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2d14e382e0 Merge remote-tracking branch 'origin/master' 2025-07-04 15:08:51 +08:00
186ed181ce vault backup: 2025-07-04 15:08:43 2025-07-04 15:08:44 +08:00
3 changed files with 72 additions and 20 deletions

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@ -266,7 +266,7 @@
},
{
"name": "Translate to Chinese",
"prompt": "<instruction>Translate the text below into Chinese:\n 1. Preserve the meaning and tone\n 2. Maintain appropriate cultural context\n 3. Keep formatting and structure\n 4. Blade翻译为叶片flapwise翻译为挥舞edgewise翻译为摆振pitch angle翻译成变桨角度twist angle翻译为扭角rotor翻译为风轮turbine、wind turbine翻译为机组、风电机组span翻译为展向deflection翻译为变形mode翻译为模态normal mode翻译为简正模态jacket 翻译为导管架superelement翻译为超单元shaft翻译为主轴azimuth、azimuth angle翻译为方位角neutral axes 翻译为中性轴\n Return only the translated text.</instruction>\n\n<text>{copilot-selection}</text>",
"prompt": "<instruction>Translate the text below into Chinese:\n 1. Preserve the meaning and tone\n 2. Maintain appropriate cultural context\n 3. Keep formatting and structure\n \n </instruction>\n\n<text>{copilot-selection}</text>\n<restrictions>\n1. Blade翻译为叶片flapwise翻译为挥舞edgewise翻译为摆振pitch angle翻译成变桨角度twist angle翻译为扭角rotor翻译为风轮turbine、wind turbine翻译为机组、风电机组span翻译为展向deflection翻译为变形mode翻译为模态normal mode翻译为简正模态jacket 翻译为导管架superelement翻译为超单元shaft翻译为主轴azimuth、azimuth angle翻译为方位角neutral axes 翻译为中性轴\n2. Return only the translated text.\n</restrictions>",
"showInContextMenu": true,
"modelKey": "gemma3:12b|ollama"
},

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@ -63,7 +63,7 @@ $$
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.
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。
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 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

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@ -3,7 +3,7 @@
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 HodgesDowells 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.
@ -25,6 +25,7 @@ In aeroelastic1 and aeroservoelastic2 analyses, 2-D blade section models† are
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。
@ -183,8 +184,9 @@ 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 Hamiltons 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.
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 Hamiltons 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.
