus $600~\mathrm{kW}$ wind turbine used in the experiments by Thomsen et al.1 A Campbell diagram shows the natural frequencies of the turbine as a function of rotation speed. A mode identification study shows that some of the corresponding mode shapes change dramatically as the rotation speed is changed, owing to modal interactions. 本节包含对实验中使用的Bonus $600~\mathrm{kW}$ 风电机组的固有频率和模态的推导。一个坎贝尔图（Campbell diagram）显示了风轮的固有频率随转速的变化。**模态识别研究表明，随着转速的变化，一些对应的模态会发生剧烈变化，这是由于模态相互作用的结果**。 # Model Parameters The Bonus $600~\mathrm{kW}$ (Mk III C) wind turbine with its $\mathrm{LM}\ 19{\cdot}1\ \mathrm{m}$ blades has a rotor radius of $22\textrm{m}$ with hub and root extension. For this study, two blade modes are chosen $(N=2)$ : first flapwise and first edgewise bending modes for the non-rotating blade. The shape functions $\Phi_{n}^{x}$ and $\Phi_{n}^{y}$ of the entire blade–root–hub assembly are obtained from curve ftis to the results of a finite element model (see Figure 3). The functions are normalized so that the associated modal masses are one. The natural frequencies of the flapwise and edgewise blade modes (as measured in a test stand with the hub and root extension) are 1Ð70 and $2{\cdot}94~\mathrm{Hz}$ respectively. The remaining model parameters are obtained by tuning the natural frequencies of the stationary turbine $(\Omega\equiv0)$ ) to the frequencies measured for the particular experimental turbine. The turbine is operating at a speed of about $27.4~\mathrm{rpm}$ $\Omega\approx2{\cdot}87$ rad $\mathrm{s}^{-1}$ ). Bonus $600~\mathrm{kW}$ (Mk III C) 风电机组，其 $\mathrm{LM}\ 19{\cdot}1\ \mathrm{m}$ 叶片，风轮半径为 $22\textrm{m}$，包括轴毂和根部延伸。对于本研究，选择了两个叶片模态 $(N=2)$：第一个挥舞模态和第一个摆振模态，用于非旋转叶片。整个叶片–根部–轴毂组件的形状函数 $\Phi_{n}^{x}$ 和 $\Phi_{n}^{y}$，是通过拟合有限元模型的结果获得的（见图3）。这些函数被归一化，使得相关的模态质量为一。在测试支架上测得的挥舞模态和摆振模态的固有频率分别为 1.70 和 $2{\cdot}94~\mathrm{Hz}$。 其余模型参数是通过调整静止机组的固有频率（$\Omega\equiv0$）来匹配特定实验风电机组测得的频率而获得的。风电机组正在以约 $27.4~\mathrm{rpm}$ 的速度运行（$\Omega\approx2{\cdot}87$ rad $\mathrm{s}^{-1}$）。 ![](957376a7e579e755d01141349ef562dc417a67a945cf69141650ed12e2ba5b66.jpg) Figure 3. Mode shape functions for the blade–root–hub assembly: first flapwise mode (left) and first edgewise mode (right). The functions are curve fits to the points obtained from a finite element model with Timoshenko beam elements 叶片-根-轮毂组件的模态函数：一阶挥舞模态（左）和一阶摆振模态（右）。这些函数是对有限元模型中Timoshenko梁单元计算所得点的曲线拟合。 Table I. Modelled and measured natural frequencies of the lowest 10 modes of the stationary turbine 对固定机组的前10个模态自然频率进行建模和测量。
Mode no.NameModelled frequency (Hz)Measured frequency (Hz)
11stshafttorsion0.570.56
21st longitudinal tower bending0.730.73
31st lateral tower bending0.780.80
41st yawing fap1·241.20
51st tilting fap1.481.50
61st symmetric flap1.761.76
71stverticaledgewise2.852.90
81sthorizontaledgewise2.95
92nd yawing fap3.433.60
102nd tilting flap3.74
# Natural Frequencies and Mode Shapes Modelled and measured natural frequencies of the 10 lowest modes of the stationary turbine are listed in Table I. The measured frequencies are obtained from identification of peaks in frequency responses. For the mode pairs 7/8 and 9/10 it is only possible to identify a single response peak. A comparison of modelled and measured frequencies leads to the assumption that the model is sufficiently tuned to the experimental turbine. The naming of the 10 lowest modes is based on animations of their mode shapes and the measured frequency responses. The sequence of modes is typical for a turbine of this size, with the exception that the 1st shaft torsion may be higher than the tower bending modes. The longitudinal tower bending always lies slightly lower than