From bd449d94a8ea292686a562fc1878a9bf6cb93526 Mon Sep 17 00:00:00 2001 From: yize Date: Thu, 3 Jul 2025 16:59:35 +0800 Subject: [PATCH] vault backup: 2025-07-03 16:59:34 --- .../Kallesøe-Effect of steady deflections on the a.md | 68 +- .../Kallesøe-Equations of motion for a rotor blade.md | 778 ++++++++++++++++++ ...esøe-Equations of motion for a rotor blade_origin.pdf | Bin 0 -> 443646 bytes ...71757eb1a1259bcc67eedcad9326ef4a3690aa1ee45ebf50e.jpg | Bin 0 -> 166652 bytes ...7243e53a3c723acfb00e4e886baf0a9dbbe8cf78badde1182.jpg | Bin 0 -> 50740 bytes ...063930df0ace9b75a3b81efe5ac90b1e77b700cfc654b46ca.jpg | Bin 0 -> 110238 bytes ...ad665c9d3ad833ead7a355e73bfde4c5d791fb2c9156e1c47.jpg | Bin 0 -> 89486 bytes 7 files changed, 826 insertions(+), 20 deletions(-) create mode 100644 线性化求解器/参考文献/Kallesøe-Equations of motion for a rotor blade/auto/Kallesøe-Equations of motion for a rotor blade.md create mode 100644 线性化求解器/参考文献/Kallesøe-Equations of motion for a rotor blade/auto/Kallesøe-Equations of motion for a rotor blade_origin.pdf create mode 100644 线性化求解器/参考文献/Kallesøe-Equations of motion for a rotor blade/auto/images/1c3ac9475d7dd6471757eb1a1259bcc67eedcad9326ef4a3690aa1ee45ebf50e.jpg create mode 100644 线性化求解器/参考文献/Kallesøe-Equations of motion for a rotor blade/auto/images/1dc8ba98cb4fd297243e53a3c723acfb00e4e886baf0a9dbbe8cf78badde1182.jpg create mode 100644 线性化求解器/参考文献/Kallesøe-Equations of motion for a rotor blade/auto/images/3b76a1326c2c154063930df0ace9b75a3b81efe5ac90b1e77b700cfc654b46ca.jpg create mode 100644 线性化求解器/参考文献/Kallesøe-Equations of motion for a rotor blade/auto/images/aef42751cc21fe8ad665c9d3ad833ead7a355e73bfde4c5d791fb2c9156e1c47.jpg diff --git a/线性化求解器/参考文献/Kallesøe-Effect of steady deflections on the a/auto/Kallesøe-Effect of steady deflections on the a.md b/线性化求解器/参考文献/Kallesøe-Effect of steady deflections on the a/auto/Kallesøe-Effect of steady deflections on the a.md index dec3f5c..58856d0 100644 --- a/线性化求解器/参考文献/Kallesøe-Effect of steady deflections on the a/auto/Kallesøe-Effect of steady deflections on the a.md +++ b/线性化求解器/参考文献/Kallesøe-Effect of steady deflections on the a/auto/Kallesøe-Effect of steady deflections on the a.md @@ -41,50 +41,69 @@ Research in utilizing sweep and pre-bend blades is ongoing. The European Union f Wind turbine stability can be analysed by a variety of different model types. The most detailed description of the turbine response is given by numerical non-linear time simulation tools.14–18 These tools show instabilities as well as non-linear effects limiting the response to for instance limit cycle oscillations. They can also be used to analyse the effect of, for instance, turbulence and wind shear’s effects on turbine stability. The referenced tools use different models and different model complexity. For instance, $\mathrm{FAST^{18}}$ is a modal-based code which on the one hand does not include a torsional degree of freedom of the blade and non-linear geometric couplings, but on the other hand is relatively computationally inexpensive. A code like $\mathrm{HAWC}2^{14,15}$ has a more complex model with a structural model based on a multi-body formulation where each body is a Timoshenko beam element including torsion. The drawback of these time-simulation tools is that they are computationally intensive and they can make it difficult to extract the important aeroelastic mechanisms from the large volume of results. Another approach is to use eigenvalue analysis of a linear (or linearized) model of the turbine.11,12,19–21 The HAWCStab code19,21 offers a platform for linearization of the undeflected turbine structure, while the code TURBU11 offers a platform for aeroservoelastic stability analysis based on linearization around the def lected/curved blade state. The structural model in TURBU is based on a simple co-rotational beam element approach. Each beam element consists of a rigid body with springs and dampers in its entry point; average strains in the springs and torque-free rotation offsets between the beam elements embody the average deflected/curved blade state. Riziotis et al.12 offers a multi-body platform which finds a reference state by time integration and linearizes the aeroservoelastic equations around this reference state to provide a stability tool including closed loop control. This type of tool can give both structural eigenfrequencies and eigenmodes that describe the basic structural dynamics of the turbine and aeroelastic frequencies, damping and modes of the aeroelastic motion. The aeroelastic damping reveals any stability problems for the turbine. However, since it is linear tools, they do not give any information concerning non-linear mechanism that limit the amplitude of a linear negative damped mode. The knowledge of structural and aeroelastic frequencies and mode shapes is very useful in the analysis and in the interpretation of results from aeroelastic time simulations. However, the modes of the aeroelastic response of the whole turbine can still be complex to analyse. To reduce the complexity, and thus make the results more transparent, a blade-only analysis is used.22 This allows a clear physical interpretation and insight into the mechanisms that govern the dynamic response of the blade and many basic characteristic of turbine stability can be extract from a blade-only analysis. -This paper uses a non-linear blade model23 which includes the effect of large blade def lections, pitch action and rotor speed variations. This blade model is strongly inspired by the work of Hodges and Dowell1 First, the structural model is combined with a steady-state aerodynamic model based on beam element momentum (BEM) theory and discritized by a f inite difference scheme. The resulting algebraic non-linear aeroelastic model is employed to compute steady-state blade def lections and induced velocities of a blade from the 5 MW Reference Wind Turbine (RWT) by National Renewable Energy Laboratory (NREL)24 at normal power production conditions. The steady-state def lections are compared with the results from HAWC2 simulations, showing good agreement. Throughout this paper, the 5 MW RWT by NREL is used as an example blade. The reference turbine is an artif icial turbine based on state-of-the-art turbines on the market. The blade is strongly inspired by the $61.5\mathrm{~m~LM}$ glasf iber blade (LM Wind Power, Kolding, Denmark). This blade belongs to the mid-region of f lexible designs of state-of-the-art blades, and hence, the geometric couplings can be more pronounced for other blade designs. The big advantage of this blade however is that all data is publicly available and it has been widely used in other research work and therefore a good reference with realistic f lexibility compared with most state-of-the-art blades. A non-linear structural blade model23 and an unsteady aerodynamic model25 are then linearized about the steadystate def lected blade, preserving the main effects of the geometric non-linearities. The linear model is discritized by the f inite difference scheme which along with boundary conditions form a differential eigenvalue problem. The solution to this eigenvalue problem gives the aeroelastic frequencies and damping, but also information concerning the fundamental aeroelastic behaviour of the blade. The analysis shows that the aeroelastic damping of the edgewise modes changes when the steady-state def lection is included. The aeroelastic motion is analysed in detail for three different operation conditions in which there is large differences in the damping when including or excluding steady-state blade def lections. +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. + +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8-10，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TURBU11，考虑了几何非线性效应。Riziotis等12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13，他们通过翼梢弯曲来引入翼叶摆振-扭转耦合。 + +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14–18。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，FAST18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HAWC214,15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11,12,19–21**。HAWCStab代码19,21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11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12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22。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。 + +本文使用了一种非线性叶片模型23，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Dowell1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）24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23和非稳态气动模型25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。 + # 2. STRUCTURAL BLADE MODEL -The structural blade model described in Kallesøe23 is based on the work by Hodges and Dowell1 using secondorder Bernoulli–Euler beam theory to describe the blade motion by a non-linear partial integral-differential equation of motion +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 +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。 + + $$ -\vec{\bf M}\ddot{\bf u}+\overline{{{\bf F}}}\left(\dot{\bf u},\overline{{{\bf u}}}^{\prime\prime},\overline{{{\bf u}}}^{\prime},\overline{{{\bf u}}},\ddot{\beta},\dot{\beta},\beta,\ddot{\phi},\dot{\phi},\phi\right)\!=\overline{{{\bf f}}}\left({\bf f}_{a e r o},M_{a e r o},u^{\prime},\nu^{\prime}\right) +\overline{\bf M}\ddot{\overline{\bf u}}+\overline{{{\bf F}}}\left(\dot{\overline{\bf u}},\overline{{{\bf u}}}^{\prime\prime},\overline{{{\bf u}}}^{\prime},\overline{{{\bf u}}},\ddot{\beta},\dot{\beta},\beta,\ddot{\phi},\dot{\phi},\phi\right)\!=\overline{{{\bf f}}}\left({\bf f}_{a e r o},M_{a e r o},u^{\prime},\nu^{\prime}\right) $$ -where $\bar{\bf M}$ is the mass matrix, $\bar{\mathbf{F}}$ is a non-linear function that includes stiffness, damping, gyroscopic terms together with centrifugal force-based integral terms. The state vector $\bar{\mathbf{u}}=[u\left(t,s\right),\nu\left(t,s\right),\theta\left(t,s\right)]$ holds edgewise, f lapwise and torsional deformations, respectively. +where $\bar{\bf M}$ is the mass matrix, $\bar{\mathbf{F}}$ is a non-linear function that includes stiffness, damping, gyroscopic terms together with centrifugal force-based integral terms. The state vector $\bar{\mathbf{u}}=[u\left(t,s\right),\nu\left(t,s\right),\theta\left(t,s\right)]$ holds edgewise, flapwise and torsional deformations, respectively. -Flapwise is def ined 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 +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。 + +Flapwise is defined as the direction normal to the rotor plane (positive downwind) and edgewise as in the rotor plane (positive towards leading edge) for a blade at zero pitch. When the blade pitches, the $(u,\,\nu)$ frame follows the blade. The position along the blades elastic axis is denoted $s$ , $t$ is the time, $\beta=\beta(t)$ is the global pitch of the blade, $\phi=\phi(t)$ is the azimuth angle of the rotor and the right hand side force function $\bar{\mathbf{f}}$ holds the effect of the aerodynamic forces $\mathbf{f}_{a e r o}$ and aerodynamic moment $M_{a e r o}$ on the blade. The dots denote time derivatives and the primes denote derivatives with respect to the longitudinal coordinate $s$ . As an example, the equation of motion for edgewise blade bending is given by + +叶片挥舞方向定义为垂直于风轮平面的方向（顺风方向为正），摆振方向定义为在风轮平面内（前缘方向为正），当叶片变桨角度为零时。当叶片变桨时，$(u,\,\nu)$坐标系跟随叶片运动。叶片弹性轴上的位置用$s$表示，$t$表示时间，$\beta=\beta(t)$表示叶片的全局变桨角度，$\phi=\phi(t)$表示风轮的方位角，右侧力函数$\bar{\mathbf{f}}$包含作用于叶片的气动力$\mathbf{f}_{a e r o}$和气动力矩$M_{a e r o}$的影响。点表示时间导数，撇号表示对纵向坐标$s$的导数。例如，摆振叶片弯曲的运动方程为： $$ \begin{array}{r l}&{m\big(\ddot{u}-\ddot{\theta}l_{c g}\sin\big(\overline{{\theta}}\big)\big)+F_{u,1}\big(\ddot{\beta},\dot{\beta},\dot{\phi},\dot{\nu},\dot{\theta},u^{\prime},u,\theta,\beta\big)+F_{u,2}\big(\dot{\phi},\dot{u},\dot{\nu},u^{\prime},\nu,\theta,\beta\big)}\\ &{\quad+\,F_{u,3}(\phi,\beta,\theta,u^{\prime},\nu^{\prime})+F_{u,4}(u^{\prime\prime},\nu^{\prime\prime},\theta)+F_{u,5}\big(\ddot{\phi},\beta\big)}\\ &{\quad=f_{u}+\Big((u^{\prime}+l_{p i}^{\prime})\int_{s}^{R}f_{w}\mathrm{d}\rho\Big)^{\prime}}\end{array} $$ -where the f irst term is the inertia forces, the second term $F_{u,1}$ describes the inf luence 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 inf luence form gravity, which is neglected in this work. The f ifth term describes the restoring forces - +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 +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。 $$ \begin{array}{r l}&{F_{u,4}\!=\!\big(E\big(I_{\xi}\!\cos^{2}\!(\tilde{\theta})\!+I_{\eta}\sin^{2}\!(\tilde{\theta})\big)u^{\prime\prime}\big)^{\prime\prime}+\!\big(E(I_{\xi}\!-I_{\eta})\!\cos\!(\tilde{\theta})\!\sin\!(\tilde{\theta})\nu^{\prime\prime}\big)^{\prime\prime}}\\ &{\qquad-\big(E(I_{\xi}\!-I_{\eta})\theta\big(u^{\prime\prime}\!\sin\!\big(2\tilde{\theta}\big)\!-\nu^{\prime\prime}\!\cos\!\big(2\tilde{\theta}\big)\!+I_{p i}^{\prime\prime}\sin\!\big(\tilde{\theta}\big)\!\cos\!(\tilde{\theta})\!\big)\big)^{\prime\prime}}\end{array} $$ -where the f irst 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 inf luence 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 +where the first term is the bending stiffness in the $x$ -direction, the second term is the coupling to the $y$ -direction and the last term in equation (3) is the coupling to the twist. The sixth term in equation (2) describes the influence of rotor speed variations, which is assumed constant in this work, so the term is not active. The right hand side holds the external forces, which in this case will be aerodynamic forces. Longitudinal forces on and in the blade, for example the centrifugal force, lead to integral terms in the equations of motion. A detailed description of all terms are found in Kallesøe.23 -The boundary conditions employed in this paper are for simplif ication derived for blades without pre-curvature. The boundary conditions for the root of the blade are given by the geometric constraints +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 +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 +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。 $$ u(0,t)=u^{\prime}(0,t)=\nu(0,t)=\nu^{\prime}(0,t)=\theta(0,t)=0 $$ because the frame used to describe the blade follows the root of the blade. The boundary conditions for the tip of the blade are23 +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. - +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 To determine the steady-state def lection for the blade, a non-linear steady-state aeroelastic model i.s derived. Steady-state conditions are def ined as uniform inf low, zero gravity, constant rotor speed and pitch an.gle $\ddot{\phi}=\dot{\beta}=0$ whereby all time derivatives in the structural equations of motion (1) become zero $\ddot{u}=\ddot{\nu}=\ddot{\theta}=\dot{u}=\dot{\nu}=\dot{\theta}=0$ . These uniform conditions remove the periodicity of the system. The steady-state aerodynamic model is based on blade element momentum (BEM) theory including Prendtl’s tip loss correction.26 The BEM theory computes a balance between the forces on the blade and the momentum change in the wind. The aerodynamic model is coupled to the structural model through the local wind speed and angle of attack and the structural model is coupled to the aerodynamic model through the aerodynamic forces acting on the blade. - +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. @@ -93,32 +112,41 @@ The boundary conditions for the f inite difference formulation are derived by in 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 f inite difference discretized steady-state equation (equation (6)) has 3N unknown blade def lections (f lapwise, 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. +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。