通过要求任何可接受的作用积分变分 $\delta H\equiv\int_{\mathbf{\Omega}_{t_{1}}}^{\mathbf{\Omega}_{t_{2}}}(\delta T-\delta V_{e l a}-\delta V_{g r a}+\delta Q)d t$ 为零,可以推导出运动的偏微分方程组和边界条件方程组(扩展哈密顿原理)。$w_{1}$ 项的变分会导致运动方程中的积分项,而 $w_{1}$ 本身并不出现,因为它相对较小。首先,给出叶片摆振和扭转运动的偏微分方程,随后给出相应的边界条件。其次,给出风轮方位角和变桨角度的运动方程。
Blade Bending Motion
The equation of motion of the $x$ - and $y$ -directions becomes
@ -195,6 +197,8 @@ $$
The direction of the $x_{\mathrm{{}}}$ - and $y$ -axis can be swapped by changing the $\beta$ angle, hence the only differences between the terms in equations (12a) and (12b) are the directions of projection of the forces. In the following, the individual terms in equation (12) are shown and the physical interpretation of them is discussed. Because of the similarity between the terms from equations (12a) and (12b), only the terms from equation (12a) will be discussed. The influence of pitch action is described by
通过改变β角,可以互换$x_{\mathrm{{}}}$ - 和 $y$ -轴的方向,因此方程(12a)和(12b)中各项的唯一区别在于力的投影方向。以下将展示方程(12)中的各项,并讨论它们的物理意义。由于方程(12a)和(12b)中的各项相似,仅讨论方程(12a)中的各项。俯仰动作的影响由:
$$
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}
$$
@ -204,30 +208,46 @@ F_{u,1}\!=\!\ddot{\beta}m u_{c g}-\dot{\beta}^{2}m\nu_{c g}+2\dot{\beta}m\dot{u}
$$
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
其中,$u_{c g}=u+l_{p i}+l_{c g}\mathrm{cos}(\overline{{\theta}})-l_{c g}\theta\mathrm{sin}(\overline{{\theta}})$ 和 $\nu_{c g}=\nu+l_{c g}\mathrm{sin}(\overline{{\theta}})+l_{c g}\theta\mathrm{cos}(\overline{{\theta}})$ 分别是重力中心在 $(x,y,z)$ 坐标系中的 $x$ 和 $y$ 坐标,$\Gamma_{1}=2m\dot{\beta}\dot{\phi}((u+l_{p i})\mathrm{sin}(\beta)+\nu\mathrm{cos}(\beta)+l_{c g}\mathrm{sin}(\overline{{{\theta}}}+\beta))$ 是与 $(x,y,z)$ 坐标系绕 $z$ 轴和 $\hat{y}$ 轴的角速度相关的 $z$ 方向的科里奥利力。 方程 (13a) 中的第一项是与 $(x,\,y,\,z)$ 坐标系绕 $z$ 轴的角加速度相关的虚假力\*。第二项是与 $(x,y,z)$ 坐标系绕 $z$ 轴的角速度和 $c g$ 在 $x$ 方向上的偏移相关的虚假离心力。第三项是与 $(x,y,z)$ 坐标系绕 $z$ 轴的角速度和 $c g$ 在 $y$ 方向上的速度相关的科里奥利力。 方程 (13a) 中的第四项是由于 $c g$ 的偏移和科里奥利力 $T_{1}$ 引起的弯矩的空间导数。最后一项是科里奥利力 $T_{1}$ 对从该点到叶片末端的剩余部分的弯矩。风轮转速的影响由...描述。
$$
\begin{array}{l}{{F_{u,2}=-\dot{\phi}^{2}m\hat{u}_{c g}\cos(\beta)-\dot{\phi}^{2}\big(m l_{c g}w_{0}\big(\cos(\overline{{\theta}})\big)-\theta\sin(\overline{{\theta}})\big)\big)^{\prime}-\left(l_{c g}T_{2}\right)^{\prime}\cos(\overline{{\theta}})}}\\ {{\mathrm{}}}\\ {{\mathrm{}}\qquad-2\dot{\phi}m l_{c g}\big(\dot{u}^{\prime}\cos(\overline{{\theta}})+\dot{\nu}^{\prime}\sin(\overline{{\theta}})\big)\cos(\beta)\mathrm{-}\Big((u^{\prime}+l_{p i}^{\prime})\Big)_{s}^{R}\big(\dot{\phi}^{2}m w_{0}+T_{2}\big)\mathrm{d}\rho\Big)^{\prime}}}\\ {{F_{\nu,2}=\dot{\phi}^{2}m\hat{u}_{c g}\sin(\beta)-\dot{\phi}^{2}\big(m l_{c g}w_{0}\big(\sin(\overline{{\theta}})+\theta\cos(\overline{{\theta}})\big)\big)^{\prime}-\left(l_{c g}T_{2}\right)^{\prime}\sin(\overline{{\theta}})}}\\ {{\mathrm{}}}\\ {{\mathrm{}}\qquad+2\dot{\phi}m l_{c g}\big(\dot{u}^{\prime}\cos(\overline{{\theta}})+\dot{\nu}^{\prime}\sin(\overline{{\theta}})\big)\sin(\beta)\mathrm{-}\Big(\nu^{\prime}\int_{s}^{R}\big(\dot{\phi}^{2}m w_{0}+T_{2}\big)\mathrm{d}\rho\Big)^{\prime}}}\end{array}
\begin{align}
F_{u,2} &= -\dot{\phi}^{2}m\hat{u}_{cg}\cos(\beta) - \left[\dot{\phi}^{2}\left(m l_{cg}w_{0}\cos(\overline{\theta}) - \theta\sin(\overline{\theta})\right)\right]' \\
&\quad - \left(l_{cg}T_{2}\right)' \cos(\overline{\theta}) \\
&\quad - 2\dot{\phi}m l_{cg}\left(\dot{u}'\cos(\overline{\theta}) + \dot{\nu}'\sin(\overline{\theta})\right)\cos(\beta) \\
&\quad - \left[\Big((u' + l_{pi}')\Big)_{s}^{R}\left(\dot{\phi}^{2}m w_{0} + T_{2}\right)\mathrm{d}\rho\right]' \\
F_{\nu,2} &= \dot{\phi}^{2}m\hat{u}_{cg}\sin(\beta) - \left[\dot{\phi}^{2}\left(m l_{cg}w_{0}\sin(\overline{\theta}) + \theta\cos(\overline{\theta})\right)\right]' \\
&\quad - \left(l_{cg}T_{2}\right)' \sin(\overline{\theta}) \\
&\quad + 2\dot{\phi}m l_{cg}\left(\dot{u}'\cos(\overline{\theta}) + \dot{\nu}'\sin(\overline{\theta})\right)\sin(\beta) \\
&\quad - \left[\nu'\int_{s}^{R}\left(\dot{\phi}^{2}m w_{0} + T_{2}\right)\mathrm{d}\rho\right]'
\end{align}
$$
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
其中 $\hat{u}_{c g}=(u+l_{p i})\mathrm{cos}(\beta)-\nu\mathrm{sin}(\beta)+l_{c g}\mathrm{cos}(\overline{{{\theta}}}+\beta)-l_{c g}\theta\mathrm{sin}(\overline{{{\theta}}}+\beta)$ 是在 $(\hat{x},\hat{y},\hat{z})$ 坐标系中给出,重力中心 $\hat{x}$ 坐标,$T_{2}=2m\dot{\phi}(\dot{u}\cos(\upbeta)-\dot{\nu}\sin(\beta))$ 是与风轮平面内的旋转和 $c g$ 在 $\boldsymbol{\hat{x}}$ 方向上的速度相关的 $z$ 方向上的科里奥利力。方程 (14a) 中的第一项是与风轮平面内的旋转和 $c g$ 在 $x_{\mathrm{{}}}$ 方向上的偏移相关的虚构离心力,投影到 $x_{\mathrm{{}}}$ 方向上。方程 (14a) 中的第二项和第三项分别是由于 $c g$ 到 $e a$ 在 $x$ 方向上的距离引起的弯矩的空间导数,以及虚构的离心力和科里奥利力 $T_{2}$,分别。离心力与风轮平面内的旋转和 $c g$ 从转动中心的偏移相关联。第四项是与叶片在风轮平面内的旋转和 $c g$ 在 $\hat{z}$ 方向上的速度相关的虚构科里奥利力。方程 (14a) 中的最后一项是由于从该点到叶片末端的剩余部分上的虚构离心力和科里奥利力 $T_{2}$ 引起的弯矩。重力影响由描述。
$$