the lateral tower bending mode, because it contains some tilting of the rotor, which has a large inertia. Modes 4 and 5 are the 1st tilt/yaw modes involving the flapwise blade mode. The yaw mode most often lies lower than the tilt mode, because towers are stiffer in tilt than in yaw. The sixth mode is the 1st symmetrical flap mode, where the blades vibrate simultaneously in the flapwise blade mode in counter-phase with a longitudinal tower vibration. This coupling causes the symmetric flap frequency to lie slightly above the frequency of the flapwise blade mode. Modes 7 and 8 involve the edgewise blade mode, and their frequencies are close to the edgewise blade frequency of $2{\cdot}94~\mathrm{Hz}$ (with an average slightly below). In both modes the blades vibrate edgewise against each other so that they cancel out the torsional moment at the rotor centre. The two modes differ in the direction of the reactive force at the rotor centre, as indicated by their names: 1st vertical and 1st horizontal edgewise mode. The sequence of these two modes is given by the vertical and horizontal stiffness of the rotor support. The last two modes listed in Table I are the 2nd tilt/yaw modes, where the rotor blades are tilting and yawing in counter-phase with the tilt and yaw of the nacelle. Again the yaw mode lies below the tilt mode because of the lower yaw than tilt stiffness of the tower. Figure 4 shows how the natural frequencies $\omega_{k}$ of these 10 lowest modes change with the rotation speed $\Omega$ from standstill to the operation speed. The natural frequencies of the tower bending modes and the shaft torsion mode are constant with rotation speed. The frequency of the symmetric flap mode increases owing to centrifugal stiffening of flapwise bending. The natural frequencies of the asymmetric rotor modes change with rotation speed owing to gyroscopic effects. The naming of the modes at operation listed in the table in Figure 4 requires some introduction. All pairs of asymmetric rotor modes at standstill (modes 4/5, 7/8 and 9/10) become pairs of rotor whirling modes owing to the rotation, e.g. the 1st tilt/yaw modes become the 1st flapwise whirling modes. The frequencies of the backward whirling (BW) modes decrease with rotation speed, whereas the frequencies of the forward whirling (FW) modes increase with rotation speed. 表I列出了静止机组前10阶模态的固有频率，这些频率是通过识别频率响应中的峰值获得的。对于7/8阶和9/10阶模态对，仅能识别出一个响应峰值。 对模拟和测量频率进行比较，表明模型已经足够地针对实验机组进行了调整。前10阶模态的命名是基于其模态形状的动画和测量频率响应。这些模态的顺序对于这种尺寸的风电机组来说是典型的，但1阶主轴扭转可能高于塔架弯曲模态。纵向塔架弯曲总是略低于横向塔架弯曲模态，因为横向塔架弯曲包含一些叶片摆动，而叶片具有很大的惯性。 第4阶和第5阶模态是第1阶倾转/偏航模态，涉及挥舞叶片模态。偏航模态通常低于倾转模态，因为塔架在倾转方向上比在偏航方向上更刚。第6阶模态是第1阶对称挥舞模态，其中叶片以逆相位与纵向塔架振动同时在挥舞方向上振动。这种耦合导致对称挥舞频率略高于挥舞叶片模态的频率。 第7阶和第8阶模态涉及摆振叶片模态，它们的频率接近于摆振叶片频率的$2{\cdot}94~\mathrm{Hz}$（平均值略低于此值）。在这两个模态中，叶片相互作用，在摆振方向上振动，从而抵消了风轮中心的扭转力矩。这两个模态的区别在于风轮中心反作用力的方向，这由它们的名称所示：第1阶垂直摆振模态和第1阶水平摆振模态。这两个模态的顺序由风轮支撑的垂直和水平刚度决定。 表I中列出的最后两个模态是第2阶倾转/偏航模态，其中风轮叶片以与机舱倾转和偏航相反的相位倾转和偏航。同样，由于塔架在偏航方向上的刚度低于倾转方向，偏航模态低于倾转模态。 图4显示了这些前10阶简正模态的固有频率$\omega_{k}$随转速$\Omega$从静止到额定转速的变化情况。塔架弯曲模态和主轴扭转模态的固有频率随转速变化而保持不变。由于挥舞方向的离心加固，对称挥舞模态的固有频率增加。由于陀螺效应，非对称风轮模态的固有频率随转速变化。 图4中列出的额定转速下模态的命名需要一些介绍。所有在静止状态下的非对称风轮模态对（模态4/5、7/8和9/10）都因转动而变成风轮回转模态对，例如，第1阶倾转/偏航模态变成第1阶挥舞回转模态。向后回转（BW）模态的频率随转速降低，而向前回转（FW）模态的频率随转速升高。 ![](1668404ad97d76291f188361e7728ca1d61b3c36e9a82d524b6e3b4ca4ff9ea2.jpg) Figure 4. Campbell diagram: natural frequencies as a function of rotation speed of the rotor. The points denote the measured natural frequencies for the turbine at standstill This splitting of the natural frequencies of a whirling mode pair is related to the co-ordinate system of observation through gyroscopic effects. The natural frequencies $\omega_{k}$ in Figure 4 are observed from the ground fixed frame. Equation (17) shows that in a co-rotating blade frame the frequency of a symmetric rotor mode remains $\omega_{k}$ , whereas the frequencies of BW and FW modes become $\omega_{k}+\Omega$ and $\omega_{k}-\Omega$ respectively. If only blade mode number $n$ is involved in the whirling modes, their frequencies split in the ground fixed frame about the natural frequency $\omega_{n}$ of this blade mode. In this ideal case the observer on the blade will measure the same frequency $\omega_{n}=\omega_{k}^{\mathrm{BW}}+\Omega=\omega_{k}^{\mathrm{FW}}-\Omega$ for both the BW and FW modes. This ideal condition is affected by structural asymmetry of the turbine and coupling of blade modes in the turbine modes. Such modal interactions may occur when the natural frequencies of two turbine modes come close. Figure 4 shows that the frequencies of the 1st FW edgewise mode (mode 8) and the 2nd BW flapwise mode (mode 9) become close at about $\Omega=2{\cdot}5$ rad $\mathrm{s}^{-1}$ . These two modes interact, which can be shown by the flapwise and edgewise whirling components in their mode shapes. Figure 5 shows the FW and BW components of the flapwise and edgewise blade modes in modes 7–10. These whirling components are computed from the eigenvectors in multi-blade co-ordinates using equation (18). The dominating amplitudes for modes 7 and 10 show that these modes are respectively a BW edgewise and an FW flapwise mode. However, there are no dominant modal amplitude for modes 8 and 9 over the whole range of rotation speeds. It seems that these modes interchange mode shapes: mode 8 can be defined as an FW edgewise mode and mode 9 as a BW flapwise mode at rotation speeds below 2Ð5 rad $\mathrm{s}^{-1}$ , and vice versa above this speed. At the operation speed, mode 9 must be characterized as the 1st FW edgewise mode with some content of flapwise whirling. This content is now suggested to explain the observed difference in aerodynamic damping of the BW and FW edgewise modes (modes 7 and 9). # Why the Measured Difference in Aerodynamic Damping? Thomsen et al.1 have estimated the total damping of the two edgewise whirling modes (see Figure 1). The results show that the FW edgewise mode (mode 9) is more damped than the BW edgewise mode (mode 7). The structural damping of these two modes is assumed to be the same, because their natural frequencies and mode shapes are almost identical. Thus the difference in total damping is assumed to be caused by a difference in aerodynamic damping. ![](b27d093788eedd61e8374571ef2a21361ad23a35d29ba894deefacc295c3a9c7.jpg) Figure 5. Flapwise and edgewise whirling components of the modal blade amplitudes for modes 7–10 computed from the eigenvectors using equation (18) The aim of the experiment was to measure the damping of edgewise blade vibrations during turbine operation by exciting the corresponding turbine modes using an exciter mounted in the nacelle. The presented theoretical modal analysis of the turbine was not available for the experiment. The excitation of the edgewise whirling modes was therefore based on tuning the excitation frequency to obtain the maximum edgewise blade response. The two initial guesses, however, were based on the assumption that the ideal condition for frequency splitting existed, i.e. the excitation frequency should be the natural frequency of the edgewise blade mode plus or minus the operation speed (1P). Figure 6 shows a zoom in the Campbell diagram (Figure 4) on the natural frequencies of modes 7–9, together with lines of the $\pm1\mathrm{P}$ splitting about the edgewise blade frequency $(2{\cdot}94\ \mathrm{Hz})$ . These lines intersect with the natural frequencies of modes 7 and 9 at the operation speed, showing that the initial guesses on the experimental excitation frequencies are close. However, the modal analysis has also shown that mode 9 is not a pure edgewise whirling mode; it has some flapwise component. This component affects the direction in which the blades are vibrating relative to the rotor plane. # Effective Direction of Blade Vibration Figure 7 shows how the blade cross-section at $90\%$ radius is moving in and out of the rotor plane during vibrations in the two edgewise whirling modes (modes 7 and 9). The motion of the cross-section includes the motion of the rotor support, i.e. the shaft and tower deformations add to the effective blade vibration. The traces show that the blades move more out of the rotor plane in the FW edgewise mode than in the ![](fbc1b019314328084a7e66eb9b851b544a45506606be095b8746d98395d8f071.jpg) Figure 6. A zoom in the Campbell diagram (Figure 4) on the natural frequencies of modes 7–9. The broken lines show the $\pm\Omega$ splitting about the edgewise blade frequency $\left(2{\cdot}94\,H z\right)$ ![](b8ac1f69e8e67bd89e848dd13bf3bc4bab5b265f8838cf15dc851116eb006242.jpg) Figure 7. Motion of a blade cross-section at $90\%$ radius for the 1st BW edgewise mode (left) and the 1st FW edgewise mode (right). The motion includes the tower and shaft deformations BW edgewise mode. This difference in blade vibration for the two modes can explain why Thomsen et al. observed that the FW mode is more damped than the BW mode. A quasi-steady aerodynamic analysis in the Appendix shows that the aerodynamic damping of a blade cross-section is lowest for vibrations close to the rotor plane. It is presumed in the analysis that the blade cross-section is performing a small elliptical motion, with the major axes being parallel and perpendicular to the rotor plane. This type of motion is assumed to characterize the qualitative pattern of the traces in Figure 7, except for a slight tilt of the major axes which originates from the direction of blade vibration for the edgewise blade mode (see Figure 3). An effective direction of blade vibration is defined for the elliptical motion of a blade cross-section as $$ \theta_{\mathrm{eff}}=\tan^{-1}\left({\frac{\mathrm{max.~amplitude~out~of~rotor~plane}}{\mathrm{max.~amplitude~in~rotor~plane}}}\right) $$ Because the elliptical motion is assumed to have the major axes parallel and perpendicular to the rotor plane, the maximum amplitudes are positive, yielding $0^{\circ}<\theta_{\mathrm{eff}}<90^{\circ}$ . Using this definition on the motion of the cross-section at $90\%$ radius in Figure 7, it is found that $\theta_{\mathrm{eff}}\approx9^{\circ}$ and $30^{\circ}$ for the 1st BW and FW edgewise modes respectively. Computations of $\theta_{\mathrm{eff}}$ along the blade show that the entire blade is vibrating more out of the rotor plane in the 1st FW edgewise mode than in the 1st BW edgewise mode. Hence, based on quasisteady aerodynamics, this behaviour can explain the measured difference in aerodynamic damping of these two modes. # Improved Modal Dynamics The out-of-plane motion of the blades in the 1st FW edgewise mode is mainly due to a component of the flapwise blade mode through the previously described modal interaction with the 2nd BW flapwise mode (see Figure 5). The 1st BW edgewise mode is not interacting with a flapwise whirling mode; the blades are mainly vibrating in the edgewise blade mode (with the additional motion due to the tower