Tablel.Second-orderfinitedifferenceformulationforuniformstepsize. fi+(t)-f-{(t)
f’(s, t) dsaf(s, t)
2h
f"(s, t) ²f(s, t)fi+1(t)-2f(t)+f-(t)
ds2h2
-f-2(t)+2f_(t)-2fi+1(t)+fi+2(t)
f"(s, t) a²f(s, t) ds32h3
f"(s, t)a4f(s, t)
ds4f-2(t)-4f-(t)+6f;(t)-4f+1(t)+f+2(t)
h4
![](ab42554a7d5e39cc6f7a030cf6653410bfdc8ab71835958c84d16e0ccaa7f153.jpg) 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 def lection 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 f lapwise 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 $(\approx\!11\ensuremath{\mathrm{~m~s~}^{-1}})$ at the blade tip def lection curve is caused by the shift from variable speed, constant pitch to constant speed, variable pitch operation. +# 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 f lapwise 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 f lapwise and torsional motion instead of edgewise and torsion as characterized by the f lapwise def lection. - +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 def lected 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 +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。 $$ \left(\left(E I_{\xi}-E I_{\eta}\right)\cos\left(\tilde{\theta}+\theta\right)\sin\left(\tilde{\theta}+\theta\right)\nu^{\prime\prime}\right)^{\prime\prime} @@ -130,7 +158,7 @@ $$ \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 $$ -when linearized about the def lected blade (using $\theta=\theta_{0}+\theta_{1}$ and $\nu=\nu_{0}+\nu_{1}$ ), whereby the important coupling between edgewise and torsional blade motion of a f lapwise def lected blade is preserved. The subscript 1 denotes the linear variation around the linearization point $\mathbf{u}_{0}$ . Likewise the non-linear term in the torsional equation +when linearized about the deflected blade (using $\theta=\theta_{0}+\theta_{1}$ and $\nu=\nu_{0}+\nu_{1}$ ), whereby the important coupling between edgewise and torsional blade motion of a flapwise deflected blade is preserved. The subscript 1 denotes the linear variation around the linearization point $\mathbf{u}_{0}$ . Likewise the non-linear term in the torsional equation $$ (E I_{\xi}-E I_{\eta})\cos(2\big(\tilde{\theta}+\theta\big)\big)u^{\prime\prime}\nu^{\prime\prime} diff --git a/线性化求解器/参考文献/Kallesøe-Equations of motion for a rotor blade/auto/Kallesøe-Equations of motion for a rotor blade.md b/线性化求解器/参考文献/Kallesøe-Equations of motion for a rotor blade/auto/Kallesøe-Equations of motion for a rotor blade.md new file mode 100644 index 0000000..df0bbb6 --- /dev/null +++ b/线性化求解器/参考文献/Kallesøe-Equations of motion for a rotor blade/auto/Kallesøe-Equations of motion for a rotor blade.md @@ -0,0 +1,778 @@ +# Equations of Motion for a Rotor Blade, Including Gravity, Pitch Action and Rotor Speed Variations + +B. S. Kallesøe\*, Department of Mechanical Engineering, Technical University of Denmark, Nils Koppels Allé, Building 404, DK-2800 Lyngby, Denmark + +# Key Words: + +horizontal axis turbines; blade dynamic + +This paper extends Hodges–Dowell’s partial differential equations of blade motion, by including the effects from gravity, pitch action and varying rotor speed. New equations describing the pitch action and rotor speeds are also derived. The physical interpretation of the individual terms in the equations is discussed. The partial differential equations of motion are approximated by ordinary differential equations of motion using an assumed mode method. The ordinary differential equations are used to simulate a sudden pitch change of a rotating blade. This work is a part of a project on pitch blade interaction, and the model will be extended to include an aerodynamic model and be used for analysis of basic properties of pitch blade interaction. +本文扩展了 Hodges–Dowell 的叶片运动偏微分方程，考虑了重力、变桨动作和风轮转速变化的影响。还推导了描述变桨动作和风轮转速的新方程。讨论了方程中各项的物理意义。利用假设模态法，将偏微分方程近似为常微分方程。常微分方程被用于模拟旋转叶片的突变桨动作。这项工作是叶片变桨相互作用项目的一部分，该模型将进一步扩展以包含气动模型，并用于分析叶片变桨相互作用的基本特性。 +Copyright $\copyright$ 2007 John Wiley & Sons, Ltd. + +Received 13 July 2006; Revised 21 November 2006; Accepted 5 December 2006 + +# Introduction + +As wind turbines become larger, the interaction between blade motion, pitch action, rotor speed variations and gravity becomes more pronounced. These interactions can result in increased fatigue loads on, for instance, blade components and pitch actuators. A fundamental analysis of the pitch blade interaction can help in the design of pitch actuators and/or solve pitch bearing problems. In further work, this structural model will be combined with an aerodynamic model. + +Analysis of, for instance, blade pitch interaction can be split into two different approaches: analytical analysis, such as closed form solutions, direct interpretation of terms and perturbation theory, and numerical analysis, such as finite element analysis and computer simulations, with a variety of combinations in between. Numerical approaches give detailed and relatively precise information about a given blade response to a given operation situation. It can, however, be comprehensive to achieve general information about trends and the physics behind the observed effects, because such information relies on a series of simulations. An analytical approaches often give less accurate result than the numerical analysis, because it has to be very simplified, but it allows for studying general trends and physical interpretation. + +In aeroelastic1 and aeroservoelastic2 analyses, 2-D blade section models† are often used. This is because the reduction in complexity especially in the aerodynamic models of a 2-D blade section model compared to a full 3-D model allows more thorough analytical analysis and much faster numerical simulations. + +随着风电机组越来越大，叶片运动、变桨动作、风轮转速变化和重力之间的相互作用变得更加显著。这些相互作用可能导致叶片部件和变桨执行器等部件的疲劳载荷增加。对变桨叶片相互作用进行基本分析有助于设计变桨执行器和/或解决变桨轴承问题。在后续工作中，该结构模型将与气动模型相结合。 + +对例如变桨叶片相互作用的分析可以分为两种不同的方法：解析分析，例如闭合式解、术语的直接解释和微扰理论；以及数值分析，例如有限元分析和计算机模拟，两者之间存在多种组合。数值方法可以提供关于给定叶片在给定运行情况下响应的详细而相对精确的信息。然而，要获得关于趋势和观察到的效应背后物理机制的通用信息，可能需要进行一系列模拟。解析方法通常比数值分析结果不准确，因为需要进行高度简化，但它允许研究一般趋势和进行物理解释。 + +在气弹振动1和气气动弹性2分析中，通常使用二维叶片剖面模型†。这是因为与完整的三维模型相比，二维叶片剖面模型的气动模型复杂性大大降低，从而允许进行更彻底的解析分析和更快的数值模拟。 +The frequently cited paper by Hodges and Dowell3develops the nonlinear partial differential equations of motion for a twisted helicopter rotor blade. Wendell4develops similar partial differential equations of motion focusing on wind turbine applications. Both of these works can handle pre-twisted isotropic blades, but they do not take the interaction with gravity, pitch action and rotor speed variations into account. Their formulation as partial differential equations makes them suitable for analytical analysis. Real turbine blades are made of composite materials, making them anisotropic, leading to internal elastic coupling between different forms of blade motion, which cannot be described by the equations discussed above. The problem by modeling composite materials can be solved by detailed 3-D finite element modeling, which can be done using commercial software. This approach, however, leads to relatively large models with considerable computation time. A turbine blade can also be modeled as a beam, e.g. the reaches code $\mathrm{HAWC}2^{5,6}$ or the commercial code CAMRAD II,7 both combining a finite element beam model with multi-body formulation. By combining a beam model with a multi-body formulation, large deflections and rigid body motion such as pitch action can be taken into account. Cesnik, Hodges and Sutyrin8present the variational asymptotic beam section analysis (VABS). A method for relating the 3-D elastic energy of a composite blade with initial twist and curvature to the strain energy of a 1-D beam description. In Wenbin et al.,9 the method is refined to produce a Timoshenkolike model for the 1-D strain energy based on the 3-D properties of a blade. Wenbin et al.10 show that using this method to describe a composite blade with a beam model produces accurate results comparable to full 3- D finite element code, but with much less computation time. + +被频繁引用的 Hodges 和 Dowell3 论文发展了扭转螺旋桨叶片（叶片）的非线性偏微分运动方程。Wendell4 发展了类似的偏微分运动方程，重点是风电机组（机组）应用。这两项工作都可以处理预扭转的各向同性叶片，但它们没有考虑与重力、变桨角度（变桨角度）和风轮（风轮）转速变化之间的相互作用。由于它们被表述为偏微分方程，因此非常适合进行解析分析。 真实的涡轮叶片由复合材料制成，使其具有各向异性，导致不同叶片运动形式之间的内部弹性耦合，而上述方程无法描述。可以通过详细的 3-D 有限元建模来解决复合材料建模问题，这可以使用商业软件完成。然而，这种方法会导致相对较大的模型和相当大的计算时间。涡轮叶片也可以建模为梁，例如代码 $\mathrm{HAWC}2^{5,6}$ 或商业代码 CAMRAD II,7，两者都将有限元梁模型与多体公式相结合。通过将梁模型与多体公式相结合，可以考虑大变形（变形）和刚体运动，例如变桨角度（变桨角度）。Cesnik、Hodges 和 Sutyrin8 提出了变分渐近梁截面分析 (VABS)。一种将具有初始扭角和曲率的复合叶片的 3-D 弹性能量与 1-D 梁描述的应变能联系起来的方法。在 Wenbin 等人9 的研究中，该方法被改进为基于叶片的 3-D 属性，生成了一种类似于 Timoshenko 的 1-D 应变能模型。Wenbin 等人10 表明，使用该方法通过梁模型描述复合叶片可以产生与全 3-D 有限元代码相当的准确结果，但计算时间大大缩短。 + +This work is a part of a project on describing the interaction between pitch action and blade motion, focusing on control applications. The contribution of this work is to present a model which will be used to analyze the basic properties of interaction between pitch action, gravity effects, rotor speed variations and blade motion. + +The blade model is similar to the partial differential equations of motion developed by Hodges3 and Dowell extended to take pitch action, rotor speed variations and gravity into account. Further, new models for the pitch action and rotor speed are derived. Because the model is intended to be used for first-hand analysis of basic properties of the blade pitch interaction, it rejects features important for a detailed description such as tower motion, yaw error and motion. As a first approach and to keep the equations transparent and simple, the elastic energy is described by Bernoulli–Euler theory, not taking anisotropic and warping effects into account. The elastic energy could instead be described with the more correct and detailed but also more comprehensive description proposed by Cesnik et al.8 and Wenbin et al.9 The formulation of the partial differential equations of motion adopted here leads to a rather comprehensive formulation, compared to that of Wenbin et al,10 but the detailed notation allows direct interpretation and analysis of individual terms in the equations. The equations are fully written out, and all terms are given a physical interpretation and discussed. A finite difference discretization of the model is used to compute frequencies and shapes for natural vibrations of a test blade. The partial differential equations of motion are transformed into approximating ordinary differential equations of motion by the method of assumed modes, which preserve the possibility of analytical analysis. The approximating ordinary differential equations of motion have a structure similar to the equations of motion for a 2- D blade section; hence, the model can be used to transform properties from a blade model to a 2-D blade section model. The pitch angle and rotor speed in the blade equations can be prescribed, given by external models or described by the pitch action and the rotor speed models derived in this work. The pitch action model can be combined with the blade model giving a pitch angle controlled by a pitch moment, or the pitch moment for a given pitch action and blade motion. The present rotor speed model can be expanded to include more blades, leading to a coupling between the individual blades motion. + +The following section presents the model. In the third section, the equations of motion are derived using Hamilton’s principle and discussed. In the fourth section, modes of natural vibrations of the blade are found and the blade is approximated by an assumed mode approximation, which is used in a test example. + +这项工作是关于描述俯仰动作和叶片运动之间相互作用的项目的一部分，重点在于控制应用。这项工作的贡献在于提出一个模型，用于分析俯仰动作、重力效应、风轮转速变化和叶片运动之间的基本特性。 + +叶片模型类似于 Hodges3 和 Dowell 开发的偏微分运动方程，并扩展到考虑俯仰动作、风轮转速变化和重力。此外，还推导了新的俯仰动作和风轮转速模型。由于该模型旨在用于对叶片俯仰相互作用的基本特性进行初步分析，因此它排除了详细描述所需的特征，例如塔架运动、偏航误差和运动。作为初步方法，并为了保持方程的透明和简单，弹性能量由 Bernoulli–Euler 理论描述，而不考虑各向异性和翘曲效应。弹性能量也可以使用 Cesnik et al.8 和 Wenbin et al.9 提出的更准确、更详细但更全面的描述来描述。此处采用的偏微分运动方程的公式化，与 Wenbin et al.10 的公式化相比，具有相当全面的特性，但详细的符号允许直接解释和分析方程中的各个项。方程被完全写出，并且所有项都给出物理意义并进行讨论。使用有限差分离散化模型来计算测试叶片的自然振动频率和形状。偏微分运动方程通过假设模态法转换为近似常微分运动方程，从而保留了进行分析分析的可能性。近似常微分运动方程的结构类似于 2-D 叶片截面的运动方程；因此，该模型可用于将属性从叶片模型转换为 2-D 叶片截面模型。叶片方程中的俯仰角度和风轮转速可以被规定，由外部模型给出，或由本工作中导出的俯仰动作和风轮转速模型描述。俯仰动作模型可以与叶片模型结合，从而实现由俯仰力矩控制的俯仰角度，或者在给定的俯仰动作和叶片运动下实现俯仰力矩。目前的风轮转速模型可以扩展以包含更多叶片，从而导致各个叶片的运动耦合。 + +以下部分介绍该模型。在第三部分，使用 Hamilton 原则推导运动方程并进行讨论。在第四部分，找到叶片的自然振动模态，并使用假设模态近似对叶片进行近似，该近似用于测试示例中。 + + +# Model Description + +The system consists of a rotating inextensible blade with flapwise, edgewise and torsional degrees of freedom. The blade is exposed to pitch action, varying rotor speed and nonconservative forces (e.g. aerodynamic forces). The rotor speed is associated with a torque and a rotational moment of inertia, describing the generator and drive train without gearing and drive train flexibility. A pitch moment is associated with the pitch action, offering the possibility of controlling the pitch by a pitch moment or monitoring the pitch moment having a prescribed pitch. + +The system does not include the influence from tower and yaw motion, drive train flexibility, precone blade, shaft tilt and warping. The shear center\* and the tension center† of the blade are assumed to coincide. These simplifications are justified because the focus is on analyzing pitch blade interaction, not to give a complete description of a wind turbine. + +Figure 1 (a) shows the blade rotating in the rotor plane. The $Y\cdot$ -axis of the $(X,\,Y,\,Z)$ -frame points downwind and the $(X,Z)$ -axis spans the rotor plane, with the $Z$ -axis pointing upward. Since the tower-top and yaw position are assumed fixed, the $(X,\,Y,\,Z)$ -frame becomes an inertial frame. The $(\hat{x},\hat{y},\hat{z})$ -frame rotates with the hub, such that the $\hat{z}\cdot$ -axis is aligned with the pitch axis $p i$ of the blade and the $\hat{y}$ -axis is aligned with the Y-axis. The angle between the two frames is denoted $\phi$ (the azimuth angle of the rotor). + +Figure 1 (b) shows a cross section of the blade looking outward along the ˆz-axis. The position of the blade is described in the $(x,\,y,\,z)$ -frame, which is rotated $\beta$ (the pitch angle) around the $\hat{z}$ -axis. The elastic principle $(\eta,\,\xi)$ -axis of each blade section, is rotated the angle ${\bar{\theta}}+{\bar{\theta}}$ relative to the $(x,\,z)$ -plane, where $\tilde{{\boldsymbol{\theta}}}=\tilde{{\boldsymbol{\theta}}}\bar{(\mathbf{s})}$ is the pre-twist of the elastic properties and $\theta=\theta(s,\,t)$ is the time-dependent twist of the blade section. + +该系统由一个旋转且不可伸长的叶片组成，具有挥舞、摆振和扭转自由度。叶片暴露于变桨动作、变化的转轮速度和非保守力（例如，气动力）。转轮速度与一个扭矩和一个转动惯量相关，描述了不含齿轮和驱动系统柔顺性的发电机和驱动系统。变桨动作与一个变桨力矩相关，提供了通过变桨力矩控制变桨或监测具有规定变桨的变桨力矩的可能性。 + +该系统不包括来自塔架和偏航运动的影响、驱动系统柔顺性、预锥叶片、主轴倾斜和翘曲。叶片的剪切中心\*和张力中心†被假设重合。这些简化是合理的，因为重点是分析变桨叶片相互作用，而不是提供风电机组的完整描述。 + +图 1 (a) 显示了叶片在转轮平面内旋转。$(X,\,Y,\,Z)$坐标系的$Y\cdot$轴指向迎风方向，$(X,Z)$轴展向转轮平面，$Z$轴指向上方。由于假设塔顶和偏航位置固定，$(X,\,Y,\,Z)$坐标系成为惯性坐标系。$(\hat{x},\hat{y},\hat{z})$坐标系与轮毂一起旋转，使得$\hat{z}\cdot$轴与叶片的变桨轴$p i$对齐，$\hat{y}$轴与Y轴对齐。这两个坐标系之间的角度表示为$\phi$（风轮的方位角）。 + +图 1 (b) 显示了沿$\hat{z}$轴向外的叶片截面图。叶片的位置在$(x,\,y,\,z)$坐标系中描述，该坐标系绕$\hat{z}$轴旋转了$\beta$（变桨角度）。每个叶片截面的弹性主轴 $(\eta,\,\xi)$ 坐标系，相对于 $(x,\,z)$ 平面旋转了 ${\bar{\theta}}+{\bar{\theta}}$ 角度，其中 $\tilde{{\boldsymbol{\theta}}}=\tilde{{\boldsymbol{\theta}}}\bar{(\mathbf{s})}$ 是弹性特性的预扭角，$\theta=\theta(s,\,t)$ 是叶片截面的随时间变化的扭角。 + +The position of the elastic axis ea in the $(x,\ y,\ z)$ -frame is given by $(u+l_{p i},\,\nu,\,w)$ , where $u=u(s,\,t)$ and $\nu=\nu(s,\,t)$ are the deflection from the undeformed position in the $x\cdot$ - and $y_{\mathrm{~\,~}}$ -direction, respectively, and $l_{p i}=$ $l_{p i}(s)$ is the undeformed position of ea on the $x$ -axis. The position in the $z$ -direction is given by $w=\int_{r}^{s}\sqrt{\left(1-\left(l_{p i}^{\prime}+u^{\prime}\right)^{2}-{\nu^{\prime}}^{2}\right)}\mathrm{d}s,$ , based on the inextensibility of the blade. The $\boldsymbol{w}$ coordinate is split into a static part $w_{0}=\int_{r}^{s}\sqrt{1-l_{p i}^{\prime2}}\,\mathrm{d}s$ and an approximation to the time-dependent part $\begin{array}{r}{w_{1}\!=-\!\frac{1}{2}\!\int_{r}^{s}\sqrt{{u^{\prime}}^{2}+{\nu^{\prime}}^{2}+2l_{p i}^{\prime}u^{\prime}}\mathrm{d}s}\end{array}$ . The independent variables $t$ and $s$ are the time and the distance from the root of the blade measured along $e a$ , respectively. The radius of the hub is $r$ and the radius of the rotor is $R$ , measured along the elastic axis. + +在 $(x,\ y,\ z)$ 坐标系中，弹性轴 ea 的位置由 $(u+l_{p i},\,\nu,\,w)$ 给出，其中 $u=u(s,\,t)$ 和 $\nu=\nu(s,\,t)$ 分别是相对于未变形位置在 $x$ 和 $y$ 方向上的变形，而 $l_{p i}=$ $l_{p i}(s)$ 是 ea 在 $x$ 轴上的未变形位置。在 $z$ 方向上的位置由 $w=\int_{r}^{s}\sqrt{\left(1-\left(l_{p i}^{\prime}+u^{\prime}\right)^{2}-{\nu^{\prime}}^{2}\right)}\mathrm{d}s$ 给出，基于叶片不可伸长的条件。$\boldsymbol{w}$ 坐标被分解为静态部分 $w_{0}=\int_{r}^{s}\sqrt{1-l_{p i}^{\prime2}}\,\mathrm{d}s$ 和对随时间变化的量的近似值 $\begin{array}{r}{w_{1}\!=-\!\frac{1}{2}\!\int_{r}^{s}\sqrt{{u^{\prime}}^{2}+{\nu^{\prime}}^{2}+2l_{p i}^{\prime}u^{\prime}}\mathrm{d}s}\end{array}$ 。 独立的变量 $t$ 和 $s$ 分别是时间以及沿 $e a$ 测量的从根到距离，叶轮的半径为 $R$， +![](aef42751cc21fe8ad665c9d3ad833ead7a355e73bfde4c5d791fb2c9156e1c47.jpg) +Figure 1. (a) The inertial $\left(X,\,Y,\,Z\right)$ -frame and the rotating $(\hat{x},\hat{y},\hat{z})$ -frame with the ˆz-axis aligned with the pitch axis of the blade. The external forces $(f_{w}\,f_{\nu},f_{w})$ act at the elastic axis in the $(\hat{x},\,\hat{y},\,\hat{z})$ -directions, respectively. (b) Cross section of the blade looking outward along the $\hat{z}$ -axis +图 1. (a) 惯性坐标系 (X, Y, Z) 和旋转坐标系 (ˆx, ˆy, ˆz)，其中ˆz轴与叶片变桨角度轴对齐。外力 (fw, fν, fw) 分别沿 (ˆx, ˆy, ˆz) 方向作用于弹性轴。(b) 沿 ˆz 轴向外看的叶片截面 + +The sum of rotational inertia of the hub, gearbox and generator is described by $J_{g e n}$ . The inertia of the blade is described by a concentrated mass $m=m(s)$ and a moment of rotational inertia $I_{c g}=I_{c g}(s)$ (for rotation in the cross section plane) for each blade section, both related to the center of gravity $c g$ . The center of gravity is assumed to be located on the chord, the distance $l_{c g}=l_{c g}(s)$ from ea. The chord is rotated, the angle ${\overline{{\theta}}}+\theta$ relative to the $(x,z)$ -plane, where $\overline{{\theta}}=\overline{{\theta}}(\mathrm{s})$ is the pre-twist of the chord. + +The external forces, such as aerodynamic forces, on the blade are described by four components; three forces $(f_{u},f_{\nu},f_{w})=(f_{u}(s,\,t),f_{\nu}(s,\,t),f_{w}(s,\,t))$ in the $(x,y,z)$ -directions, respectively and a twisting moment $M=M(s,\,t)$ . The forces act at the elastic axis of the blade. + +The pitch moment $M_{p i t c h}$ is associated with this pitch angle rotation and the generator torque is given by $T_{g e n}$ . +In summary, the state of the system is given by $(u,\nu,\,\theta,\,\beta,\,\phi)$ where $(\phi,\beta)$ can be prescribed, given by external models or described by the derived equations. The system is exposed to the external loads $(f_{u},f_{\nu},f_{w},\,M_{}$ , $T_{g e n},M_{p i t c h})$ , where $(T_{g e n},\,M_{p i t c h})$ only affects the $(\phi,\beta)$ equations, respectively. +叶片旋转惯性矩之和由 $J_{g e n}$ 描述。每个叶片段的惯性由集中质量 $m=m(s)$ 和旋转惯性矩 $I_{c g}=I_{c g}(s)$ (在截面平面内旋转) 描述，两者均与重心 $c g$ 相关。假设重心位于弦线上，弦线到 ea 的距离为 $l_{c g}=l_{c g}(s)$。弦线旋转，相对于 $(x,z)$ 平面的角度为 ${\overline{{\theta}}}+\theta$，其中 $\overline{{\theta}}=\overline{{\theta}}(\mathrm{s})$ 是弦线的预扭角。 + +叶片上的外部力，例如气动力，由四个分量描述；三个力 $(f_{u},f_{\nu},f_{w})=(f_{u}(s,\,t),f_{\nu}(s,\,t),f_{w}(s,\,t))$ 分别作用于 $(x,y,z)$ 方向，以及一个扭矩 $M=M(s,\,t)$。这些力作用在叶片的弹性轴上。 + +俯仰力矩 $M_{p i t c h}$ 与此俯仰角度旋转相关，发电机扭矩由 $T_{g e n}$ 给出。 +总而言之，系统的状态由 $(u,\nu,\,\theta,\,\beta,\,\phi)$ 给出，其中 $(\phi,\beta)$ 可以由外部模型规定，或由推导出的方程描述。系统暴露于外部载荷 $(f_{u},f_{\nu},f_{w},\,M_{}$ , $T_{g e n},M_{p i t c h})$，其中 $(T_{g e n},\,M_{p i t c h})$ 分别仅影响 $(\phi,\beta)$ 方程。 +# Derivation of the Equations of Motion + +The derivation of the equations of motion follows the method used in Hodges and Dowell.3 First, the potential and kinetic energies for the system are set-up, then the equations of motion and boundary condition equations are derived from these energy expressions using the extended Hamilton’s principle.11 +运动方程的推导遵循 Hodges 和 Dowell 所用的方法。³ 首先，建立系统的势能和动能，然后利用扩展的哈密顿原理，从这些能量表达式中推导出运动方程和边界条件方程。¹¹ +# Order Scheme + +To avoid unnecessary complications of the equations of motion, relatively small terms are neglected. This is done in a consistent manner by introducing an ordering scheme, assuming $\left(\frac{u}{R},\frac{\nu}{R},\frac{l_{p i}}{R},\frac{l_{c g}}{R},\theta,c\tilde{\theta}^{\prime},c\overline{{{\theta}}}^{\prime},\frac{m^{\prime}l_{c g}}{m l_{c g}^{\prime}}\right)$ to be of order $\varepsilon,$ , where $c=c(s)$ is the local chord, $\varepsilon<<1$ is a bookkeeping parameter denoting the smallness of terms, $(\mathbf{\partial})\equiv{\frac{\mathrm{d}}{\mathrm{d}t}}\,$ and $(\mathbf{\omega})^{'}\equiv\frac{\mathrm{d}\mathbf{\omega}}{\mathrm{d}s}$ . The angular acceleration of the rotor is assumed to be $\ddot{\phi}{\cal R}\sim i i$ . The ordering scheme is applied such that terms of order $\varepsilon^{\mathrm{n+2}}$ or higher are neglected, where $n$ is the lowest order of a term in the expression. +为了避免运动方程的过度复杂化，相对较小的项被忽略。这通过引入排序方案以一致的方式进行，假设 $\left(\frac{u}{R},\frac{\nu}{R},\frac{l_{p i}}{R},\frac{l_{c g}}{R},\theta,c\tilde{\theta}^{\prime},c\overline{{{\theta}}}^{\prime},\frac{m^{\prime}l_{c g}}{m l_{c g}^{\prime}}\right)$ 为 $\varepsilon$ 阶，其中 $c=c(s)$ 是局部叶片弦长，$\varepsilon<<1$ 是一个记账参数，表示项的微小程度，$(\mathbf{\partial})\equiv{\frac{\mathrm{d}}{\mathrm{d}t}}\,$ 和 $(\mathbf{\omega})^{'}\equiv\frac{\mathrm{d}\mathbf{\omega}}{\mathrm{d}s}$ 。风轮的角加速度被假定为 $\ddot{\phi}{\cal R}\sim i i$ 。排序方案应用于使阶数为 $\varepsilon^{\mathrm{n+2}}$ 或更高阶的项被忽略，其中 $n$ 是表达式中项的最低阶数。 + + +# Transformations + +Before deriving the equations of motion, a transformation between the rotating $(x,\,y,\,z)$ -frame in which the blade deflection is described and the inertial $(X,\,Y,\,Z)$ -frame is found: +在推导运动方程之前，需要找到旋转坐标系 $(x,\,y,\,z)$——用于描述叶片变形的坐标系，与惯性坐标系 $(X,\,Y,\,Z)$ 之间的转换关系： +$$ +[\mathbf{i},\mathbf{j},\mathbf{k}]^{\mathrm{T}}\,{=}\,\mathbf{T}_{\beta}\mathbf{T}_{\phi}[\mathbf{I},\mathbf{J},\mathbf{K}]^{T} +$$ + +where $[\mathbf{i},\mathbf{j},\mathbf{k}]^{\mathrm{T}}$ and $[\mathbf{I},\mathbf{J},\mathbf{K}]^{\mathrm{T}}$ are the unit vectors in the $(x,y,z)$ and $(X,Y,Z)$ -frames, respectively. The matrices ${\bf{T}}_{\beta}$ and $\mathbf{T}_{\phi}$ are the transformations from the $(\hat{x},\,\hat{y},\,\hat{z})$ -frame to the $(x,\,y,\,z)$ -frame and from the $\left(X,\,Y,\,Z\right)$ - frame to the $\left(\hat{x},\hat{y},\hat{z}\right)$ -frame, respectively. Both matrices are given in Appendix A. + +The transformation between the principle axis and the $(x,y,z)$ -frame is given by ${\bf{T}}_{e}$ and between the chord and the $(x,y,z)$ -frame is given by ${{\bf{T}}_{c}}$ . Both matrices are given in Appendix A. +其中 $[\mathbf{i},\mathbf{j},\mathbf{k}]^{\mathrm{T}}$ 和 $[\mathbf{I},\mathbf{J},\mathbf{K}]^{\mathrm{T}}$ 分别是 $(x,y,z)$ 和 $(X,Y,Z)$ 坐标系中的单位向量。矩阵 ${\bf{T}}_{\beta}$ 和 $\mathbf{T}_{\phi}$ 分别是从 $(\hat{x},\,\hat{y},\,\hat{z})$ 坐标系到 $(x,\,y,\,z)$ 坐标系的变换，以及从 $\left(X,\,Y,\,Z\right)$ 坐标系到 $\left(\hat{x},\hat{y},\hat{z}\right)$ 坐标系的变换。这两个矩阵见附录A。 + +主轴到 $(x,y,z)$ 坐标系的变换由 ${\bf{T}}_{e}$ 表示，弦到 $(x,y,z)$ 坐标系的变换由 ${{\bf{T}}_{c}}$ 表示。这两个矩阵见附录A。 +# Potential Energy + +The strain in the blade is measured by Green’s strain tensor (cf. Hodges and Dowell3): +叶片应变由格林应变张量测量（参阅 Hodges and Dowell3）： +$$ +2[\mathrm{d}s,\mathrm{d}\eta,\mathrm{d}\xi][\varepsilon_{i j}][\mathrm{d}s,\mathrm{d}\eta,\mathrm{d}\xi]^{\mathrm{T}}=\mathrm{d}\mathbf{r}_{\mathrm{l}}\cdot\mathrm{d}\mathbf{r}_{\mathrm{l}}-\mathrm{d}\mathbf{r}_{\mathrm{0}}\cdot\mathrm{d}\mathbf{r}_{\mathrm{0}} +$$ + +where d denotes the differential, $\varepsilon_{i j}$ is the strain tensor and +其中 d 表示微分，$\varepsilon_{i j}$ 为应变张量，且 +$$ +\mathbf{r}_{0}\!=\![\mathbf{I},\mathbf{J},\mathbf{K}]\mathbf{T}_{\phi}^{\mathrm{T}}\mathbf{T}_{\beta}^{\mathrm{T}}\!\left[[l_{p i},0,w_{0}]^{\mathrm{T}}+(\mathbf{T}_{e}|_{u=\nu=\theta=0})^{\mathrm{T}}[\eta_{0},\xi_{0},0]^{\mathrm{T}}\right] +$$ + +is a position vector describing a point in the undeformed blade, where $(\eta_{0},\,\xi_{0})$ is the position of the point in the undeformed blade section. The same point in the deformed blade is given by +是描述未变形叶片中某一点的位置向量，其中$(\eta_{0},\,\xi_{0})$是该点在未变形叶片剖面的位置。该点在变形叶片中的位置表示为： +$$ +\mathbf{r}_{\mathrm{1}}\!=\![\mathbf{I},\mathbf{J},\mathbf{K}]\mathbf{T}_{\phi}^{\mathrm{T}}\mathbf{T}_{\beta}^{\mathrm{T}}\big[[l_{p i}+u,\nu,w_{0}+w_{\mathrm{1}}]^{\mathrm{T}}+\mathbf{T}_{\mathrm{c}}^{\mathrm{T}}{[\eta_{\mathrm{1}},\xi_{\mathrm{1}},0]}^{\mathrm{T}}\big] +$$ + +where $(\eta_{1},\,\xi_{1})$ is the position of the point in the deformed blade section. +其中 $(\eta_{1},\,\xi_{1})$ 为叶片变形段中点的坐标。 + +Assuming uniaxial stress $\sigma_{22}=\sigma_{33}=\sigma_{23}=0$ , where $\sigma_{i j}$ is the stress tensor. Applying Hook’s law gives $\varepsilon_{22}$ $=\varepsilon_{33}=-\nu\varepsilon_{11}$ , where $\nu$ is Poisson’s ratio. By expanding these relations to second order of the bookkeeping parameter $\varepsilon$ , it can be shown that $\eta_{1}=\eta_{0}$ and $\xi_{1}=\xi_{0}$ to second order. Expanding the remanding strain tensor components to second order of $\varepsilon$ gives +假设单轴应力 $\sigma_{22}=\sigma_{33}=\sigma_{23}=0$ ，其中 $\sigma_{i j}$ 为应力张量。应用胡克定律得到 $\varepsilon_{22}$ $=\varepsilon_{33}=-\nu\varepsilon_{11}$ ，其中 $\nu$ 为泊松比。通过将这些关系展开到簿记参数 $\varepsilon$ 的二阶，可以证明 $\eta_{1}=\eta_{0}$ 且 $\xi_{1}=\xi_{0}$ 到二阶。将剩余应变张量分量展开到 $\varepsilon$ 的二阶，得到 +$$ +\begin{array}{l l}{{\displaystyle\varepsilon_{11}\!=\!-u^{\prime\prime}\big(\eta\cos(\overline{{\theta}})-\xi\sin\!\big(\hat{\theta}\big)\big)-\nu^{\prime\prime}\big(\eta\sin\!\big(\hat{\theta}\big)+\xi\cos\!\big(\overline{{\theta}}\big)\big)}}\\ {{\displaystyle\varepsilon_{12}\!=\!-\frac{1}{2}\xi\theta^{\prime},\quad\!\varepsilon_{13}\!=\!\frac{1}{2}\eta\theta^{\prime}}}\end{array} +$$ + +Using engineering strain $\varepsilon_{s s}=\varepsilon_{11}$ , $\varepsilon_{s\eta}=2\varepsilon_{12}$ , $\varepsilon_{s\xi}=2\varepsilon_{13}$ and stresses $\sigma_{\mathrm{ss}}=E\varepsilon_{s s},\,\sigma_{\mathrm{s}\eta}=G\varepsilon_{s\eta},\,\sigma_{s\xi}=G\varepsilon_{s\xi}$ where $E$ is the tensile modulus of elasticity (Young’s modulus) and $G$ is the shear modulus of elasticity, the elastic energy becomes +使用工程应变 $\varepsilon_{s s}=\varepsilon_{11}$ , $\varepsilon_{s\eta}=2\varepsilon_{12}$ , $\varepsilon_{s\xi}=2\varepsilon_{13}$ 和应力 $\sigma_{\mathrm{ss}}=E\varepsilon_{s s},\,\sigma_{\mathrm{s}\eta}=G\varepsilon_{s\eta},\,\sigma_{s\xi}=G\varepsilon_{s\xi}$ ，其中 $E$ 为抗拉弹性模量（杨氏模量），$G$ 为抗剪弹性模量，弹性势能变为： +$$ +\delta V_{e l a}=\int_{r}^{R}\iint_{A}{(\sigma_{s s}\delta\varepsilon_{s s}+\sigma_{s\eta}\delta\varepsilon_{s\eta}+\sigma_{s\xi}\delta\varepsilon_{s\xi})\mathrm{d}\eta\mathrm{d}\xi\mathrm{d}s} +$$ + +The potential energy associated with the gravity field measured from the inertial frame $\left(X,\,Y,\,Z\right)$ is described by +与惯性系 $\left(X,\,Y,\,Z\right)$ 测量的重力场相关的势能由以下公式描述： + + +$$ +V_{g r a}=\int_{r}^{R}\mathbf{r}_{c g}^{T}\cdot\mathbf{g}\mathrm{d}s +$$ + +where $\mathbf{g}=[0,\,0,\,-g]^{\mathrm{T}}$ is the gravity field and + +$$ +\mathbf{r}_{c g}\!