\begin{array}{r l}&{F_{u,3}\!=\!m g\sin(\phi)\cos(\beta)\!+\!\left(\!\left(l_{c g}(u^{\prime}\!+\!l_{p i}^{\prime})\right)^{\prime}\cos(\overline{{\theta}})\cos(\beta)\!+\!\left(l_{c g}\nu^{\prime}\right)^{\prime}\sin(\overline{{\theta}})\cos(\beta)m g\sin(\phi)\!\right.}\\ &{\qquad\left.-\left(m l_{c g}\!\left(\cos(\overline{{\theta}})-\theta\sin(\overline{{\theta}})\right)\right)^{\prime}g\cos(\phi)\!+\!\left((u^{\prime}\!+\!l_{p i}^{\prime})\right)_{s}^{R}m g\cos(\phi)\mathrm{d}\rho\right)^{\prime}}\\ &{F_{u,3}\!=\!-\!m g\sin(\phi)\sin(\beta)\!+\!\left(\!\left(l_{c g}\left(u^{\prime}\!+\!l_{p i}^{\prime}\right)\right)^{\prime}\sin(\overline{{\theta}})\cos(\beta)\!-\!\left(l_{c g}\nu^{\prime}\right)^{\prime}\sin(\overline{{\theta}})\sin(\beta)\!\right)\!m g\sin(\phi)}\\ &{\qquad-\left(m l_{c g}\left(\sin(\overline{{\theta}})-\theta\cos(\overline{{\theta}})\right)\right)^{\prime}g\cos(\phi)\!+\!\left(\nu^{\prime}\!\int_{s}^{R}m g\cos(\phi)\mathrm{d}\rho\right)^{\prime}}\end{array}
$$
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
在方程 (15a) 中,第一项是重力在 $x$ 方向上的分量。第二项是由于重力在 $\hat{x}$ 方向上的分量以及 $z$ 方向上的 $c g$ 偏移量引起的弯矩的空间导数。第三项是由于 $c g$ 和 $e a$ 在 $x$ 方向上的距离以及重力在 $z$ 方向上的分量引起的弯矩的空间导数。方程 (15a) 中的最后一项是由于从该点到叶片末端,重力在 $z$ 方向上的分量作用在叶片剩余部分产生的弯矩。由叶片的弯曲刚度引起的回复力由以下方式描述:
$$
\begin{array}{r l}&{F_{u,4}=\big(E\big(I_{\xi}\cos^{2}\big(\tilde{\theta}\big)+I_{\eta}\sin^{2}\big(\tilde{\theta}\big)\big)u^{\prime\prime}\big)^{\prime\prime}+\big(E(I_{\xi}-I_{\eta})\cos\big(\tilde{\theta}\big)\sin\big(\tilde{\theta}\big)\nu^{\prime\prime}\big)^{\prime\prime}}\\ &{\qquad\qquad-\big(E(I_{\xi}-I_{\eta})\theta\big(u^{\prime\prime}\sin\big(2\tilde{\theta}\big)-\nu^{\prime\prime}\cos\big(2\tilde{\theta}\big)+I_{p i}^{\prime\prime}\sin\big(\tilde{\theta}\big)\cos\big(\tilde{\theta}\big)\big)\big)^{\prime\prime}}\\ &{F_{v,4}=\big(E\big(I_{\xi}\sin^{2}\big(\tilde{\theta}\big)+I_{\eta}\cos^{2}\big(\tilde{\theta}\big)\big)\nu^{\prime\prime}\big)^{\prime\prime}+\big(E(I_{\xi}-I_{\eta})\cos\big(\tilde{\theta}\big)\sin\big(\tilde{\theta}\big)u^{\prime\prime}\big)^{\prime\prime}}\\ &{\qquad+\left(E(I_{\xi}-I_{\eta})\theta\big(u^{\prime\prime}\cos\big(2\tilde{\theta}\big)+\nu^{\prime\prime}\sin\big(2\tilde{\theta}\big)\big)\right)^{\prime\prime}-\big(I_{p i}^{\prime\prime}\theta E(I_{\xi}\sin^{2}(\theta)+I_{\eta}\cos^{2}(\theta))\big)^{\prime\prime}}\end{array}
$$
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
其中第一项为 $x$ 向弯曲刚度,第二项为与 $\nu$ 向的耦合,最后一项为与扭角的耦合。惯性矩由 $I_{\xi}=\int\int_{A}\eta^{2}\mathrm{d}\eta\mathrm{d}\xi$ 和 $I_{\eta}=\int\int_{A}\xi^{2}\mathrm{d}\eta\mathrm{d}\xi$ 给出。风轮的角加速度的影响由以下描述:
$$
F_{u,5}\!={m\ddot{\phi}w_{0}\cos(\beta)},\ \ \ F_{v,5}\!=\!-m{\ddot{\phi}w_{0}}\sin(\beta)
$$
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.