and shaft deformations). Is it possible to add a flapwise component to the BW edgewise mode without removing the flapwise component in the FW edgewise mode, thereby increasing the overall aerodynamic damping of the turbine? An example shows that such improvement of the modal dynamics of the Bonus $600~\mathrm{kW}$ turbine may be possible. The natural frequency of the edgewise blade mode is chosen to be the design parameter in this optimization problem. Note that in reality an improved design must be the result of an integrated process towards a ‘global optimum’, where not only a single design parameter (the edgewise blade stiffness) is considered to achieve a single objective (the effective direction of blade vibration). Figure 8 shows the effective directions of blade vibration defined by equation (20) at the $90\%$ radius for modes 7–10 as function of the edgewise blade frequency. The modes are here named by their numbers, because their modal interactions prevent a unique mode identification over the entire range of design variation. These modes are all interesting, because they involve the edgewise blade mode. Shaft torsion and lateral tower bending modes also involve blade vibrations close to the rotor plane; however, the resulting low aerodynamic damping is compensated by high structural damping due to the generator slip of an asynchronous machine. A local optimum is marked at an increase of $22\%$ in the edgewise blade frequency, from 2Ð94 to $3{\cdot}59\ \mathrm{Hz}$ . For this design the effective direction of blade vibration for mode 7 has tripled to approximately $27^{\circ}$ . The effective direction of blade vibration has also increased for mode 8; however, the blades are now vibrating closer to the rotor plane in modes 9 and 10. In fact, mode 10 would be the most critical mode for higher edgewise blade frequencies. ![](e36f60f30c18c521b819097fdc94747ac3d39dfc9c912bc63f4000e25e395b40.jpg) Figure 8. Effective directions of blade vibration at $90\%$ radius with respect to the rotor plane for modes 7–10 as a function of the edgewise blade frequency The improved design (with respect to the effective direction of blade vibration) uses the modal interaction with 2nd flapwise whirling modes to increase the flapwise component in the 1st edgewise whirling modes. It is not a coincidence that the optimal edgewise blade frequency is almost identical to the average frequency of the 2nd tilt and yaw modes at standstill (see Table I and Figure 4). Figure 9 shows a Campbell diagram for modes 7–10 of the improved turbine, together with the $\pm1P$ splitting about the edgewise blade frequency. The ideal condition with a single blade mode involved in the rotor whirling mode is not present, and the modal interaction of edgewise and flapwise whirling is complete. For this particular turbine it may be impossible to increase the edgewise blade frequency by $22\%$ ; however, by changing other design parameters, it may be possible to lower the frequencies of the 2nd flapwise whirling modes to obtain the desired modal interaction. A rule of thumb in structural design for maximum out-ofrotor-plane blade vibration can be formulated as: Make sure the 1st edgewise blade frequency lies between the frequencies of the 2nd tilt and yaw modes. Whether and how this design guideline can be followed will depend on the wind turbine. # Conclusion A theoretical modal analysis of a three-bladed wind turbine shows that the blades cannot be considered as separate dynamic components of the turbine. Blade vibrations in the different modes of the turbine are strongly affected by the dynamics of the shaft, nacelle