=\![\mathbf{I},\mathbf{J},\mathbf{K}]\mathbf{T}_{\phi}^{\mathrm{T}}\mathbf{T}_{\beta}^{\mathrm{T}}\big[[l_{p i}+u,\nu,w_{0}+w_{1}]^{\mathrm{T}}+\mathbf{T}_{\mathrm{c}}^{\mathrm{T}}[l_{c g},0,0]^{\mathrm{T}}\big] +$$ + +is a position vector describing the center of gravity. + +# Kinetic Energy + +The inertia of the system is described by a mass pr. length $m$ , a moment of rotational inertia pr. length $I_{c g}$ of the blade and a moment of rotational inertia $J_{g e n}$ that describes the hub, gear box and generator. The use of concentrated mass description of the blade inertia, instead of a more general description integration over the cross section, leads much to less complexity in the derivation. A general description will lead to extra terms, such as rotational inertiae about $x-$ and $y_{\mathrm{~\,~}}$ -axis, but these terms turn out to be relatively small anyway. The kinetic energy of the system is given by +系统的惯性由单位长度的质量 $m$，叶片（blade）的单位长度转动惯量 $I_{c g}$ 以及描述（hub）、齿轮箱和发电机的转动惯量 $J_{g e n}$ 来描述。采用集中质量描述叶片的惯性，而不是更一般的横截面积分描述，可以大大简化推导过程。更一般的描述会导致额外的项，例如关于 $x-$ 和 $y_{\mathrm{~\,~}}$ -轴的转动惯量，但这些项最终来说相对较小。系统的动能由以下公式给出： +$$ +T\!=\!\frac{1}{2}J_{g c n}\dot{\phi}^{2}+\int_{r}^{R}\!\Big(\frac{1}{2}m\mathbf{r}_{c g}^{\mathrm{T}}\cdot\dot{\mathbf{r}}_{c g}+\frac{1}{2}I_{c g}\big(\dot{\beta}+\dot{\theta}\big)^{2}\Big)\mathrm{d}s +$$ + +where ${\dot{\beta}}+{\dot{\theta}}$ is the angular velocity of the blade section around the elastic axis. +其中 ${\dot{\beta}}+{\dot{\theta}}$ 是叶片在弹性轴周围的角速度。 + +# Nonconservative Forces非保守力 + +The nonconservative forces are taken into account by describing the variational work done by them for any admissible variation: +非保守力通过描述其对任何可行变动所做的变分功来考虑： +$$ +\delta Q=T_{g c n}\delta\phi+M_{p i t c h}\delta\beta+\int_{r}^{R}(\mathbf{f}^{\mathrm{{T}}}\cdot\delta\mathbf{r}_{e a}+M\delta(\theta+\beta))\mathrm{d}s +$$ + +where $\mathbf{f}=\mathbf{T}_{\phi}^{\mathrm{T}}\mathbf{T}_{\beta}^{\mathrm{T}}[f_{u},f_{\nu},f_{w}]^{\mathrm{T}}$ and + +$$ +\mathbf{r}_{e a}\!=\!\{\mathbf{I},\mathbf{J},\mathbf{K}\}\mathbf{T}_{\phi}^{\mathrm{T}}\mathbf{T}_{\beta}^{\mathrm{T}}[l_{p i}+u,\nu,w_{0}+w_{1}]^{\mathrm{T}} +$$ + +is a position vector describing the elastic axis. + +# Equations of Motion + +By demanding that any admissible variation of the action integral $\delta H\equiv\int_{\mathbf{\Omega}_{t_{1}}}^{\mathbf{\Omega}_{t_{2}}}(\delta T-\delta V_{e l a}-\delta V_{g r a}+\delta Q)d t$ is zero, a set of partial differential equations of motion and a set of boundary condition equations are derived (extended Hamilton’s principle11). The variation of the $w_{1}$ term leads to integral terms in the equations of motion, while the $w_{1}$ itself does not appear because it is relatively small. First, the partial differential equations of blade bending and torsional motion are presented, followed by the corresponding boundary conditions. Second, the equations of motion for the rotor azimuth angle and the pitch angle are presented. + +Blade Bending Motion + +The equation of motion of the $x$ - and $y$ -directions becomes + +$$ +\begin{array}{r l}&{m\big(\ddot{u}-\ddot{\theta}l_{c g}\sin(\overline{{\theta}})\big)+F_{u,1}\big(\ddot{\beta},\dot{\beta},\dot{\phi},\dot{\nu},\dot{\theta},u^{\prime},u,\nu,\theta,\beta\big)+F_{u,2}\big(\dot{\phi},\dot{u},\dot{\nu},u^{\prime},\nu,\theta,\beta\big)}\\ &{\quad+F_{u,3}(\phi,\beta,\theta,u^{\prime},\nu^{\prime})+F_{u,4}(u^{\prime\prime},\nu^{\prime\prime},\theta)+F_{u,5}\big(\ddot{\phi},\beta\big)=f_{u}+\Big((u^{\prime}+l_{p i}^{\prime})\int_{s}^{R}f_{w}\mathrm{d}\rho\Big)^{\prime}}\\ &{\quad m\big(\ddot{\nu}+\ddot{\theta}l_{c g}\cos(\overline{{\theta}})\big)+F_{\nu,1}\big(\ddot{\beta},\dot{\beta},\dot{\phi},\dot{\theta},\nu^{\prime},u,\nu,\theta,\beta\big)+F_{\nu,2}\big(\dot{\phi},\dot{u},\dot{\nu},u^{\prime},\nu,\theta,\beta\big)}\\ &{\quad+F_{\nu,3}\big(\phi,\beta,\theta,u^{\prime},\nu^{\prime}\big)+F_{\nu,4}\big(u^{\prime\prime},\nu^{\prime\prime},\theta\big)+F_{\nu,5}\big(\ddot{\phi},\beta\big)=f_{\nu}+\Big(\nu^{\prime}\int_{s}^{R}f_{w}\mathrm{d}\rho\Big)^{\prime}}\end{array} +$$ + +The direction of the $x_{\mathrm{{}}}$ - and $y$ -axis can be swapped by changing the $\beta$ angle, hence the only differences between the terms in equations (12a) and (12b) are the directions of projection of the forces. In the following, the individual terms in equation (12) are shown and the physical interpretation of them is discussed. Because of the similarity between the terms from equations (12a) and (12b), only the terms from equation (12a) will be discussed. The influence of pitch action is described by + +$$ +F_{u,1}\!=\!-\ddot{\beta}m\nu_{c g}-\dot{\beta}^{2}m u_{c g}-2\dot{\beta}m\dot{\nu}_{c g}+\left(T_{1}l_{c g}\cos(\overline{{{\theta}}})\right)^{\prime}+\left((u^{\prime}+l_{p i}^{\prime})\!\int_{s}^{R}T_{1}\!\mathrm{d}\rho\right)^{\prime} +$$ + +$$ +F_{u,1}\!=\!\ddot{\beta}m u_{c g}-\dot{\beta}^{2}m\nu_{c g}+2\dot{\beta}m\dot{u}_{c g}+\left(T_{1}l_{c g}\sin(\overline{{{\theta}}}\,)\right)^{\prime}+\left(\nu^{\prime}\!\int_{s}^{R}T_{1}\mathrm{d}\rho\right)^{\prime} +$$ + +Where $u_{c g}=u+l_{p i}+l_{c g}\mathrm{cos}(\overline{{\theta}})-l_{c g}\theta\mathrm{sin}(\overline{{\theta}})$ and $\nu_{c g}=\nu+l_{c g}\mathrm{sin}(\overline{{\theta}})+l_{c g}\theta\mathrm{cos}(\overline{{\theta}})$ are the $x$ and $y$ coordinates of the center of gravity in the $(x,y,z)$ -frame, respectively, $\Gamma_{1}=2m\dot{\beta}\dot{\phi}((u+l_{p i})\mathrm{sin}(\beta)+\nu\mathrm{cos}(\beta)+l_{c g}\mathrm{sin}(\overline{{{\theta}}}+\beta))$ is the Coriolis force in the $z-$ -direction, associated with the angular velocity of the $(x,y,z)$ -frame about the $z$ -axis and about the $\hat{y}$ -axis. The first term in equation (13a) is the fictitious force\* in the $x$ -direction associated with the angular acceleration of the $(x,\,y,\,z)$ -frame about the $z-$ -axis. The second term is the fictitious centrifugal force associated with the angular velocity of the $(x,y,z)$ -frame about the $z$ -axis and the offset of $c g$ in the $x$ - direction. The third term is the Coriolis force associated with the angular velocity of the $(x,y,z)$ -frame about the $z$ -axis and the velocity of $c g$ in the $y$ -direction. The fourth term in equation (13a) is the spatial derivative of the moment caused by the offset of $c g$ and the Coriolis force $T_{1}$ . The last term is the bending moment caused by the Coriolis force $T_{1}$ on the remaining part of the blade, from this point to the tip. The influence from the rotor speed is described by + +$$ +\begin{array}{l}{{F_{u,2}=-\dot{\phi}^{2}m\hat{u}_{c g}\cos(\beta)-\dot{\phi}^{2}\big(m l_{c g}w_{0}\big(\cos(\overline{{\theta}})\big)-\theta\sin(\overline{{\theta}})\big)\big)^{\prime}-\left(l_{c g}T_{2}\right)^{\prime}\cos(\overline{{\theta}})}}\\ {{\mathrm{}}}\\ {{\mathrm{}}\qquad-2\dot{\phi}m l_{c g}\big(\dot{u}^{\prime}\cos(\overline{{\theta}})+\dot{\nu}^{\prime}\sin(\overline{{\theta}})\big)\cos(\beta)\mathrm{-}\Big((u^{\prime}+l_{p i}^{\prime})\Big)_{s}^{R}\big(\dot{\phi}^{2}m w_{0}+T_{2}\big)\mathrm{d}\rho\Big)^{\prime}}}\\ {{F_{\nu,2}=\dot{\phi}^{2}m\hat{u}_{c g}\sin(\beta)-\dot{\phi}^{2}\big(m l_{c g}w_{0}\big(\sin(\overline{{\theta}})+\theta\cos(\overline{{\theta}})\big)\big)^{\prime}-\left(l_{c g}T_{2}\right)^{\prime}\sin(\overline{{\theta}})}}\\ {{\mathrm{}}}\\ {{\mathrm{}}\qquad+2\dot{\phi}m l_{c g}\big(\dot{u}^{\prime}\cos(\overline{{\theta}})+\dot{\nu}^{\prime}\sin(\overline{{\theta}})\big)\sin(\beta)\mathrm{-}\Big(\nu^{\prime}\int_{s}^{R}\big(\dot{\phi}^{2}m w_{0}+T_{2}\big)\mathrm{d}\rho\Big)^{\prime}}}\end{array} +$$ + +where $\hat{u}_{c g}=(u+l_{p i})\mathrm{cos}(\beta)-\nu\mathrm{sin}(\beta)+l_{c g}\mathrm{cos}(\overline{{{\theta}}}+\beta)-l_{c g}\theta\mathrm{sin}(\overline{{{\theta}}}+\beta)$ is the $\hat{x}$ coordinate of the center of gravity given in the $(\hat{x},\hat{y},\hat{z})$ -frame, $T_{2}=2m\dot{\phi}(\dot{u}\cos(\upbeta)-\dot{\nu}\sin(\beta))$ is the Coriolis force in the $z$ -direction associated with the rotation in the rotor plane and the velocity of $c g$ in the $\boldsymbol{\hat{x}}$ -direction The first term in equation (14a) is the fictitious centrifugal force associated with the rotation in the rotor plane and the offset of $c g$ in the $x_{\mathrm{{}}}$ -direction projected onto the $x_{\mathrm{{}}}$ -direction. The second and third terms in equation (14a) are the spatial derivative of the moment caused by the distance from $c g$ to $e a$ in the $x$ -direction and the fictitious centrifugal and the Coriolis force $T_{2}$ , respectively. The centrifugal force is associated with the rotation in the rotor plane and the offset of $c g$ from the center of rotation. The fourth term is the fictitious Coriolis force associated with the rotation of the blade in the rotor plane and the velocity of $c g$ in the $\hat{z}$ -direction The last term in equation (14a) is the bending moment from the fictitious centrifugal and the Coriolis force $T_{2}$ on the remaining part of the blade from this point to the tip. The influence from gravity is described by + +$$ +\begin{array}{r l}&{F_{u,3}\!=\!m g\sin(\phi)\cos(\beta)\!+\!\left(\!\left(l_{c g}(u^{\prime}\!+\!l_{p i}^{\prime})\right)^{\prime}\cos(\overline{{\theta}})\cos(\beta)\!+\!\left(l_{c g}\nu^{\prime}\right)^{\prime}\sin(\overline{{\theta}})\cos(\beta)m g\sin(\phi)\!\right.}\\ &{\qquad\left.-\left(m l_{c g}\!\left(\cos(\overline{{\theta}})-\theta\sin(\overline{{\theta}})\right)\right)^{\prime}g\cos(\phi)\!+\!\left((u^{\prime}\!+\!l_{p i}^{\prime})\right)_{s}^{R}m g\cos(\phi)\mathrm{d}\rho\right)^{\prime}}\\ &{F_{u,3}\!=\!-\!m g\sin(\phi)\sin(\beta)\!+\!\left(\!\left(l_{c g}\left(u^{\prime}\!+\!l_{p i}^{\prime}\right)\right)^{\prime}\sin(\overline{{\theta}})\cos(\beta)\!-\!\left(l_{c g}\nu^{\prime}\right)^{\prime}\sin(\overline{{\theta}})\sin(\beta)\!\right)\!m g\sin(\phi)}\\ &{\qquad-\left(m l_{c g}\left(\sin(\overline{{\theta}})-\theta\cos(\overline{{\theta}})\right)\right)^{\prime}g\cos(\phi)\!+\!\left(\nu^{\prime}\!\int_{s}^{R}m g\cos(\phi)\mathrm{d}\rho\right)^{\prime}}\end{array} +$$ + +where the first term in equation (15a) is the $x$ -component of the gravity force. The second term is the spatial derivative of the moment caused by the $\hat{x}$ -component of the gravity force and the offset of $c g$ in the $z$ -direction. The third term is the spatial derivative of the moment caused by the distance between $c g$ and $e a$ in the $x$ -direction and the $z$ -component of the gravity force. The last term in equation (15a) is the bending moment from the $z-$ -component of the gravity force on the remaining part of the blade, from this point to the tip. The restoring force caused by the bending stiffness of the blade is described by + +$$ +\begin{array}{r l}&{F_{u,4}=\big(E\big(I_{\xi}\cos^{2}\big(\tilde{\theta}\big)+I_{\eta}\sin^{2}\big(\tilde{\theta}\big)\big)u^{\prime\prime}\big)^{\prime\prime}+\big(E(I_{\xi}-I_{\eta})\cos\big(\tilde{\theta}\big)\sin\big(\tilde{\theta}\big)\nu^{\prime\prime}\big)^{\prime\prime}}\\ &{\qquad\qquad-\big(E(I_{\xi}-I_{\eta})\theta\big(u^{\prime\prime}\sin\big(2\tilde{\theta}\big)-\nu^{\prime\prime}\cos\big(2\tilde{\theta}\big)+I_{p i}^{\prime\prime}\sin\big(\tilde{\theta}\big)\cos\big(\tilde{\theta}\big)\big)\big)^{\prime\prime}}\\ &{F_{v,4}=\big(E\big(I_{\xi}\sin^{2}\big(\tilde{\theta}\big)+I_{\eta}\cos^{2}\big(\tilde{\theta}\big)\big)\nu^{\prime\prime}\big)^{\prime\prime}+\big(E(I_{\xi}-I_{\eta})\cos\big(\tilde{\theta}\big)\sin\big(\tilde{\theta}\big)u^{\prime\prime}\big)^{\prime\prime}}\\ &{\qquad+\left(E(I_{\xi}-I_{\eta})\theta\big(u^{\prime\prime}\cos\big(2\tilde{\theta}\big)+\nu^{\prime\prime}\sin\big(2\tilde{\theta}\big)\big)\right)^{\prime\prime}-\big(I_{p i}^{\prime\prime}\theta E(I_{\xi}\sin^{2}(\theta)+I_{\eta}\cos^{2}(\theta))\big)^{\prime\prime}}\end{array} +$$ + +where the first term is the bending stiffness in the $x$ -direction, the second term is the coupling to the $\nu$ -direction, and the last term is the coupling to the twist. The principle moments of inertia are given by $I_{\xi}=\int\!\int_{A}\!\eta^{2}\mathrm{d}\eta\mathrm{d}\xi$ and $I_{\eta}=\int\int_{A}\!\xi^{2}\mathrm{d}\eta\mathrm{d}\xi.$ . The effect of an angular acceleration of the rotor is described by + +$$ +F_{u,5}\!={m\ddot{\phi}w_{0}\cos(\beta)},\ \ \ F_{v,5}\!=\!-m{\ddot{\phi}w_{0}}\sin(\beta) +$$ + +which is the fictitious angular acceleration of $c g$ associated with the angular acceleration of the $(x,y,z)$ -frame about the $Y.$ -axis. The right hand side of equations (12a) and (12b) describes the external forces, $f_{u}$ and $f_{\nu}$ are the forces in the $x\cdot$ - and $y$ -directions, respectively. The last term is the bending moment from the external force in the $z$ -direction on the remaining part of the blade, from this point to the tip. + +Blade Torsional Motion + +The equation of torsional motion is + +$$ +\begin{array}{r l}&{\bigl(I_{c g}+m l_{c g}^{2}\bigr)\ddot{\theta}-m l_{c g}\bigl(\ddot{u}\sin(\overline{{\theta}})-\ddot{\nu}\cos(\overline{{\theta}})\bigr)+F_{\theta,1}\bigl(\dot{\phi},u^{\prime},\nu^{\prime},u,\nu,\beta\bigr)+F_{\theta,2}\bigl(\ddot{\beta},\dot{\beta},\dot{u},\dot{\nu},u,\nu\bigr)}\\ &{\quad+\,F_{\theta,3}\bigl(\ddot{\phi},\beta\bigr)+F_{\theta,4}\bigl(\phi,u^{\prime},\nu^{\prime},\theta,\beta\bigr)+F_{\theta,5}\bigl(u^{\prime\prime},\nu^{\prime\prime},\theta^{\prime}\bigr)+F_{\theta,6}\bigl(\theta^{\prime}\bigr)=M}\end{array} +$$ + +where the rotor speed leads to the fictitious centrifugal forces: + +$$ +F_{\theta,1}\!