这与关于 $(x,y,z)$ -坐标系绕 $Y$-轴旋转产生的虚构角加速度 $c g$ 相关联。方程 (12a) 和 (12b) 的右侧描述了外部力,$f_{u}$ 和 $f_{\nu}$ 分别是 $x$ 和 $y$ 方向上的力。最后一个项是由于外部力在 $z$ 方向上作用于叶片剩余部分而产生的弯矩,从该点延伸至叶片末端。
Blade Torsional Motion
@ -238,42 +258,45 @@ $$
$$
where the rotor speed leads to the fictitious centrifugal forces:
其中风轮转速导致虚构离心力:
$$
F_{\theta,1}\!=\!m l_{c g}{\dot{\phi}}^{2}{\hat{u}}_{c g}\sin({\overline{{\theta}}}+\beta)+m l_{c g}w_{0}\phi^{2}\big(\nu^{\prime}\cos({\overline{{\theta}}})\!-\!(u^{\prime}\!+\!l_{p i}^{\prime})\sin({\overline{{\theta}}})\big)
$$
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
其中第一项与 $c g$ 在 $\hat{x}$ 方向上的偏移量相关联,第二项与旋转中心到 $c g$ 的距离相关联。俯仰作用的影响由以下描述:
$$
\begin{array}{l}{{F_{\theta,2}=(I_{c g}+m l_{c g}^{2})\ddot{\beta}-\ddot{\beta}m l_{c g}(u\cos(\overline{{{\theta}}})+\nu\sin(\overline{{{\theta}}}))+m l_{c g}\dot{\beta}^{2}\big((u+l_{p i})\sin(\overline{{{\theta}}})-\nu\cos(\overline{{{\theta}}})\big)}}\\ {{\mathrm{}}}\\ {{\mathrm{}\qquad+\,2m l_{c g}\dot{\beta}(\dot{u}\cos(\overline{{{\theta}}})+\dot{\nu}\sin(\overline{{{\theta}}}))}}\end{array}
$$
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:
其中第一项是与 $(x,y,z)$ -坐标系绕 $z$ 轴旋转相关的虚构角加速度;第二项是与 $(x,y,z)$ -坐标系绕 $z$ 轴旋转相关的虚构离心力;最后一项是与 $(x,y,z)$ -坐标系绕 $z$ 轴旋转以及 $c g$ 在弦向上的速度相关的虚构科里奥利力。风轮的加速度导致以下项:
$$
F_{\theta,3}\,{=}\,{-}m\ddot{\phi}w_{0}l_{c g}\sin(\overline{{\theta}}+\beta)
$$
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
这与关于 $(x,y,z)$ 坐标系绕 $Y$ 轴的角加速度相关的,$cg$ 的虚构角加速度。重力效应由以下描述:
$$
F_{\theta,4}=-l_{c g}\big(\mathrm{sin}\big(\beta+\overline{{\theta}}\big)+\theta\cos\big(\beta+\overline{{\theta}}\big)m g\sin(\phi)+l_{c g}\big(\nu^{\prime}\cos\big(\overline{{\theta}}\big)-(u^{\prime}+l_{p i}^{\prime})\sin(\overline{{\theta}}\big)\big)m g\cos(\phi)
$$
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
其中第一项是由于重力 $\hat{x}$ 分量引起的扭转力矩,以及 $c g$ 和 $e a$ 在 ${\hat{y}}$ 方向上的距离;最后一项是由于 $c g$ 和 $e a$ 之间的距离以及重力在变形叶片截面上投影的 $z$ 分量引起的扭转力矩。叶片弯曲和扭转之间的弹性耦合由…描述。
$$
\begin{array}{r l}&{F_{\theta,5}\!=\!-\!\bigg(E I_{\eta\eta\xi}\big(\tilde{\theta}+\theta\big)^{\prime}\big(u^{\prime\prime}\sin(\overline{{\theta}})\!-\!\nu^{\prime\prime}\cos(\theta)\big)\bigg)^{\prime}\!+\!\bigg(E I_{\eta\xi\xi}\big(\tilde{\theta}+\theta\big)^{\prime}\big(u^{\prime\prime}\cos(\tilde{\theta})\!+\nu^{\prime\prime}\sin(\tilde{\theta})\big)\bigg)^{\prime}}\\ &{\qquad-\left(E I_{\xi}-E I_{\eta}\right)\!\big((u^{\prime\prime2}-\nu^{\prime\prime2})\cos(\tilde{\theta})\sin(\tilde{\theta})\!-u^{\prime\prime}\nu^{\prime\prime}\cos(2\tilde{\theta})\big)}\\ &{\qquad-\left.E I_{\xi}l_{p i}^{\prime\prime}\big(u^{\prime\prime}\cos(\tilde{\theta})\!+\nu^{\prime\prime}\sin(\tilde{\theta})\big)\sin(\tilde{\theta})\!+E I_{\eta}l_{p i}^{\prime\prime}\big(u^{\prime\prime}\sin(\tilde{\theta})\!-\!\nu^{\prime\prime}\cos(\tilde{\theta})\big)\right)\!\cos(\tilde{\theta})}\end{array}
$$
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
其中,$I_{\eta\eta\xi}=\ \int\!\int_{A}\!\eta(\eta^{2}\,+\,\xi^{2})\mathrm{d}\eta\mathrm{d}\xi$ 且 $I_{\eta\xi\xi}=\ \int\int_{A}^{}\!\xi(\eta^{2}\,+\,\xi^{2})\mathrm{d}\eta\mathrm{d}\xi$ 。由扭转刚度引起的回复力为:
$$
F_{\theta,6}\,{=}\,{-}(G J(\theta^{\prime}\,{+}\,\nu^{\prime}(u^{\prime\prime}\,{+}\,l_{p i}^{\prime\prime})))^{\prime}
$$
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 $M$ .