and tower. It is therefore necessary to consider the dynamics of the complete turbine when estimating the risk of stall-induced blade vibrations. The analysis of a stall-controlled $600~\mathrm{kW}$ turbine suggests why the forward and backward edgewise whirling modes have been observed to have different aerodynamic damping.1 The explanation is a difference in the blade vibration for the two edgewise whirling modes. The forward whirling mode interacts with a flapwise whirling mode, whereby the blades vibrate more out of the rotor plane, yielding an increased aerodynamic damping. Variations of the turbine design indicate that the structural dynamics of a wind turbine can be tailored to obtain a higher aerodynamic damping of its critical modes. A rule of thumb in structural design to lower the risk of stall-induced edgewise vibrations can be formulated as: Make sure the 1st edgewise blade frequency lies between the frequencies of the 2nd tilt and yaw modes. Whether and how this design guideline can be followed will depend on the wind turbine. ![](43f05a925486cdde7941cb8be05a93435038eb584261ba1bd5fff97bffbf8dc4.jpg) Figure 9. Natural frequencies of modes 7–10 for the improved turbine. The broken lines show the $\pm\Omega$ splitting about the new edgewise blade frequency of $3{\cdot}59\,H z$ # Acknowledgements This work was partly funded by the Danish Energy Agency under ENS 1363/00-0006, which is gratefully acknowledged. The author would also like to thank Kenneth Thomsen and Jørgen Thirstrup Petersen for their many invaluable discussions and suggestions. # Appendix: Aerodynamic Damping This appendix deals with the aerodynamic damping of blade vibrations based on quasi-steady aerodynamics. Unsteady aerodynamic effects can be dominating when the blades operate in stall; however, the basic qualitative behaviour of the aerodynamic damping is assumed to be described by quasi-steady aerodynamics (see e.g. References 8 and 16 for details on dynamic stall effects). The aim is to determine the aerodynamic damping of a vibrating blade cross-section. The cross-section is assumed to translate in and out of the rotor plane with the velocities $\dot{u}_{x}$ and $\dot{u}_{y}$ respectively, as shown in Figure 10. Torsional motion is neglected. The velocity triangle shows that the relative flow angle with respect to the rotor plane and the relative flow speed are $$ \phi=\tan^{-1}\left(\frac{V-\dot{u}_{y}}{r\Omega+\dot{u}_{x}}\right),\qquad W=\sqrt{(V-\dot{u}_{y})^{2}+(r\Omega+\dot{u}_{x})^{2}} $$ where $V$ is the wind speed, and the product of radius and rotation speed, $r\Omega$ , is the steady tangential speed of the cross-section. Note that induced velocities from the wake behind the turbine are neglected, which has no qualitative influence on the derivations. The angle of attack is the difference between the relative flow angle and the twist of the cross-section: $\alpha=\phi-\theta$ . The assumption of quasi-steady conditions (neglecting added mass effects) means that the lift and drag on the cross-section can be written as $$ L(\dot{u}_{x},\dot{u}_{y})=\frac{1}{2}c\rho W^{2}C_{\mathrm{L}}(\alpha),\quad D(\dot{u}_{x},\dot{u}_{y})=\frac{1}{2}c\rho W^{2}C_{\mathrm{D}}(\alpha) $$ where $c$ is the chord length, $\rho$ is the air density, and $C_{\mathrm{{L}}}$ and $C_{\mathrm{D}}$ are the steady aerodynamic lift and drag coefficients given as functions of the angle of attack. The lift and drag forces are non-linear functions of the velocities $\dot{u}_{x}$ and $\dot{u}_{y}$ of the cross-section. Assuming that these velocities are small compared with the steady flow velocities $\dot{u}_{x}<