=\!m l_{c g}{\dot{\phi}}^{2}{\hat{u}}_{c g}\sin({\overline{{\theta}}}+\beta)+m l_{c g}w_{0}\phi^{2}\big(\nu^{\prime}\cos({\overline{{\theta}}})\!-\!(u^{\prime}\!+\!l_{p i}^{\prime})\sin({\overline{{\theta}}})\big) +$$ + +where the first term is associated with the offset of $c g$ in the $\hat{x}$ -direction and the second is associated with the distance from the center of rotation to $c g$ . The influence from the pitch action is described by + +$$ +\begin{array}{l}{{F_{\theta,2}=(I_{c g}+m l_{c g}^{2})\ddot{\beta}-\ddot{\beta}m l_{c g}(u\cos(\overline{{{\theta}}})+\nu\sin(\overline{{{\theta}}}))+m l_{c g}\dot{\beta}^{2}\big((u+l_{p i})\sin(\overline{{{\theta}}})-\nu\cos(\overline{{{\theta}}})\big)}}\\ {{\mathrm{}}}\\ {{\mathrm{}\qquad+\,2m l_{c g}\dot{\beta}(\dot{u}\cos(\overline{{{\theta}}})+\dot{\nu}\sin(\overline{{{\theta}}}))}}\end{array} +$$ + +where the first term is the fictitious angular acceleration associated with the angular acceleration of the $(x,y,$ , $z)$ -frame about the $z$ -axis. The second term is the fictitious centrifugal force associated with the rotation of the $(x,y,z)$ -frame about the $z-$ -axis. The last term is the fictitious Coriolis force associated with the rotation of the $(x,y,z)$ -frame about the $z$ -axis and the velocity of $c g$ in the chord direction. The acceleration of the rotor leads to the following term: + +$$ +F_{\theta,3}\,{=}\,{-}m\ddot{\phi}w_{0}l_{c g}\sin(\overline{{\theta}}+\beta) +$$ + +which is the fictitious angular acceleration of $c g$ associated with the angular acceleration of the $(x,y,z)$ -frame about the $Y\cdot$ -axis. The effect of gravity is described by + +$$ +F_{\theta,4}=-l_{c g}\big(\mathrm{sin}\big(\beta+\overline{{\theta}}\big)+\theta\cos\big(\beta+\overline{{\theta}}\big)m g\sin(\phi)+l_{c g}\big(\nu^{\prime}\cos\big(\overline{{\theta}}\big)-(u^{\prime}+l_{p i}^{\prime})\sin(\overline{{\theta}}\big)\big)m g\cos(\phi) +$$ + +where the first term is the twisting moment caused by the $\hat{x}$ -component of the gravity force and the distance between $c g$ and $e a$ in the ${\hat{y}}.$ -direction. The last term is the twisting moment caused by the distance between $c g$ and $e a$ and the $z-$ -component of the gravity force projected onto the cross section of the deformed blade. The elastic coupling between the bending and twisting of the blade is described by + +$$ +\begin{array}{r l}&{F_{\theta,5}\!=\!-\!\bigg(E I_{\eta\eta\xi}\big(\tilde{\theta}+\theta\big)^{\prime}\big(u^{\prime\prime}\sin(\overline{{\theta}})\!-\!\nu^{\prime\prime}\cos(\theta)\big)\bigg)^{\prime}\!+\!\bigg(E I_{\eta\xi\xi}\big(\tilde{\theta}+\theta\big)^{\prime}\big(u^{\prime\prime}\cos(\tilde{\theta})\!+\nu^{\prime\prime}\sin(\tilde{\theta})\big)\bigg)^{\prime}}\\ &{\qquad-\left(E I_{\xi}-E I_{\eta}\right)\!\big((u^{\prime\prime2}-\nu^{\prime\prime2})\cos(\tilde{\theta})\sin(\tilde{\theta})\!-u^{\prime\prime}\nu^{\prime\prime}\cos(2\tilde{\theta})\big)}\\ &{\qquad-\left.E I_{\xi}l_{p i}^{\prime\prime}\big(u^{\prime\prime}\cos(\tilde{\theta})\!+\nu^{\prime\prime}\sin(\tilde{\theta})\big)\sin(\tilde{\theta})\!+E I_{\eta}l_{p i}^{\prime\prime}\big(u^{\prime\prime}\sin(\tilde{\theta})\!-\!\nu^{\prime\prime}\cos(\tilde{\theta})\big)\right)\!\cos(\tilde{\theta})}\end{array} +$$ + +where $I_{\eta\eta\xi}=\ \int\!\int_{A}\!\eta(\eta^{2}\,+\,\xi^{2})\mathrm{d}\eta\mathrm{d}\xi$ and $I_{\eta\xi\xi}=\ \int\int_{A}^{}\!\xi(\eta^{2}\,+\,\xi^{2})\mathrm{d}\eta\mathrm{d}\xi$ . The restoring force caused by torsional stiffness is given by + +$$ +F_{\theta,6}\,{=}\,{-}(G J(\theta^{\prime}\,{+}\,\nu^{\prime}(u^{\prime\prime}\,{+}\,l_{p i}^{\prime\prime})))^{\prime} +$$ + +where the polar moment of inertia is $J=\int\int_{\cal A}{\left(\eta^{2}+\xi^{2}\right)}d\eta d\xi$ . The right hand side describes the external moment on the blade $M$ . + +Boundary Conditions + +The boundary conditions for the root of the blade are given by the geometric constraints: + +$$ +u(0,t)\!=\!u^{\prime}(0,t)\!=\!\nu(0,t)\!=\!\nu^{\prime}(0,t)\!=\!\theta(0,t)\!=\!0 +$$ + +because the coordinate frame used to describe the blade follows the root of the blade. + +The boundary conditions for the tip of the blade are determined by the boundary condition equations derived by demanding any admissible variation of the action integral to be zero. The boundary conditions become + +$$ +\begin{array}{l}{{u^{\prime\prime}(R,t)=\nu^{\prime\prime}(R,t)\,{=}\,\theta^{\prime}(R,t)=0}}\\ {{E I_{\xi}I_{\eta}u^{\prime\prime\prime}(R,t)=m l_{c g}\big(\dot{\phi}^{2}w_{0}-g\cos(\phi)\big)\big(I_{\eta}\sin\!\big(\tilde{\theta}-\overline{{\theta}}\big)\!\sin\!\big(\overline{{\theta}}\big)+I_{\xi}\cos\!\big(\tilde{\theta}-\overline{{\theta}}\big)\!\cos\!\big(\tilde{\theta}\big)\big)}}\\ {{E I_{\xi}I_{\eta}\nu^{\prime\prime\prime}(R,t)=m l_{c g}\big(\dot{\phi}^{2}w_{0}-g\cos(\phi)\big)\big(I_{\eta}\cos\!\big(\tilde{\theta}-\overline{{\theta}}\big)\!\sin\!\big(\tilde{\theta}\big)-I_{\xi}\sin\!\big(\tilde{\theta}-\overline{{\theta}}\big)\!\cos\!\big(\tilde{\theta}\big)\big)}}\end{array} +$$ + +If $l_{c g}(R)\neq0$ , the boundary conditions for the tip are functions of rotor speed $\dot{\phi}$ and rotor position $\phi$ and therefore time-varying. This is because an offset of the center of gravity from the elastic axis at the blade tip leads to a bending moment at the tip, caused by the gravity and centrifugal force. Most modern wind turbine blades, however, are tapered at the tip, leading to $l_{c g}(R)/R<<\varepsilon_{*}$ , making the time variation of the boundary conditions negligible. + +# Pitch Action + +The equation of motion for the pitch angle is + +$$ +\begin{array}{l}{{\displaystyle\int_{r}^{R}\big(\big(I_{c g}+m I_{c g}^{2}\big)\big(\ddot{\theta}+\ddot{\beta}\big)+m\big(I_{p i}^{2}+2I_{c g}I_{p i}\cos(\overline{{\theta}})\big)\ddot{\beta}-m\ddot{u}I_{c g}\sin(\overline{{\theta}})+m\ddot{\nu}\big(I_{p i}+l_{c g}\cos(\overline{{\theta}})\big)\big)\mathrm{d}s+F_{\beta,1}\big(\ddot{\beta},\ddot{\kappa}\big)}\ ~}\\ {{\displaystyle~~~+F_{\beta,2}\big(\dot{\beta},\dot{\nu},\nu\big)+F_{\beta,3}\big(\dot{\phi},u,\nu,\beta\big)+F_{\beta,4}\big(\ddot{\phi},u,\nu,\beta\big)+F_{\beta,5}\big(u,\nu,\theta,\beta,\phi\big)+F_{\beta,6}\big(\ddot{\beta},\ddot{u},\ddot{\nu},u,\nu\big)}\ ~}\\ {{\displaystyle=M_{p i r c h}+\int_{r}^{R}\big(M+f_{\nu}\big(u+I_{p i}\big)-f_{u}\nu\big)\mathrm{d}s}}\end{array} +$$ + +where + +$$ +F_{\beta,1}\big(\dot{\beta},\dot{\nu},u\big)\!=\!2\dot{\beta}\!\int_{r}^{R}m\dot{u}u_{c g}\mathrm{d}s,\quad F_{\beta,2}\big(\dot{\beta},\dot{\nu},u\big)\!=\!2\dot{\beta}\!\int_{r}^{R}m\dot{\nu}\nu_{c g}\mathrm{d}s +$$ + +are the moments caused by the fictitious Coriolis force associated with the relative velocity of the blade and rotation of the $(x,y,z)$ -frame about the $z_{i}$ -axis. The effect of the fictitious centrifugal force associated with the rotation of the $(x,y,z)$ -frame about the $\hat{y}_{}$ -axis is described by + +$$ +F_{\beta,3}\big(\dot{\phi},u,\nu,\beta\big)\!=\dot{\phi}^{2}\int_{r}^{R}m\hat{u}_{c g}\hat{\nu}_{c g}\mathrm{d}s +$$ + +where $\hat{\nu}_{c g}=(u+l_{p i})\mathrm{sin}(\beta)+\nu\mathrm{cos}(\beta)+l_{c g}\mathrm{sin}(\overline{{{\theta}}}+\beta)$ is the $\hat{y}$ coordinate of the center of gravity in the (ˆx,ˆy,ˆz)- frame. The effect of an angular acceleration of the $(x,y,z)$ -frame about the $\hat{y}$ -axis is described by + +$$ +F_{\beta,4}\big(\ddot{\phi},u,\nu,\beta\big)\!=-\ddot{\phi}\!\!\int_{r}^{R}m w_{0}\hat{\nu}_{c g}\mathrm{d}s +$$ + +The gravity force is described by + +$$ +F_{\beta,5}(u,\nu,\beta,\phi)=g\sin(\phi)\!\!\int_{r}^{R}m\hat{\nu}_{c g}\mathrm{d}s +$$ + +and + +$$ +F_{\beta,6}(\vec{\beta},\ddot{u},\ddot{\nu},u,\nu)=\ddot{\beta}\int_{r}^{R}m(u^{2}+\nu^{2}+2l_{c g}(u\cos(\overline{{\theta}})+\nu\sin(\overline{{\theta}}))+2l_{p i}u)\mathrm{d}s+\int_{r}^{R}m(\ddot{\nu}u-\ddot{u}\nu)\mathrm{d}s +$$ + +is nonlinear inertia. + +If the pitch angle is prescribed or given by an external model, equation (27) can be used to compute the pitch moment, by solving for $M_{p i t c h}$ and feed in the blade motion and pitch action. + +# Rotor Position + +Assuming a rigid drive train and no gearing, the rotor position is described by + +$$ +\begin{array}{l}{{\displaystyle J_{g c n}{\ddot{\phi}}+\int_{r}^{R}m w_{0}\big(w_{0}{\ddot{\phi}}+u\cos(\beta)-{\ddot{\nu}}\sin(\beta)\big)\mathrm{d}s}\ ~}\\ {{\displaystyle\quad+\;F_{\phi,1}\big({\dot{\beta}},u,\nu,\beta\big)+F_{\phi,2}\big({\dot{\beta}},{\dot{u}},{\dot{\nu}},{\dot{\theta}},\beta\big)+F_{\phi,3}\big(u,\phi\big)+F_{\phi,4}\big({\ddot{\beta}},u,\nu,\beta\big)\ ~}}\\ {{\displaystyle=T_{g c n}+\int_{r}^{R}\big(\big(f_{u}\cos(\beta)-f_{\nu}\sin(\beta)\big)w_{0}+f_{w}\big(\nu\sin(\beta)-\big(u+l_{p i}\big)\cos(\beta)\big)\big)\mathrm{d}s}\ ~}\end{array} +$$ + +The effect of the fictitious centrifugal force associated with rotation of the $(x,\,y,\,z)$ -frame about the $z$ -axis is described by + +$$ +F_{\phi,1}\big(\dot{\beta},u,\nu,\beta\big)\!=-\dot{\beta}^{2}\!\int_{r}^{R}m w_{0}\hat{u}_{c g}\mathrm{d}s +$$ + +and + +$$ +F_{\phi,2}\big(\dot{\beta},\dot{u},\dot{\nu},\dot{\theta},\beta\big)\!=-2\dot{\beta}\!\!\int_{r}^{R}\!m w_{0}(\dot{u}\sin(\beta)\!+\!\dot{\nu}\cos(\beta))\mathrm{d}s +$$ + +describes the fictitious Coriolis force associated with the rotation of the $(x,y,z)$ -frame about the $z$ -axis and the relative velocity of the blade. The effect of gravity is described by + +$$ +F_{\phi,3}(u,\phi)=g\sin(\phi)\!\!\int_{r}^{R}m w_{0}\mathrm{d}s+g\cos(\phi)\!\!\int_{r}^{R}m{\hat{u}}_{c g}\mathrm{d}s +$$ + +and + +$$ +F_{\phi,4}\big(\ddot{\beta},u,\nu,\beta\big)\!=-\ddot{\beta}\!\!\int_{r}^{R}m w_{0}\hat{\nu}_{c g}\mathrm{d}s +$$ + +describes the fictitious acceleration associated with an angular acceleration of the $\left(x,\,y,\,z\right)$ -frame about the $z$ -axis. + +The effect from the forces on the blade and the motion of the blade on the rotor speed is described by the two integral terms in equation (33) and by equations (34) to (37). + +The rotor speed equation (33) only includes the effects from one blade, but it can be extended to include the effects from more blades by adding an extra of the two integral terms in equation (33) and one of equations (34) to (37) for each extra blade. + +# Discussion + +Comparing the partial differential equations of motion (equations (12) and (18)) with Hodges and Dowell’s,3 it is noticed that the gravity terms (equations (15) and (22)), the pitch action terms (equations (13a) and (20)) and the terms involving varying rotor speed (equations (17) and (24)) are new. On the other hand, the terms involving warp effects in Hodges and Dowell3are not included here because this effect can be neglected without essential loss of accuracy for most applications.3 + +In the following discussion, the $x\cdot$ - and $y$ -direction will be denoted edgewise and flapwise to help the physical interpretation. The inertia terms in equation (12) are seen to couple the edgewise and flapwise motions to the torsional motion of the blade. The degree of coupling is seen to depend on the pre-twist of the chord. The first term in equation (13) shows that an acceleration of the pitch angle excites the edgewise and flapwise motion depending on the flapwise and edgewise deflection, respectively. That is, an acceleration of the pitch angle of a flapwise deflected blade excites the edgewise motion of the blade. The first term in the integral in equation (14) is a restoring force dependent on the rotation speed of the rotor, known as centrifugal stiffness. The effect of gravity (equations (15) and (22)) is seen to vary with the $\phi$ -angle as expected. The restoring force (equation (16)) couples the bending motion to the torsional motion. The degree of coupling is dependent on the edgewise and flapwise deflection of the blade. An acceleration of the rotor excites the edgewise and flapwise motion (equation (17)), the excitation is dependent on the pitch angle. The inertia term from equation (18) couples the torsional motion to the edgewise and flapwise motion. The degree of coupling to the edgewise and the flapwise motion is dependent on the pre-twist of the chord. The first term in equation (20) shows a strong coupling between pitch acceleration and torsional motion. The effect of rotor acceleration (equation (21)) on the torsional motion is dependent on the pitch setting and the pre-twist of the blade. The bending motion is coupled to the torsional motion through the bending stiffness (equation (23)). + +The first term in equation (27) shows the strong coupling between torsional motion and pitch motion. The first term in equation (32) shows the effect of blade deflection on the pitch inertia, and the second term in equation (32) shows how the motion of a deflected blade affects the pitch equation. + +To avoid unnecessary complications, structural damping is not included in the derivation of the equations of motion, but a damping term e.g. viscus damping could easily be added to the equations describing the structural damping. + +Extra degrees of freedom like tower, yaw motion or tilt can be included by introducing a new inertial frame, defining a transformation from the new inertial frame to the present inertial frame, and using this new transformation in the description of the energies before applying Hamilton’s method. This will lead to extra equations for the each extra degree of freedom and to periodic coefficients (like the gravity term). + +# Application Example + +In this section, a finite difference discretization of the blade model is used to compute the modes of natural vibrations of a particular $63\,\mathrm{m}$ blade.12 The frequencies and shapes of the natural modes of vibrations are compared to results from HAWCstab\*,13 showing good agreement. The modes are used as basic for an assumed mode discretization of the partial differential equations of motion, approximating them by three ordinary differential equations. The modes of natural vibrations of the assumed mode approximated model are compared with the previously derived modes, showing a reasonable agreement. To illustrate and test the pitch model, the assumed mode approximated model is used for time simulations of a rapid 2deg pitch change. The response is compared to $\mathrm{HAWC}2^{\dagger5,6}$ showing good agreement. + +# Finite Difference Discretization + +The spatial derivatives of an unforced and linearized version of the partial differential equations of motion (equations (12) and (18)) are approximated by a second-order finite difference approximation. The resulting approximating ordinary differential equations can be written as + +$$ +\hat{\mathbf{M}}\ddot{\tilde{\mathbf{q}}}+\tilde{\mathbf{D}}\ddot{\tilde{\mathbf{q}}}+\tilde{\mathbf{K}}\tilde{\mathbf{q}}=\mathbf{0} +$$ + +where $\tilde{\textbf{M}}$ , $\tilde{\mathbf{D}}$ and $\tilde{\bf K}$ hold the constant coefficients from the discretization and $\tilde{\mathbf{q}}=[u_{I},\nu_{I},\theta_{I}$ , … $,u_{n},\nu_{n},\theta_{n}]$ holds the deformations at the $n$ discretization points. Equation (38) is a differential eigenvalue problem where the eigenvalues give the frequency and damping of natural vibrations of the blade and the corresponding eigenvectors give the shape of the natural vibrations. + +Table I compares the six lowest eigenfrequencies for the blade with results from HAWCstab.13A good agreement is seen for all frequencies. Figure 2 shows the shape of first, second and sixth modes. The shapes are compared to results from HAWCstab showing a good agreement. + +# Assumed Mode Approximation + +The partial differential equations of motion are transformed into three approximating ordinary differential equations by the assumed mode method.11,14 The time- and spatial-dependent state variables for the blade are approx + +Table I. Frequencies for the first six natural modes of the test blade + + +