其中极惯性矩为 $J=\int\int_{\cal A}{\left(\eta^{2}+\xi^{2}\right)}d\eta d\xi$ 。右侧描述了作用在叶片上的外力矩 $M$ 。
Boundary Conditions
@ -286,35 +309,37 @@ $$
because the coordinate frame used to describe the blade follows the root of the blade.
The boundary conditions for the tip of the blade are determined by the boundary condition equations derived by demanding any admissible variation of the action integral to be zero. The boundary conditions become
因为用于描述叶片的坐标系跟随叶片根部。
叶片尖部的边界条件由要求作用量积分的任何可行变分均为零所推导出的边界条件方程决定。边界条件变为
$$
\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}
$$
If $l_{c g}(R)\neq0$ , the boundary conditions for the tip are functions of rotor speed $\dot{\phi}$ and rotor position $\phi$ and therefore time-varying. This is because an offset of the center of gravity from the elastic axis at the blade tip leads to a bending moment at the tip, caused by the gravity and centrifugal force. Most modern wind turbine blades, however, are tapered at the tip, leading to $l_{c g}(R)/R<<\varepsilon_{*}$ , making the time variation of the boundary conditions negligible.
如果 $l_{c g}(R)\neq0$ ,则叶片尖部的边界条件是风轮转速 $\dot{\phi}$ 和风轮位置 $\phi$ 的函数,因此是随时间变化的。这是因为重力中心相对于叶片尖部的弹性轴存在偏移,导致尖部产生弯矩,由重力和离心力共同作用造成。然而,大多数现代风电机组叶片在尖部采用锥度设计,导致 $l_{c g}(R)/R<<\varepsilon_{*}$ 使得边界条件的随时间变化可以忽略不计
# Pitch Action
The equation of motion for the pitch angle is
$$
\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}
\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},\dot{u},u\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
F_{\beta,1}\big(\dot{\beta},\dot{\nu},u\big)=2\dot{\beta}\int_{r}^{R}m\dot{u}u_{c g}{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}{d}s
$$
are the moments caused by the fictitious Coriolis force associated with the relative velocity of the blade and rotation of the $(x,y,z)$ -frame about the $z_{i}$ -axis. The effect of the fictitious centrifugal force associated with the rotation of the $(x,y,z)$ -frame about the $\hat{y}_{}$ -axis is described by
这些是由于与叶片相对速度和 $(x,y,z)$ -坐标系绕 $z_{i}$ -轴旋转相关的虚构科里奥利力所造成的。与 $(x,y,z)$ -坐标系绕 $\hat{y}_{}$ -轴旋转相关的虚构离心力效应由以下内容描述:
$$
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
$$
where $\hat{\nu}_{c g}=(u+l_{p i})\mathrm{sin}(\beta)+\nu\mathrm{cos}(\beta)+l_{c g}\mathrm{sin}(\overline{{{\theta}}}+\beta)$ is the $\hat{y}$ coordinate of the center of gravity in the (ˆx,ˆy,ˆz)- frame. The effect of an angular acceleration of the $(x,y,z)$ -frame about the $\hat{y}$ -axis is described by
其中 $\hat{\nu}_{c g}=(u+l_{p i})\mathrm{sin}(\beta)+\nu\mathrm{cos}(\beta)+l_{c g}\mathrm{sin}(\overline{{{\theta}}}+\beta)$ 是在 (ˆx,ˆy,ˆz) 坐标系中重心位置的 $\hat{y}$ 坐标。关于 $\hat{y}$ 轴的 (x,y,z) 坐标系角加速度的影响由以下描述:
$$
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
$$
@ -384,14 +409,24 @@ The first term in equation (27) shows the strong coupling between torsional moti
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 Hamiltons method. This will lead to extra equations for the each extra degree of freedom and to periodic coefficients (like the gravity term).