FinitedifferenceAssumedmode
ModenumberHAWC freq. [Hz]freq. [Hz]%J!P[ZH] badiff. %
10·690.7010.637
21.081·1461·044
31·961·971
43.974.052一一
54·514·551二二
67.837.7917.972
+ +The results from HAWCstab,13 the finite difference approximation of the present model and for the assumed mode approximation. Both the frequencies and the relative difference to the HAWCstab results are given. + +![](1c3ac9475d7dd6471757eb1a1259bcc67eedcad9326ef4a3690aa1ee45ebf50e.jpg) +Figure 2. Modes of natural vibrations computed by the finite difference approximated model ‘- -’ and the assumed mode approximated model ‘-’ compared to the modes computed by HAWCstab13 $\surd$ ’. (a) First mode, (b) second mode, (c) sixth mode + +imated by one edgewise $u(s,\,t)=u_{s}(s)u_{t}(t)$ , one flapwise $\nu(s,\,t)=\nu_{s}(s)\nu_{t}(t)$ and one torsional $\theta(s,\,t)=\theta_{s}(s)\theta_{t}(t)$ mode. The mode shapes $(u_{s},\,\nu_{s},\,\theta_{s})$ are the edgewise, flapwise and torsional contents of the second, first and sixth modes, respectively (the first modes dominated by edgewise, flapwise and torsional motion). The timedependent wight functions $(u_{t},\,\nu_{t},\,\theta_{t})$ are the new state variables of the system. The external forces on the blade are also split into a spatial part and a time-dependent part $f_{u}(s,\ t)=f_{u,s}(s)f_{u,t}(t),\ f_{\nu}(s,\ t)=f_{\nu,s}(s)f_{\nu,t}(t),\ f_{w}(s,\ t)=$ $f_{w,s}(s)f_{w,t}(t)$ and $M(s,t)=M_{s}(s)M_{t}(t).$ . The approximations are inserted into equations (12) and (18); the equations are wight by the corresponding spatial variable and integrated over the blade length, removing the spatial dependency. + +The ordinary differential equation of blade motion becomes + +$$ +\begin{array}{r l}&{\mathbf{M}\ddot{\mathbf{q}}+2\mathbf{D}\big(\dot{\phi},\dot{\beta},\beta\big)\dot{\mathbf{q}}+\mathbf{K}\big(\dot{\phi},\dot{\beta},\beta\big)\mathbf{q}+\mathbf{L}\big(\ddot{\beta},\phi,\beta\big)\mathbf{q}+\mathbf{N}\big(\dot{\phi},\dot{\beta},\beta,\mathbf{q}\big)+\mathbf{F}_{e x t,1}\mathbf{q}f_{w,t}}\\ &{\quad=\mathbf{F}\big(\ddot{\beta},\ddot{\phi},\dot{\beta},\dot{\phi},\beta,\phi\big)+\mathbf{F}_{e x t,0}[f_{u,t}f_{\nu,t}f_{w,t}M_{t}]^{\mathrm{T}}}\end{array} +$$ + +where $\mathbf{q}=[u_{t},\,\nu_{t},\,\theta_{t}]^{\mathrm{T}}$ and the rest of the terms are given in equation (50) in Appendix B. Inserting the expansions into equation (27), the integrals can be computed and the equation of pitch action becomes + +$$ +\begin{array}{r l}&{\big(I_{\beta,0}+I_{\beta,1}(\mathbf{q})\big)\ddot{\beta}+D_{\beta}(\dot{\mathbf{q}},\mathbf{q})\dot{\beta}=M_{p i t c h}+\big(\mathbf{M}_{e x t,0}+\mathbf{q}^{\mathrm{T}}\mathbf{M}_{e x t,1}\big)[f_{u,t}f_{\nu,t}M_{t}]^{\mathrm{T}}}\\ &{\quad+\,\mathbf{I}_{\beta,\mathbf{q}}\ddot{\mathbf{q}}+f_{\beta,\ddot{\mathbf{q}}}(\ddot{\mathbf{q}},\mathbf{q})+f_{\beta,\phi}\big(\ddot{\phi},\dot{\phi},\mathbf{q}\big)+f_{\beta,g r a v}(\mathbf{q})\sin(\phi)}\end{array} +$$ + +The individual terms are given in equation (51) in Appendix B. Inserting the expansions into equation (33) and computing the integrals, the equation of rotor position becomes + +$$ +\begin{array}{r l}&{(J_{g c n}+I_{\phi})\ddot{\phi}+f_{\phi,g}(\phi,\beta,\mathbf{q})\!=T_{g c n}+f_{\phi,\mathbf{q}}(\ddot{\mathbf{q}},\beta)\!+I_{\phi,\beta}(\mathbf{q},\beta)\ddot{\beta}+f_{\phi,\beta}\big(\dot{\beta},\dot{\mathbf{q}},\beta,\mathbf{q}\big)}\\ &{\quad+\,\mathbf{f}_{e x t,0}(\beta)[f_{u,t}f_{\nu,t}f_{w,t}]^{\mathrm{T}}+f_{e x t,1}(\mathbf{q},\beta)f_{w,t}}\end{array} +$$ + +The individual terms are given in equation (52) in Appendix B. An unforced and linearized version of equation (39) gives a differential eigenvalue problem: + +$$ +\mathbf{M}_{l i n}{\ddot{\mathbf{q}}}+\mathbf{D}_{l i n}{\dot{\mathbf{q}}}+\mathbf{K}_{l i n}\mathbf{q}=\mathbf{0} +$$ + +where the eigenvalue gives the frequency of natural vibrations of the assumed mode approximated model, and the eigenvectors give the coupling of the assumed modes in the natural vibrations. The found frequencies are compared with the previously found frequencies in Table I showing a good agreement. Figure 2 shows the natural mode shapes together with the previously found mode shapes. The edgewise and flapwise contents of the first and second modes are seen to agree very well with previous results. The torsional contents of the first mode are seen to disagree slightly from the previous results. The torsional contents of the second mode are seen to disagree with the previous result, but the value of the torsional contents is small compared to the edgewise and flapwise contents, hence the error is acceptable. The edgewise and flapwise contents of the sixth mode (first torsional mode) are seen to disagree quite a lot with the previous results. This is because the edgewise and flapwise contents are dominated by higher order edgewise and flapwise motion, which cannot be captured by this low order model. The value of the edgewise and flapwise contents is, however, small compared to the torsional contents, hence the error is acceptable. + +# Test Example + +The pitch model is illustrated and tested by a numerical simulation where the rotor is rotating with a constant angular velocity $\dot{\Phi}$ rad $\mathrm{s}^{-1}$ , and at $70\,\mathrm{s}$ , a 2deg pitch change is imposed. The pitch change has a rise time of 0·2 s and $1\!\cdot\!5\%$ overshoot. No aerodynamic forces are included in this example. The pitch moment is computed by feeding (equation (40)) with the prescribed pitch action and the computed blade motion. The results from the simulations are compared with results from HAWC2\*,5,6 showing a good agreement. + +Figure 3 shows the blade tip deflection and pitch moment from the present model and from HAWC2. The edgewise and flapwise motion are dominated by gravity, which is seen as the oscillations on the scale of 5s (corresponding to the rotor speed on $0.79\,\mathrm{rad\s^{-1}}$ ). A small excitation of the flapwise motion is seen at the pitch action at $70\,\mathrm{s}$ . The torsional motion of the blade is strongly excited by the pitch action at 70s. The pitch moment is high during the pitch action, and strongly effected by the torsional motion of the blade afterward. The flap motions agree very well for the two models. The amplitude of the flapwise motion on the scale of $5\,\mathrm{s}$ is a bit smaller for the present model than for HAWC2, and the excitation at $70\,\mathrm{s}$ is a bit more pronounced for the HAWC2 results, but still the two models agree well. The torsional motion agrees very well in amplitude, but there is a small disagreement in frequency. There is a good agreement between the pitch moment from the two models. + +![](3b76a1326c2c154063930df0ace9b75a3b81efe5ac90b1e77b700cfc654b46ca.jpg) +Figure 3. Tip deflection and pitch moment of a blade rotating with a constant speed of $2\pi$ and with a 2deg pitch change at $70s$ . ‘- - -’ the present model, $\cdot\cdot^{\prime}H A W C2^{5,6}$ + +# Discussion + +The results from the finite difference discretized model show that the present model captures the fundamental properties of the blade as well as HAWCstab.13 The results from the assumed mode model show that even with only three ordinary differential equations, important basic properties of the blade can be described, and that the pitch blade interaction can be modeled very well. + +The relative simple structure of the equations of motion (equation (39)) makes them suitable for qualitative analysis of interaction between pitch action and blade motion and/or fast simulation. The structure of equation (39) is similar to the structure of the equations of motion of a 2-D blade section model (as those used in Chaviaropoulos et al.1 and Block and Strganac2), therefore, the model has the same beneftis as the 2-D blade section model, but with a clear connection to the real turbine blade. The rotor position model (equation (41)) can be used to analyze how the motion of one blade effects the rotor speed, but more important, it can easily be extended with more blades, giving a coupling between the motion of the individual blades. The rotor position model is extended with more blades by adding one of each term in equation (52) for each blade involved. An improved description of the blade motion can be achieved if more mode shapes or coupled mode shapes are used. The drawback of this is a more complicated system, making analytical analysis and interpretation harder. + +# Conclusion + +This work extends the nonlinear partial differential equations of motion originally derived from Hodges and Dowell, taking pitch action, rotor speed variations and gravity into account. New equations are derived for the pitch action and rotor speed. Frequencies and shapes of natural vibrations of the blade are computed and compared to results from HAWCstab, showing a good agreement. The partial differential equations of motion are transformed into approximating ordinary differential equations of motion (equation (39)) by an assumed mode discretization. This model is suitable for basic analysis of interaction between pitch action and blade motion. The approximating ordinary differential equations of motion are used to simulate the response and pitch moment for a rotating turbine blade with a rapid 2deg pitch change. The results from the simulation are compared to the results from HAWC2, showing a good agreement. + +This work is a part of a project on pitch blade interaction, and the model will be extended to include an aerodynamic model and be used for analysis of basic properties of pitch blade interaction. + +# Acknowledgements + +The author thanks Morten Hartvig Hansen, Risø National Laboratory for his inspiring ideas and helpful discussions related to this work. This work is founded partly by The Technical University of Denmark and Risø National Laboratory. + +# Appendix A + +Coordinate Transformations + +The derivation of the transformation matrices follows the method used in Hodges and Dowell.3 The major difference between these matrices and those of Hodges and Dowell3 is the inclusion of the pitch angle $\beta$ . The transformation between the initial $\left(X,\,Y,\,Z\right)$ -frame and the $\left(\hat{x},\,\hat{y},\,\hat{z}\right)$ -frame is given by + +$$ +\begin{array}{r}{\left[\hat{\mathbf{i}}\right]}\\ {\left[\hat{\mathbf{j}}\right]=\mathbf{T}_{\phi}\left[\mathbf{J}\right]=\left[\begin{array}{c c c}{\cos(\phi(t))}&{0}&{-\sin(\phi(t))}\\ {0}&{1}&{0}\\ {\sin(\phi(t))}&{0}&{\cos(\phi(t))}\end{array}\right]\left[\mathbf{J}\right]}\end{array} +$$ + +and between the $(\hat{x},\hat{y},\hat{z})$ -frame and the $(x,\,y,\,z)$ -frame is given by + +$$ +\begin{array}{r l}&{\left[\!\!\begin{array}{c}{\mathbf{\hat{i}}}\\ {\mathbf{j}}\\ {\mathbf{k}}\end{array}\!\!\right]=\mathbf{T}_{\beta}\left[\!\!\begin{array}{c}{\hat{\mathbf{i}}}\\ {\hat{\mathbf{j}}}\\ {\hat{\mathbf{k}}}\end{array}\!\!\right]=\left[\!\!\begin{array}{c c c}{\cos(\beta(t))}&{\sin(\beta(t))}&{0\!\!\sqrt{\!\!\dot{\mathbf{i}}}}\\ {-\sin(\beta(t))}&{\cos(\beta(t))}&{0\!\!\sqrt{\!\!\dot{\mathbf{j}}}}\\ {0}&{0}&{1\!\!\sqrt{\!\!\dot{\mathbf{k}}}}\end{array}\!\!\right]\!\!\right]_{\mathbf{\hat{k}}}}\end{array} +$$ + +The principle axis of each cross section of the blade is described by the $(\mathfrak{n},\xi,\zeta)$ -frame with origin at $e a$ , where $\eta$ and $\zeta$ are the principle axes of the cross section and the $\zeta.