通过对比运动偏微分方程(方程(12)和(18))与 Hodges 和 Dowell 的方程3 发现引力项(方程(15)和(22))、俯仰作用项(方程(13a)和(20))以及涉及风轮转速变化的项(方程(17)和(24)是新引入的。另一方面Hodges 和 Dowell3 中的涉及翘曲效应的项在此处未包含因为对于大多数应用忽略该效应不会造成本质上的精度损失。3
在以下讨论中,将 $x$ 方向和 $y$ 方向分别称为摆振方向和挥舞方向,以帮助物理解释。可以看出,方程(12)中的惯性项将摆振和挥舞运动与叶片的扭转运动耦合在一起。耦合程度取决于弦的预扭角。方程(13)中的第一项表明,变桨角度的加速会激发摆振和挥舞运动,分别取决于挥舞和摆振变形。也就是说,一个挥舞变形的叶片的变桨角度加速会激发叶片的摆振运动。方程(14)中的积分项的第一项是与风轮转速相关的恢复力,称为离心刚度。引力效应(方程(15)和(22))被发现随 $\phi$ 角变化,正如预期。恢复力(方程(16))将弯曲运动与扭转运动耦合在一起。耦合程度取决于叶片的摆振和挥舞变形。风轮的加速会激发摆振和挥舞运动(方程(17)),激发程度取决于变桨角度。方程(18)中的惯性项将扭转运动与摆振和挥舞运动耦合在一起。与摆振和挥舞运动的耦合程度取决于弦的预扭角。方程(20)中的第一项显示了变桨加速度和扭转运动之间的强耦合。风轮加速度(方程(21))对扭转运动的影响取决于变桨角度和叶片的预扭角。弯曲运动通过弯曲刚度(方程(23))与扭转运动耦合在一起。
方程(27)中的第一项显示了扭转运动和变桨运动之间的强耦合。方程(32)中的第一项显示了叶片变形对变桨惯性的影响,而方程(32)中的第二项显示了变形叶片的运动如何影响变桨方程。
为了避免不必要的复杂性,在推导运动方程时没有包含结构阻尼,但可以轻松地将例如粘性阻尼项添加到描述结构阻尼的方程中。
可以通过引入新的惯性系,定义从新的惯性系到当前惯性系的变换,并在应用 Hamilton 方法之前,将新的变换应用于能量描述,从而包含塔架、偏航运动或倾斜等额外的自由度。这将导致每个额外的自由度都有额外的方程,并且会产生周期系数(例如引力项)。
# 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.
在本节中,采用有限差分离散化方法对叶片模型进行计算,以获得特定$63\,\mathrm{m}$叶片的固有振动模态。12 将计算得到的固有振动模态的频率和形状与HAWCstab\*13的结果进行比较结果吻合良好。这些模态被用作假设模态离散化方法的基础用于将运动的偏微分方程近似为三个常微分方程。假设模态近似模型的固有振动模态与先前推导出的模态进行比较结果显示出合理的吻合度。为了说明和测试变桨角度模型使用假设模态近似模型进行快速2°变桨角度的时间模拟。将响应与$\mathrm{HAWC}2^{\dagger5,6}$的结果进行比较,结果吻合良好。
# 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
无外力作用且线性化的运动偏微分方程(方程 (12) 和 (18))的空间导数,被近似为二阶有限差分近似。由此得到的近似常微分方程可以写成:
$$
\hat{\mathbf{M}}\ddot{\tilde{\mathbf{q}}}+\tilde{\mathbf{D}}\ddot{\tilde{\mathbf{q}}}+\tilde{\mathbf{K}}\tilde{\mathbf{q}}=\mathbf{0}
@ -400,55 +435,68 @@ $$
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.