$ -axis points outward along the elastic axis of the deformed blade. This frame has the unit vectors $(\tilde{\mathrm{i}},\tilde{\mathrm{j}},\tilde{\mathrm{k}})$ given by the following transformation: + +$$ +\begin{array}{r l}&{\left[\begin{array}{l}{\overline{{\mathbf{i}}}}\\ {\overline{{\mathbf{j}}}}\\ {\overline{{\mathbf{k}}}}\end{array}\right]=\mathbf{T}_{\mathrm{e}}\left[\begin{array}{l l l}{{\mathbf{i}}}\\ {{\bf{j}}}\\ {{\bf{k}}}\end{array}\right]=\left[\begin{array}{l l l}{\cos(\hat{\theta}(s,t))}&{\sin(\hat{\theta}(s,t))}&{0}\\ {-\sin(\hat{\theta}(s,t))}&{\cos(\hat{\theta}(s,t))}&{0}\\ {0}&{0}&{1}\end{array}\right]}\\ &{\begin{array}{r l}{{\mathbf{\theta}}_{1}}&{0}\\ {0}&{\sqrt{1-\nu^{\prime}(s,t)^{2}}}&{-\nu^{\prime}(s,t)}\\ {0}&{0}&{1}\end{array}\right]}\\ &{\begin{array}{r l}{\sqrt{\bigg[\frac{1-(l_{p}^{\prime}(s)+u^{\prime}(s,t))^{2}-\nu^{\prime}(s,t)^{2}}{1-\nu^{\prime}(s,t)^{2}}}}&{0}&{\cdots\frac{l_{p}^{\prime}(s)+u^{\prime}(s,t)}{\sqrt{1-\nu^{\prime}(s,t)^{2}}}}\\ {0}&{0}&{1}\end{array}\right]\times\left[\begin{array}{l}{1}\\ {1}\\ {\overline{{\mathbf{k}}}}\\ {\overline{{\mathbf{k}}}}\end{array}\right]}\\ &{\begin{array}{r l}{\frac{L_{p}^{\prime}(s)+u^{\prime}(s,t)}{\sqrt{1-\nu^{\prime}(s,t)^{2}}}}&{0}\\ {0}&{\sqrt{\frac{1-\big(l_{p}^{\prime}(s)+u^{\prime}(s,t)\big)^{2}-\nu^{\prime}(s,t)^{2}}{1-\nu^{\prime}(s,t)^{2}}}}\end{array}\right]\mathbf{k},}\end{array}}\end{array} +$$ + +where $\tilde{\theta}$ is the rotation of the blade around the elastic axis. The first matrix in equation (45) is the rotation about the $\hat{z}$ -axis, the next matrix is the rotation about the $x$ -axis and the last matrix is the rotation about the $y$ -axis. + +The chord is described by the $(\overline{{\mathbf{i}}},\overline{{\mathbf{j}}},\overline{{\mathbf{k}}})$ unit vectors parallel to the chord, normal upward from the chord and parallel to the elastic axis, respectively. This set of unit vectors is given by + +$$ +[\bar{\mathbf{i}}\quad\bar{\mathbf{j}}\quad\overline{{\mathbf{k}}}]^{\mathrm{T}}=\mathbf{T}_{\mathrm{c}}[\mathbf{i}\quad\mathbf{j}\quad\mathbf{k}]^{\mathrm{T}} +$$ + +where ${{\bf{T}}_{c}}$ is equal to ${\bf{T}}_{e}$ except that the twist qˆ includes the aerodynamic pre-twist instead of the elastic pre-twist. + +The rotation of the principle axis of the blade sections as a function of the $s$ coordinate is given by the differential equation: + +$$ +\begin{array}{r}{\mathbf{T}_{e}^{\prime}\!=\!\!\left[\!\!\begin{array}{c c c}{0}&{\tilde{\omega}_{k}}&{-\tilde{\omega}_{j}}\\ {-\tilde{\omega}_{k}}&{0}&{\tilde{\omega}_{i}}\\ {\tilde{\omega}_{j}}&{-\tilde{\omega}_{i}}&{0}\end{array}\!\!\right]\!\mathbf{T}_{c}\!\Rightarrow\!\!\left[\!\!\begin{array}{c c c}{0}&{\tilde{\omega}_{k}}&{-\tilde{\omega}_{j}}\\ {-\tilde{\omega}_{k}}&{0}&{\tilde{\omega}_{i}}\\ {\tilde{\omega}_{j}}&{-\tilde{\omega}_{i}}&{0}\end{array}\!\!\right]\!=\!T_{\mathrm{c}}\mathbf{T}_{\mathrm{c}}^{-1}}\end{array} +$$ + +where $(\tilde{w}_{i},\,\tilde{w}_{j},\,\tilde{w}_{k})$ is the rotation about the $(\tilde{\mathbf{i}},\tilde{\mathbf{j}},\tilde{\mathbf{k}})$ -directions respectively, and it is utilized that $\mathbf{T}_{\phi}^{\prime}=\mathbf{T}_{\beta}^{\prime}=0$ . The rotation about the $\tilde{\mathbf{k}}$ -direction is also measured by changes in the twist coordinates of the blade, (the pre-twist $\tilde{\theta}=\tilde{\theta}(s)$ and the elastic twist $\theta_{e l a}=\theta_{e l a}(s,\,t))$ . Hence, + +$$ +\left(\tilde{\theta}+\theta_{e l a}\right)^{\prime}=\tilde{\omega}_{k}=\hat{\theta}+\nu^{\prime}(u^{\prime\prime}+l_{p i}^{\prime\prime})+O(\varepsilon^{3}) +$$ + +using the order scheme (see previous discussion). + +Rearranging and intergrading equation (48) lead to an expression for the rotation of each blade section around the elastic axis: + +$$ +\hat{\theta}=\tilde{\theta}+\theta_{e l a}-\int_{0}^{s}\nu^{\prime}\big(\boldsymbol{u}^{\prime\prime}+\boldsymbol{l}_{p i}^{\prime}\big)\mathrm{d}\rho=\tilde{\theta}+\theta,\quad0=\theta\big(\boldsymbol{s},t\big)=\theta_{e l a}-\int_{0}^{s}\nu^{\prime}\big(\boldsymbol{u}^{\prime\prime}+\boldsymbol{l}_{p i}^{\prime\prime}\big)\mathrm{d}\rho +$$ + +where $\theta$ is the time-dependent twist of the blade relative to the $\left(x,\,y,\,z\right)$ -frame. Inserting equation (49) into the expression for ${\bf{T}}_{e}$ leads to the transformation matrix of the elastic properties. Replacing $\tilde{\theta}$ with $\bar{\theta}$ in ${\bf\delta T}_{e}$ gives the transformation matrix ${{\bf{T}}_{c}}$ of the chord. + +Note that $\mathbf{T}^{\mathrm{T}}\mathbf{T}=\mathbf{I}$ holds for all the transformation matrices. + +# Appendix B + +# Blade Model + +The individual terms in the assumed mode approximated blade model (equation (39)) are + +$$ +\begin{array}{r l}&{\mathbb{D}(\boldsymbol{\phi},\boldsymbol{\beta},\boldsymbol{\beta})=\hat{\wp}(\mathbb{D}_{\boldsymbol{\ell}\times\boldsymbol{\sin}}(\boldsymbol{\beta})+\mathbb{D}_{\boldsymbol{\ell}\times\boldsymbol{\cos}}(\boldsymbol{\beta}))+\hat{\varrho}(\mathbb{D}_{\boldsymbol{\ell}\times\boldsymbol{\sin}}(\boldsymbol{\beta})+\mathbb{D}_{\boldsymbol{\ell}\times\boldsymbol{\cos}}\cos(\beta))+\hat{\jmath}\mathfrak{D}_{\boldsymbol{\mathcal{N}}_{\boldsymbol{\ell}}}(\boldsymbol{\beta})}\\ &{\mathrm{K}(\boldsymbol{\phi},\boldsymbol{\beta},\boldsymbol{\beta})=\mathbf{K}_{\boldsymbol{\ell}}+\boldsymbol{\mathrm{K}}+\boldsymbol{\beta}^{2}\mathbb{K}_{\boldsymbol{\ell}}+2\hat{\beta}\hat{\boldsymbol{\psi}}(\mathbb{R}_{\boldsymbol{\ell}\times\boldsymbol{\sin}}\sin(\beta)+\mathbb{K}_{\boldsymbol{\ell}\times\boldsymbol{\cos}}(\boldsymbol{\beta}))}\\ &{\qquad\qquad\quad+\vec{\varrho}^{2}\left(\mathbb{K}_{\boldsymbol{\ell}\times\boldsymbol{\sin}}+\mathbb{K}_{\boldsymbol{\ell}\times\boldsymbol{\sin}}\sin^{2}(\beta)+\mathbb{K}_{\boldsymbol{\ell}\times\boldsymbol{\cos}}\cos^{2}(\beta)+\mathbb{K}_{\boldsymbol{\ell}\times\boldsymbol{\sin}}\sin(\beta)\cos(\beta)\right)}\\ &{L(\vec{\beta},\boldsymbol{\phi},\boldsymbol{\beta})=\hat{\jmath}\mathcal{R}_{\boldsymbol{\ell}\times\boldsymbol{\sin}}(\boldsymbol{\phi})(\mathbb{D}_{\boldsymbol{\ell}\times\boldsymbol{\sin}},\sin(\beta)+\mathbb{F}_{\boldsymbol{\ell}\times\boldsymbol{\cos}}(\boldsymbol{\beta}))+\boldsymbol{\mathrm{g}}\sin(\phi)\boldsymbol{\mathrm{F}}_{\boldsymbol{\ell}},}\\ &{\mathrm{N}(\boldsymbol{\phi},\boldsymbol{\beta},\boldsymbol{\beta},\boldsymbol{\mathrm{q}})=\varrho,\mathbb{F}_{\boldsymbol{\ell}}(\boldsymbol{\mathrm{q}})+\mathbb{F}_{\boldsymbol{\ell}}\left[L_{\boldsymbol{\ell}\times\boldsymbol{\sin}}^{\prime}(\boldsymbol{\mu})^{2}\right]^{\dagger}+2\hat{\varrho}(\mathbb{F}_{\boldsymbol{\ell}},\sin(\beta)+\mathbb{F}_{\boldsymbol{\ell}\times\boldsymbol{\cos}}(\boldsymbol{\beta}))\boldsymbol{\mathrm{q}}}\\ &{\qquad +$$ + +where the constants for the linear terms are + +$$ +{\bf M}=\int_{\nu}^{R}{\left[\begin{array}{c c c}{m u_{s}^{2}}&{0}&{-u_{s}\theta_{s}m l_{c g}\sin(\overline{{\theta}})}\\ {0}&{m\nu_{s}^{2}}&{\nu_{s}\theta_{s}m l_{c g}\cos(\overline{{\theta}})}\\ {-u_{s}\theta_{s}m l_{c g}\sin(\overline{{\theta}})}&{\nu_{s}\theta_{s}m l_{c g}\cos(\overline{{\theta}})}&{(I_{c g}+m l_{c g}^{2})\theta_{s}^{2}}\end{array}\right]}\mathrm{d}s +$$ + +$$ +\int_{r}^{R}\left[\frac{E\big(I_{\xi}\;\mathrm{cos}^{2}(\tilde{\theta})+I_{\eta}\;\mathrm{sin}^{2}(\tilde{\theta})\big)u_{*}^{\prime\prime}u_{*}^{\prime\prime}}{E\big(I_{\xi}\;I_{\eta}\big)\cos(\tilde{\theta})\sin(\tilde{\theta})u_{*}^{\prime\prime}v_{*}^{\prime\prime}}\right.\qquad E\big(I_{\xi}\;-\;I_{\eta}\big)\cos(\tilde{\theta})\sin(\tilde{\theta})u_{*}^{\prime\prime}v_{*}^{\prime\prime}\qquad-E\big(I_{\xi}\;I_{\eta}\big)\cos(\tilde{\theta})\sin(\tilde{\theta})\big)d\tilde{\eta}=0,\quad\mathrm{for~o~r~o~r~}\quad R\in\mathbb{R}^{3}, +$$ + +$$ +\mathbf{K}_{\phi,0}=\int_{r}^{R}\left[\begin{array}{c c c}{\left(u_{s}^{\prime}\right)^{2}\int_{s}^{R}m w_{0}\mathrm{d}\rho}&{0}&{-m l_{c g}w_{0}\sin(\overline{{\theta}})u_{s}^{\prime}\theta_{s}}\\ {0}&{\left(\nu_{s}^{\prime}\right)^{2}\int_{s}^{R}m w_{0}\mathrm{d}\rho}&{m l_{c g}w_{0}\cos(\overline{{\theta}})\nu_{s}^{\prime}\theta_{s}}\\ {-m l_{c g}w_{0}\sin(\overline{{\theta}})u_{s}^{\prime}\theta_{s}}&{m l_{c g}w_{0}\cos(\overline{{\theta}})\nu_{s}^{\prime}\theta_{s}}&{0}\end{array}\right]\mathrm{d}s +$$ + +$$ +\mathbf{K}_{\phi,c c}=\int_{r}^{R}\!\left[\begin{array}{c c c c}{-m u_{s}^{2}}&{0}&{m l_{c g}\sin(\overline{{\theta}})u_{s}\theta_{s}}\\ {0}&{0}&{0}\\ {m l_{c g}\sin(\overline{{\theta}})u_{s}\theta_{s}}&{0}&{0}\end{array}\right]\!\mathrm{d}s +$$ + +$$ +\mathbf{K}_{\phi,s s}\!=\!\int_{r}^{R}\!\!\left[\!\!\begin{array}{c c c}{0}&{0}&{0}\\ {0}&{-m\nu_{s}^{2}}&{-l_{c g}m\cos(\overline{{\theta}})\nu_{s}\theta_{s}}\\ {0}&{-l_{c g}m\cos(\overline{{\theta}})\nu_{s}\theta_{s}}&{0}\end{array}\!\!\right]\!\mathrm{d}s +$$ + +$$ +{\bf K}_{\dot{\phi},s c}=\int_{r}^{R}\!\left[\begin{array}{c c c}{0}&{m u_{s}\nu_{s}}&{m l_{c g}\cos(\overline{{\theta}})u_{s}\theta_{s}}\\ {m u_{s}\nu_{s}}&{0}&{-l_{c g}m\sin(\overline{{\theta}})\nu_{s}\theta_{s}}\\ {m l_{c g}\cos(\overline{{\theta}})u_{s}\theta_{s}}&{-l_{c g}m\sin(\overline{{\theta}})\nu_{s}\theta_{s}}&{0}\end{array}\right]\mathrm{d}s +$$ + +$$ +\mathbf{D}_{\phi,a,\mathrm{s}}\!=\!\int_{r}^{R}\!\left[m l_{c g}\cos(\overline{{\theta}})u_{s}^{\prime}\nu_{s}\right.\left.\right.-m l_{c g}\cos(\overline{{\theta}})\nu_{s}u_{s}^{\prime}\left.\right.\left.0\right]\!\!\!\!\mathrm{d}s +$$ + +$$ +\mathbf{D}_{\phi,a,\varsigma}=\int_{r}^{R}\left[\begin{array}{c c c}{m l_{c g}\cos(\overline{{\theta}})u_{s}u_{s}^{\prime}}&{-m l_{c g}\sin(\overline{{\theta}})\nu_{s}^{\prime}u_{s}}&{0}\\ {-m l_{c g}\cos(\overline{{\theta}})u_{s}u_{s}^{\prime}}&{0}&{0}\\ {m l_{c g}\sin(\overline{{\theta}})u_{s}\nu_{s}^{\prime}}&{0}&{0}\\ {0}&{0}&{0}\\ {0}&{0}&{0}\end{array}\right]\mathrm{d}s +$$ + +$$ +\mathbf{k}_{\beta}=\int_{r}^{R}{\left[\begin{array}{c c c c}{-m u_{s}^{2}}&{0}&{m l_{c g}\sin(\overline{{\theta}})u_{s}\theta_{s}}\\ {0}&{-m\nu_{s}^{2}}&{-m l_{c g}\cos(\overline{{\theta}})\nu_{s}\theta_{s}}\\ {m l_{c g}\sin(\overline{{\theta}})u_{s}\theta_{s}}&{-m l_{c g}\cos(\overline{{\theta}})\nu_{s}\theta_{s}}&{0}\end{array}\right]}\mathrm{d}s +$$ + +$$ +\mathbf{D}_{\beta}\!=\!\int_{r}^{R}\!\left[\begin{array}{c c c}{0}&{-m\nu_{s}u_{s}}&{-m l_{c g}\cos(\overline{{\theta}})u_{s}\theta_{s}}\\ {m\nu_{s}u_{s}}&{0}&{-m l_{c g}\sin(\overline{{\theta}})\nu_{s}\theta_{s}}\\ {m l_{c g}\cos(\overline{{\theta}})u_{s}\theta_{s}}&{m l_{c g}\sin(\overline{{\theta}})\nu_{s}\theta_{s}}&{0}\end{array}\right]\!\mathrm{d}s +$$ + +$$ +\mathbf{K}_{\beta\phi,s}=\int_{r}^{R}\left[\begin{array}{c c c}{-m l_{c g}u_{s}u_{s}^{\prime}\cos(\overline{{\theta}})-l_{p i}^{\prime}u_{s}^{\prime}\biggr\int_{s}^{R}m u_{s}\mathrm{d}\rho}&{0}&{0}\\ {-u_{s}^{\prime2}\int_{s}^{R}m(l_{p i}+l_{c g}\cos(\overline{{\theta}}))m u_{s}\mathrm{d}\rho}&{0}&{0}\\ {-m l_{c g}u_{s}\nu_{s}^{\prime}\sin(\overline{{\theta}})}&{-u_{s}^{\prime2}\int_{s}^{R}m(l_{p i}+l_{c g}\cos(\overline{{\theta}}))\mathrm{d}\rho}&{0}\\ {0}&{0}&{0}\\ {0}&{0}&{0}\end{array}\right]\mathrm{d}s +$$ + +$$ +\mathbf{K}_{\beta\phi,c}=\int_{r}^{R}\left[\begin{array}{c c c c}{-{u_{s}^{\prime}}^{2}\displaystyle\int_{s}^{R}m l_{c g}\sin(\overline{{\theta}}){\mathrm{d}}\rho}&{-{m l_{c g}}{\nu_{s}}{u_{s}^{\prime}}\cos(\overline{{\theta}})}&{0}\\ {0}&{-{l_{p}^{\prime}}{u_{s}^{\prime}}\displaystyle\int_{s}^{R}m{\nu_{s}}{\mathrm{d}}\rho}&{0}\\ {0}&{-{m l_{c g}}{\nu_{s}}{\nu_{s}^{\prime}}\sin(\overline{{\theta}}){\mathrm{d}}\rho}&{0}\\ {0}&{-{\nu_{s}^{\prime}}^{2}\displaystyle\int_{r}^{R}m l_{c g}\sin(\overline{{\theta}}){\mathrm{d}}\rho}&{0}\\ {0}&{0}&{0}\end{array}\right]{\mathrm{d}}s +$$ + +$$ +\mathbf{F}_{g,1,s}=\int_{r}^{R}\!\left[\begin{array}{c c c}{0}&{0}&{0}\\ {0}&{m l_{c g}\nu_{s}^{\prime2}\sin(\overline{{\theta}})}&{0}\\ {0}&{0}&{m l_{c g}\theta_{s}^{2}\sin(\overline{{\theta}})\right]\!\mathrm{d}s}\end{array} +$$ + +$$ +\mathbf{F}_{g,1,c}=\int_{r}^{R}\!\left[\!\!{\begin{array}{c c c}{-m l_{c g}u_{s}^{\prime\,2}\cos(\overline{{\theta}})}&{-m l_{c g}u_{s}^{\prime}u_{s}^{\prime}\sin(\overline{{\theta}})}&{0}\\ {-m l_{c g}{\nu}_{s}^{\prime}u_{s}^{\prime}\sin(\overline{{\theta}})}&{0}&{0}\\ {0}&{0}&{-m l_{c g}{\theta}_{s}^{2}\cos(\overline{{\theta}})}\end{array}}\!\!\right]\!\mathrm{d}s +$$ + +$$ +\mathbf{F}_{g,2}=\int_{r}^{R}{\left[\begin{array}{c c c}{-{u_{s}^{\prime}}^{2}{\int_{s}^{R}}m\mathrm{d}\rho}&{0}&{-m l_{c g}\theta_{s}{u_{s}^{\prime}}\sin{(\overline{{\theta}})}}\\ {0}&{-{\nu_{s}^{\prime}}^{2}{\int_{s}^{R}}m\mathrm{d}\rho}&{m l_{c g}\theta_{s}{\nu_{s}^{\prime}}\cos(\overline{{\theta}})}\\ {-m l_{c g}\theta_{s}{u_{s}^{\prime}}\sin(\overline{{\theta}})}&{m l_{c g}\theta_{s}{\nu_{s}^{\prime}}\cos(\overline{{\theta}})}&{0}\end{array}\right]}\mathrm{d}s +$$ + +$$ +\mathbf{D}_{\phi,s}=\int_{r}^{R}\left[\begin{array}{c c c c}{0}&{-l_{p i}^{\prime}u_{s}^{\prime}\int_{s}^{R}m\nu_{s}\mathrm{d}\rho}&{0}\\ {0}&{0}&{0}\\ {0}&{0}&{0\Biggr]\mathrm{d}s,}&{\mathbf{D}_{\phi,c}=\int_{r}^{R}\left[\begin{array}{c c c c}{l_{p i}^{\prime}u_{s}^{\prime}\int_{s}^{R}m u_{s}\mathrm{d}\rho}&{0}&{0}\\ {0}&{0}&{0}\\ {0}&{0}&{0}\end{array}\right]\mathrm{d}s +$$ + +$$ +\mathbf{K}=\int_{r}^{\kappa}\left[\begin{array}{c c c c}{0}&{0}&{0}&{0}\\ {0}&{0}&{0}\\ {E I_{\eta\eta\xi}\overline{{\theta}}^{\prime}\theta_{s}^{\prime}u_{s}^{\prime\prime}\mathrm{sin}(\overline{{\theta}})-E I_{\eta\xi\overline{{\xi}}}\overline{{\theta}}^{\prime}\theta_{s}^{\prime}u_{s}^{\prime\prime}\mathrm{cos}(\overline{{\theta}})}&{-E I_{\eta\eta\overline{{\xi}}}\overline{{\theta}}^{\prime}\theta_{s}^{\prime}u_{s}^{\prime\prime}\mathrm{cos}(\overline{{\theta}})-E I_{\eta\xi\overline{{\xi}}}\overline{{\theta}}^{\prime}\theta_{s}^{\prime}\nu_{s}^{\prime\prime}\mathrm{sin}(\widetilde{\theta})}&{0}\end{array}\right]^{0}, +$$ + +and for constants for the nonlinear terms: + +$$ +\mathbf{F}_{1}=\int_{r}^{R}{\left[\begin{array}{c c c}{-E(I_{\xi}-I_{\eta})u_{s}^{\prime\prime}u_{s}^{\prime}\theta_{s}\sin(2\tilde{\theta})}&{E(I_{\xi}-I_{\eta})\nu_{s}^{\prime\prime}u_{s}^{\prime\prime}\theta_{s}\cos(2\tilde{\theta})}&{0}\\ {E(I_{\xi}-I_{\eta})u_{s}^{\prime\prime}\nu_{s}^{\prime\prime}\theta_{s}\cos(2\tilde{\theta})}&{E(I_{\xi}-I_{\eta})\nu_{s}^{\prime\prime}\nu_{s}^{\prime\prime}\theta_{s}\sin(2\tilde{\theta})}&{0}\\ {\theta_{s}^{\prime2}u_{s}^{\prime\prime}E(I_{\eta\eta\xi}\sin(\overline{{\theta}})-E I_{\eta\xi\xi}\cos(\overline{{\theta}}))}&{-\theta_{s}^{\prime2}\nu_{s}^{\prime\prime}E(I_{\eta\eta\xi}\cos(\tilde{\theta})+I_{\eta\xi\xi}\sin(\tilde{\theta}))}&{0}\end{array}\right]}\mathrm{d}s +$$ + +$$ +\mathbf{F}_{2}=\int_{r}^{R}\left[\begin{array}{c c c}{0}&{0}&{0}\\ {0}&{0}&{0}\\ {-E(I_{\xi}-I_{\eta})\theta_{s}u_{s}^{\prime\prime}u_{s}^{\prime\prime}\mathrm{cos}(\tilde{\theta})\mathrm{sin}(\tilde{\theta})}&{E(I_{\xi}-I_{\eta})\theta_{s}\nu_{s}^{\prime\prime}\nu_{s}^{\prime\prime}\mathrm{cos}(\tilde{\theta})\mathrm{sin}(\tilde{\theta})}&{0}\end{array}\right]\mathrm{d}s +$$ + +$$ +\mathbf{F}_{3,s}=\int_{r}^{R}{\left[\begin{array}{l l l l}{0}&{-u_{s}^{\prime\,2}\displaystyle\int_{s}^{R}m\nu_{s}\mathrm{d}\rho}&{0}\\ {0}&{-\nu_{s}^{\prime\,2}\displaystyle\int_{s}^{R}m\nu_{s}\mathrm{d}\rho}&{0}\\ {0}&{0}&{0}\\ {0}&{\qquad0}&{0}\end{array}\right]}\mathrm{d}s,\quad\mathbf{F}_{3,c}=\int_{r}^{R}{\left[\begin{array}{l l l}{u_{s}^{\prime\,2}\displaystyle\int_{s}^{R}m u_{s}\mathrm{d}\rho}&{0}&{0}\\ {\nu_{s}^{\prime\,2}\displaystyle\int_{s}^{R}m u_{s}\mathrm{d}\rho}&{0}&{0}\\ {0}&{0}&{0}\\ {0}&{0}&{0}\end{array}\right]}\mathrm{d}s +$$ + +$$ +\mathbf{F}_{4,s}=\int_{r}^{R}\left[-\nu_{s}^{\prime2}\int_{s}^{R}m u_{s}\mathrm{d}\rho\quad0\quad0\right]\mathrm{d}s,\quad\mathbf{F}_{4,c}=\int_{r}^{R}\left[0\quad-u_{s}^{\prime2}\int_{s}^{R}m\nu_{s}\mathrm{d}\rho\quad0\right]}\\ {0\quad\qquad\qquad\quad0\quad0\quad0\quad0\right]\mathrm{d}s,\quad\mathbf{F}_{4,c}=\int_{r}^{R}\left[0\quad-\nu_{s}^{\prime2}\int_{s}^{R}m\nu_{s}\mathrm{d}\rho\quad0\right]\mathrm{d}s}\end{array} +$$ + +$$ +\mathbf{f}_{5}=\int_{r}^{R}[0\quad0\quad E(I_{\xi}-I_{\eta})u_{s}^{\prime\prime}\nu_{s}^{\prime\prime}\mathrm{cos}\big(2\tilde{\theta}\big)]^{\mathrm{T}}\mathrm{d}s +$$ + +and for the forcing terms: + +$$ +\mathbf{F}_{\beta,0}\!=\!\int_{r}^{R}\!\left[-m(l_{r g}u_{s}\sin(\overline{{\theta}}))\!\!\begin{array}{c c c}{m(l_{p i}+l_{c g}\cos(\overline{{\theta}}))u_{s}}\\ {m{l}_{c g}\nu_{s}\sin(\overline{{\theta}})}\\ {-(I_{c g}+m l_{c g}^{2})\theta_{s}}&{-m l_{c g}l_{p i}\theta_{s}\sin(\overline{{\theta}})}\end{array}\right]\!\mathrm{d}s +$$ + +$$ +\mathbf{f}_{\beta{_{\phi,s}}}\!=\!\int_{r}^{R}\!\left[\!\!{\begin{array}{c}{\!\!\!{m l_{{c}g}}(l_{p i}+l_{{c}g}\cos(\overline{{\theta}}))u_{s}^{\prime}\cos(\overline{{\theta}})\!+\!l_{p i}^{\prime}u_{s}^{\prime}\displaystyle\int_{s}^{R}\!\!{m(l_{p i}+l_{{c}g}\cos(\overline{{\theta}}))}\mathrm{d}\rho}\\ {\!\!{m l_{{c}g}}(l_{p i}+l_{{c}g}\cos(\overline{{\theta}}))\nu_{s}^{\prime}\sin(\overline{{\theta}})}\\ {0}\end{array}}\!\!\right]\!\mathrm{d}s +$$ + +$$ +\mathbf{f}_{\beta\phi,c}=\int_{r}^{R}\!\!\left[{m}l_{c g}l_{c g}\cos(\beta)\sin(\overline{{\theta}})u_{s}^{\prime}\cos(\overline{{\theta}})+l_{p i}^{\prime}u_{s}^{\prime}\!\int_{s}^{R}{m}l_{c g}\sin(\overline{{\theta}})\mathrm{d}\rho\right]_{\mathrm{d}s}}\\ {\quad{m}l_{c g}l_{c g}\sin^{2}(\overline{{\theta}})\nu_{s}^{\prime}}\\ {0}\end{array}\!\!\!\! +$$ + +$$ +{\bf F}_{\phi,0}={\int_{r}^{R}}\Bigg[0\quad-m l_{c g}w_{0}u_{s}^{\prime}\cos(\overline{{\theta}})\\ {0\quad-m l_{c g}w_{0}\nu_{s}^{\prime}\sin(\overline{{\theta}})}\\ {0\quad-m l_{c g}l_{p i}^{\prime}w_{0}\theta_{s}\sin(\overline{{\theta}})\Bigg]\mathrm{d}s +$$ + +$$ +\mathbf{F}_{\phi,s}=\int_{r}^{R}{\left[\begin{array}{c c c c}{0}&{0}\\ {m w_{0}u_{s}}&{0}\\ {m w_{0}l_{c g}\theta_{s}\cos({\overline{{\theta}}})}&{0}\end{array}\right]}\mathrm{d}s,\ \ \ \mathbf{F}_{\phi,c}=\int_{r}^{R}{\left[\begin{array}{c c c c}{-m w_{0}u_{s}}&{0}\\ {0}&{0}\\ {m w_{0}l_{c g}\theta_{s}\sin({\overline{{\theta}}})}&{0}\end{array}\right]}\mathrm{d}s +$$ + +$$ +\mathbf{F}_{\phi,s s}=\int_{r}^{R}\left[\overset{\displaystyle0}{\underset{\displaystyle0}{0}}\right.\qquad m l_{c g}\sin(\overline{{\theta}})\nu_{s}\quad\quad\quad\Biggl]\mathrm{d}s +$$ + +$$ +\mathbf{F}_{\phi,c c}=\int_{r}^{R}\!\!\left[\begin{array}{c c c}{0}&{m(l_{p i}+l_{c g}\cos(\overline{{\theta}}))u_{s}}\\ {0}&{0}\\ {0}&{-m l_{c g}(l_{p i}+l_{c g}\cos(\overline{{\theta}}))\theta_{s}\sin(\overline{{\theta}})\!\right]\!\mathrm{d}s}\end{array} +$$ + +$$ +\mathbf{F}_{\phi,\mathrm{sc}}=\int_{r}^{R}\!\left[\!\!\begin{array}{c c}{0}&{-m l_{c g}\sin(\overline{{\theta}})u_{s}}\\ {0}&{-m\big(l_{p i}+l_{c g}\cos(\overline{{\theta}})\big)\nu_{s}}\\ {0}&{-m l_{c g}\big(l_{p i}\cos(\theta)+l_{c g}\cos(\overline{{\theta}})-l_{c g}\sin^{2}(\overline{{\theta}})\big)\theta_{s}\,\!}\end{array}\!\!\right]\!\mathrm{d}s +$$ + +$$ +\mathbf{F}_{g,0}=\int_{r}^{R}{\left[\begin{array}{l l}{0}&{-m l_{c g}u_{s}^{\prime}\cos(\overline{{\theta}})\!-\!l_{p i}^{\prime}u_{s}^{\prime}\displaystyle\int_{s}^{R}m\mathrm{d}\rho}\\ {0}&{-m l_{c g}\nu_{s}^{\prime}\sin(\overline{{\theta}})}\\ {0}&{m l_{c g}l_{p i}^{\prime}\theta_{s}\sin(\overline{{\theta}})}\end{array}\right]}\mathrm{d}s +$$ + +$$ +\mathbf{F}_{g,s}(\beta)\!=\!\int_{r}^{R}\!\left[\begin{array}{c c c}{0}&{0}\\ {m\nu_{s}(0}&{0\Biggr]\mathrm{d}s,}&{\mathbf{F}_{g,c}(\beta)\!=\!\int_{r}^{R}\!\left[\begin{array}{c c c}{-m u_{s}+m l_{c g}l_{p i}^{\prime}u_{s}^{\prime}\cos(\overline{{\theta}})}&{0}\\ {m l_{c g}l_{p i}^{\prime}\nu_{s}^{\prime}\sin(\overline{{\theta}})}&{0}\\ {m l_{c g}\theta_{s}\sin(\overline{{\theta}})}&{0}\end{array}\right]\mathrm{d}s +$$ + +$$ +\mathbf{f}_{\phi}=\int_{r}^{R}\Bigl[-l_{p i}^{\prime}u_{s}^{\prime}\Bigr]_{s}^{R}m w_{0}\mathrm{d}\rho\quad0\quad0\Bigr]^{\mathrm{T}}\mathrm{d}s +$$ + +$$ +\mathbf{F}_{e x t,0}=\int_{r}^{R}{\left[\begin{array}{c c c c}{f_{u,s}u_{s}}&{0}&{-l_{p i}^{\prime}u_{s}^{\prime}{\int_{s}^{R}}f_{w,s}\mathrm{d}\rho}&{0}\\ {0}&{f_{v,s}\nu_{s}}&{0}&{0}\\ {0}&{0}&{0}&{M_{s}\theta_{s}}\end{array}\right]}^{\mathrm{T}}\mathrm{d}s +$$ + +$$ +\mathbf{F}_{e x t,1}\!=\!\int_{r}^{R}\!\left[\begin{array}{c c c}{\!\!\!u_{s}^{\prime2}\!\int_{s}^{R}f_{w,s}\mathrm{d}\rho}&{\!\!\!0}&{\!\!\!0}\\ {\!\!\!0}&{\!\!\!u_{s}^{\prime2}\!\int_{s}^{R}f_{w,s}\mathrm{d}\rho}&{\!\!\!0}\\ {\!\!\!0}&{\!\!\!0}&{\!\!\!0}\end{array}\right]^{\mathrm{T}}\! +$$ + +# Pitch Model + +The individual terms in the assumed mode approximated pitch model (equation (40)) are + +$$ +\begin{array}{r l}&{I_{\beta,1}(\mathbf{q})=\mathbf{I}_{\beta,1}\mathbf{q}+\mathbf{I}_{\beta,2}{[u_{t}^{2}{\nu}_{t}^{2}]}^{\top}}\\ &{D_{\beta}(\mathbf{\dot{q}},\mathbf{q})=2\mathbf{f}_{\beta,0}\mathbf{\dot{q}}+2\mathbf{I}_{\beta,2}{[\dot{u}_{i}{u_{t}}{\dot{\nu}}_{t}{]}^{\top}}}\\ &{f_{\beta,4}(\mathbf{\dot{q}},\mathbf{q})=m_{w}(\dot{u_{t}}{\nu}_{t}-\dot{\nu}_{t}{u_{t}})}\\ &{f_{\beta,\phi}(\vec{\phi},\vec{\phi},\mathbf{q})=\vec{\phi}(f_{\beta,\phi}{u_{t}}\sin(\beta)+f_{\beta,\vec{\phi},C}{V_{t}}\cos(\beta)+I_{\beta,\phi},\sin(\beta)+I_{\beta,\phi,c}\cos(\beta))}\\ &{\qquad\qquad\qquad-2\dot{\phi}^{2}(f_{\beta,\phi,\sin}(2\beta)+f_{\beta,\phi,c}\cos(2\beta)+(\mathbf{f}_{\beta,\phi,\sin}(2\beta)+f_{\beta,\phi,c}\cos(2\beta))\mathbf{q}}\\ &{\qquad\qquad\qquad+\sin(2\beta)\mathbf{f}_{\beta,\phi,2}\big[u_{t}^{2}{\nu}_{t}^{2}\big]^{\top}+\cos(2\beta)f_{\beta,\phi,2}u_{t}{\nu}_{t}\big)}\\ &{f_{\beta,\varepsilon\mu\nu}(\mathbf{q})=-g((\mathbf{f}_{\beta,\varepsilon\mathrm{2},s}\sin(\beta)+\mathbf{f}_{\beta,\varepsilon\mathrm{2},c}\cos(\beta))\mathbf{q}+f_{\beta,\varepsilon\mathrm{3}}\sin(\beta)+f_{\beta,\varepsilon\mathrm{c}}\cos(\beta)))}\end{array} +$$ + +where the constants are + +$$ +I_{\beta,0}=\int_{r}^{R}\big(I_{c g}+m\big(I_{c g}^{2}+I_{p i}^{2}+2I_{p i}I_{c g}\cos(\overline{{\theta}})\big)\big)\mathrm{d}s,\quad\mathbf{I}_{\beta,1}=\int_{r}^{R}[2m\big(I_{p i}+l_{c g}\cos(\theta)\big)\quad2m l_{c g}\sin(\overline{{\theta}})\quad0]\mathrm{d}s, +$$ + +$$ +\mathbf{I}_{\beta,2}=\int_{r}^{R}[m u_{s}^{2}\quad m\nu_{s}^{2}]\mathrm{d}s,\quad\mathbf{I}_{\beta,\mathbf{q}}=\int_{r}^{R}[m u_{s}l_{c g}\sin(\overline{{\theta}})\quad-m\nu_{s}(l_{p i}+l_{c g}\sin(\overline{{\theta}}))\quad-I_{c g}-m l_{c g}^{2}]\mathrm{d}s +$$ + +$$ +I_{\beta,\phi,s}=\int_{r}^{R}m w_{0}(I_{p i}+l_{c g}\cos(\overline{{\theta}}))\mathrm{d}s,\quad I_{\beta,\phi,c}=\int_{r}^{R}m w_{0}l_{c g}\sin(\overline{{\theta}})\mathrm{d}s,\quad m_{u\nu}=\int_{r}^{R}m u_{s}\nu_{s}\mathrm{d}s +$$ + +$$ +\mathbf{f}_{\beta,\mathbf{q}}=\int_{r}^{R}[m u_{s}(l_{p i}+l_{c g}\cos(\overline{{\theta}}))\quad m v_{s}l_{c g}\sin(\overline{{\theta}})\quad0]\mathrm{d}s,\quad f_{\beta,\vec{\circ},\vec{\circ}}=\int_{r}^{R}m u_{s}w_{0}\mathrm{d}s,\quad f_{\beta,\vec{\circ},\vec{\circ}}=\int_{r}^{R}m v_{s}w_{0}\mathrm{d}s +$$ + +$$ +\mathbf{M}_{e x t,0}\!=\!\int_{r}^{R}[0\quad l_{p i}f_{\nu,s}\quad M_{s}]\mathrm{d}s,\quad\mathbf{M}_{e x t,1}\!=\!\int_{r}^{R}\!\left[{\!\!\begin{array}{c c c}{0}&{u_{s}f_{\nu,s}}&{0}\\ {-\nu_{s}f_{u,s}}&{0}&{0}\\ {0}&{0}&{0}\end{array}\!\!\right]\!\mathrm{d}s +$$ + +$$ +f_{\beta,\phi,s}=\int_{r}^{R}m\Big(\frac{1}{2}(l_{p i}^{2}-l_{c g}^{2})+l_{c g}^{2}\cos^{2}(\overline{{\theta}}\big)+l_{c g}l_{p i}\cos(\overline{{\theta}})\Big)\mathrm{d}s +$$ + +$$ +f_{\beta,\phi,c}=\int_{r}^{R}m(l_{c g}^{2}\cos(\overline{{\theta}})\sin(\overline{{\theta}}\,)\,+l_{c g}l_{p i}\sin(\overline{{\theta}}\,))\mathrm{d}s +$$ + +$$ +\mathbf{f}_{\beta,\phi,s}=\int_{r}^{R}{\left[\begin{array}{c}{m u_{s}(l_{p i}+l_{e g}\cos(\overline{{\theta}}))}\\ {-m\nu_{s}l_{c g}\sin(\overline{{\theta}})}\\ {0}\end{array}\right]}^{\mathrm{T}}{\mathrm{d}}s,\quad\mathbf{f}_{\beta,\phi,c}=\int_{r}^{R}{\left[\begin{array}{c}{m u_{s}l_{c g}\sin(\overline{{\theta}})}\\ {m\nu_{s}(l_{p i}+l_{c g}\cos(\overline{{\theta}}))}\\ {0}\end{array}\right]}^{\mathrm{T}}{\mathrm{d}}s +$$ + +$$ +\mathbf{f}_{\beta,\phi,2}=\int_{r}^{R}{\left[\begin{array}{l}{1/2m u_{s}^{2}}\\ {-1/2m v_{s}^{2}}\end{array}\right]}^{\mathrm{T}}\mathrm{d}s,\quad\mathbf{f}_{\beta,g,2,s}=\int_{r}^{R}{\left[\begin{array}{c}{m u_{s}}\\ {0}\\ {-m\theta_{s}l_{c g}\sin(\overline{{\theta}})\mathrm{.}}\end{array}\right]}^{\mathrm{T}}\mathrm{d}s,\quad\mathbf{f}_{\beta,g,2,c}=\int_{r}^{R}{\left[\begin{array}{c}{0}\\ {m v_{s}}\\ {m\theta_{s}l_{c g}\cos(\overline{{\theta}})\mathrm{.}}\end{array}\right]}^{\mathrm{T}}\mathrm{d}s +$$ + +$$ +f_{\beta,\phi,2}\!=\!\int_{r}^{R}m u_{s}\nu_{s}\mathrm{d}s,\quad f_{\beta,g,s}\!=\!\int_{r}^{R}m(l_{p i}+l_{c g}\cos(\overline{{\theta}}))\mathrm{d}s,\quad f_{\beta,g,c}\!=\!\int_{r}^{R}m l_{c g}\sin(\overline{{\theta}})\mathrm{d}s +$$ + +# Rotor Speed Model + +The individual terms in the assumed mode approximated pitch model (equation (41)) are + +$$ +\begin{array}{r l}&{f_{\phi,z}(\phi,\beta,\mathbf{q})=f_{\phi,z,0}\sin(\phi)+(f_{\phi,z,\mu,0}\cos(\beta)-f_{\phi,z,\nu,0}\sin(\beta)}\\ &{\qquad\qquad\qquad+f_{\phi,z,\mu,1}u_{t}\cos(\beta)-f_{\phi,z,\nu,1}\nu_{t}\sin(\beta))\cos(\phi)}\\ &{I_{\phi,\beta}(\mathbf{q},\beta)=I_{\phi,z,0}\sin(\beta)+I_{\phi,z,0}\cos(\beta)+I_{\phi,u,1}u_{t}\sin(\beta)+I_{\phi,z,1}\nu_{t}\cos(\beta)}\\ &{f_{\phi,\alpha_{1}}(\vec{\mathbf{q}},\beta)=I_{\phi,u,1}\vec{u}_{t}\cos(\beta)-I_{\phi,x,1}\vec{\nu}_{t}\sin(\beta)}\\ &{f_{\phi,\beta}(\vec{\mathbf{\alpha}},\mathbf{i},\beta,\mathbf{q})=(I_{\phi,u,0}\cos(\beta)-I_{\phi,\nu,0}\sin(\beta)+I_{\phi,u,1}u_{t}\cos(\beta)-I_{\phi,z,1}\nu_{t}\sin(\beta))\dot{\beta}^{2}}\\ &{\qquad\qquad\qquad\qquad\qquad+2\dot{\beta}(I_{\phi,u,1}\dot{u}_{t}\sin(\beta)+I_{\phi,v,1}\dot{u}_{t}\cos(\beta))}\\ &{\mathbf{f}_{\mathrm{ext},0}(\beta)=\mathbf{f}_{\mathrm{ext},0,x}\cos(\beta)+\mathbf{f}_{\mathrm{ext},0,x}\sin(\beta)}\\ &{\mathbf{f}_{\mathrm{ext},\left(\mathbf{q},\beta\right)}(\mathbf{q},\beta)=f_{\mathrm{ext},1,\nu_{t},\sin}(\beta)-f_{\mathrm{ext},\left\mathbf{u},\mu_{t}\right.\cos(\beta)}}\end{array} +$$ + +where the constants are + +$$ +I_{\phi}=\int_{r}^{R}m w_{0}^{2},\quad I_{\phi,u,0}=\int_{r}^{R}m w_{0}\,(l_{p i}+l_{c g}\cos(\overline{{\theta}}))\mathrm{d}s,\quad I_{\phi,v,0}=\int_{r}^{R}m w_{0}l_{c g}\sin(\overline{{\theta}})\mathrm{d}s +$$ + +$$ +I_{\phi,u,1}\!=\!\int_{r}^{R}m w_{0}u_{s}\mathrm{d}s,\quad I_{\phi,v,1}\!=\!\int_{r}^{R}m w_{0}\nu_{s}\mathrm{d}s,\quad f_{\phi,g,u,1}\!=\!\int_{r}^{R}g m u_{s}\mathrm{d}s,\quad f_{\phi,g,v,1}\!=\!\int_{r}^{R}g m\nu_{s}\mathrm{d}s +$$ + +$$ +f_{\phi,g,0}=\int_{r}^{R}g m w_{0}\mathrm{d}s,\quad f_{\phi,g,u,0}=\int_{r}^{R}g m(l_{p i}+l_{c g}\cos(\overline{{\theta}}))\mathrm{d}s,\quad f_{\phi,g,\nu,0}=\int_{r}^{R}g m l_{c g}\sin(\overline{{\theta}})\mathrm{d}s +$$ + +$$ +f_{e x t,0,s}=\int_{r}^{R}[w_{0}\,f_{u,s}\quad0\quad-l_{p i}\,f_{w,s}]\mathrm{d}s,\quad f_{e x t,0,c}=\int_{r}^{R}[0\quad-w_{0}\,f_{v,s}\quad0]\mathrm{d}s +$$ + +$$ +f_{e x t,1,u}=\int_{r}^{R}u_{s}\,f_{w,s}\mathrm{d}s,\quad f_{e x t,1,v}=\int_{r}^{R}u_{s}\,f_{w,s}d s,\quad f_{e x t,1,u}=\int_{r}^{R}u_{s}\,f_{w,s}\mathrm{d}s,\quad f_{e x t,1,v}=\int_{r}^{R}\nu_{s}\,f_{w,s}\mathrm{d}s +$$ + +# References + +1. Chaviaropoulos PK, Soerensen NN, Hansen MOL, Nikolaou IG, Aggelis KA, Johansen J, Gaunaa M, Hambraus T, von Geyr HF, Hirsch C, Shun K, Voutsinas SG, Tzabiras G, Perivolaris Y, Dyrmose SZ. Viscous and aeroelastic effects on wind turbine blades. The viscel project. part II: Aeroelastic stability investigations. Wind Energy 2003; 6: 387–404. +2. Block JJ, Strganac TW. Applied active control for a nonlinear aeroelastic structure. Journal of Guidance, Control, and Dynamics 1998; 21: 838–845. +3. Hodges DH, Dowell EH. Nonlinear equations of motion for the elastic bending and torsion of twisted nonuniform rotor blades. Technical Report TN D-7818, NASA, December 1974. +4. Wendell J. Aeroelastic stability of wind turbine rotor blades. Technical Report E(11·1)-4131, U.S. Department of Energy, September 1978. +5. Larsen TJ, Hansen A, Buhl T. Aeroelastic effects of large blade deflections for wind turbines. Proceedings of the special topic conference ‘The Science of making Torque from Wind’, Delft, The Netherlands, 2004; 238–246. +6. Larsen TJ, Madsen HA, Hansen AM, Thomsen K. Investigations of stability effects of an offshore wind turbine using the new aeroelastic code HAWC2. Proceedings of the conference ‘Copenhagen Offshore Wind 2005’, Copenhagen, 2005; 25–28. +7. Johnson W. Rotorcraft aeromechanics applications of a comprehensive analysis. AHS International Meeting on Advanced Rotorcraft Technology and Disaster Relief, Japan, 1998; S5–1 to S5–14. +8. Cesnik CES, Hodges DH, Sutyrin VG. Cross-sectional analysis of composite beams including large initial twist and curvature effects. AIAA Journal 1996; 34: 1913–1920. +9. Wenbin Y, Hodges DH, Volovoi V, Cesnik CES. On timoshenko-like modeling of initially curved and twisted composite beams. International Journal of Solids and Structures 2002; 39: 5101–5121. +10. Wenbin Y, Volovoi V, Hodges DH, Hong X. Validation of the variational asymptotic beam sectional analysis. AIAA Journal 2002; 40: 2105–2112. +11. Thomsen JJ. Vibrations and Stability: Advanced Theory, Analysis, and Tools. Springer-Verlag: Berlin-Heidelberg-New York, 2003. +12. Jonkman J. Nreloffshrbsline5mw. Technical report, NREL/NWTC, 1617 Cole Boulevard; Golden, CO 80401-3393, USA, 2005. +13. Hansen MH. Aeroelastic stability analysis of wind turbines using an eigenvalue approach. Wind Energy 2004; 7: 133–143. +14. Meirovitvh L. Computational Methods in Structural Dynamics. 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