其中,$\tilde{\textbf{M}}$ , $\tilde{\mathbf{D}}$ 和 $\tilde{\bf K}$ 分别代表离散化过程中的常数系数,而 $\tilde{\mathbf{q}}=[u_{I},\nu_{I},\theta_{I}$ , … $,u_{n},\nu_{n},\theta_{n}]$ 代表 $n$ 个离散点处的变形。方程 (38) 是一个微分特征值问题,其特征值给出叶片的固有频率和阻尼,而对应的特征向量则给出固有振动的模态形状。
表 I 比较了叶片的前六个最低特征频率与 HAWCstab 的结果。可以看到,所有频率都表现出良好的吻合度。图 2 显示了第一、第二和第六模态的形状。这些形状与 HAWCstab 的结果进行比较,同样显示出良好的吻合度。
# 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
表I. 测试叶片的头六个简正模态频率
<html><body><table><tr><td></td><td></td><td colspan="2">Finitedifference</td><td colspan="2">Assumedmode</td></tr><tr><td>Modenumber</td><td>HAWC freq. [Hz]</td><td>freq. [Hz]</td><td>%J!P</td><td>[ZH] ba</td><td>diff. %</td></tr><tr><td>1</td><td>0·69</td><td>0.70</td><td>1</td><td>0.63</td><td>7</td></tr><tr><td>2</td><td>1.08</td><td>1·14</td><td>6</td><td>1·04</td><td>4</td></tr><tr><td>3</td><td>1·96</td><td>1·97</td><td>1</td><td></td><td></td></tr><tr><td>4</td><td>3.97</td><td>4.05</td><td>2</td><td></td><td></td></tr><tr><td>5</td><td>4·51</td><td>4·55</td><td>1</td><td></td><td></td></tr><tr><td>6</td><td>7.83</td><td>7.79</td><td>1</td><td>7.97</td><td>2</td></tr></table></body></html>
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.
HAWCstab的计算结果13以及基于当前模型采用的有限差分近似和假设模态近似的结果。均给出了频率以及相对于HAWCstab结果的相对差异。
![](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
图 2. 有限差分逼近模型“– ”和假设模态逼近模型“”计算出的自然振动模态与HAWCstab13 √ 计算出的模态进行比较。(a) 第一模态,(b) 第二模态,(c) 第六模态
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 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 approximated 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.
采用模态法,将运动偏微分方程转化为三个近似常微分方程。<sup>11,14</sup> 叶片的时空相关状态变量,分别用一个摆振模态 $u(s,\,t)=u_{s}(s)u_{t}(t)$,一个挥舞模态 $\nu(s,\,t)=\nu_{s}(s)\nu_{t}(t)$ 和一个扭转模态 $\theta(s,\,t)=\theta_{s}(s)\theta_{t}(t)$ 近似表示。模态形状 $(u_{s},\,\nu_{s},\,\theta_{s})$ 分别为第二、第一和第六模态的摆振、挥舞和扭转分量(第一模态分别由摆振、挥舞和扭转运动主导)。随时间变化的权系数函数 $(u_{t},\,\nu_{t},\,\theta_{t})$ 是系统的新的状态变量。叶片上的外部力也被分解为空间部分和随时间变化的函数,分别为 $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)$ 和 $M(s,t)=M_{s}(s)M_{t}(t)$。将近似值代入方程 (12) 和 (18),并用对应的空间变量对这些方程进行权系数函数乘积并沿叶片长度进行积分,从而消除空间依赖性。
The ordinary differential equation of blade motion becomes
叶片运动的常微分方程变为:
$$
\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}
$$
where $\mathbf{q}=[u_{t},\,\nu_{t},\,\theta_{t}]^{\mathrm{T}}$ and the rest of the terms are given in equation (50) in Appendix B. Inserting the expansions into equation (27), the integrals can be computed and the equation of pitch action becomes
其中 $\mathbf{q}=[u_{t},\,\nu_{t},\,\theta_{t}]^{\mathrm{T}}$其余项见附录B中的公式(50)。将展开式代入公式(27),可以计算积分,变桨角度作用方程变为
$$
\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}
$$
The individual terms are given in equation (51) in Appendix B. Inserting the expansions into equation (33) and computing the integrals, the equation of rotor position becomes
单个项的表达式见附录B中的公式(51)。将这些展开式代入公式(33)并计算积分后,风轮位置方程变为:
$$
\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}
$$
The individual terms are given in equation (52) in Appendix B. An unforced and linearized version of equation (39) gives a differential eigenvalue problem:
单个项的表达式见附录B中的公式(52)。公式(39)的一个未施加力和线性化的版本给出一个微分特征值问题:
$$
\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.
其中特征值给出了假设模态近似模型固有振动的频率而特征向量则给出了固有振动中模态之间的耦合关系。所求频率与表I中先前求得的频率进行比较结果吻合良好。图2显示了固有模态形状以及先前求得的模态形状。可以观察到第一和第二模态的摆振和挥舞分量与先前结果非常吻合。第一模态的扭转分量与先前结果略有差异。第二模态的扭转分量与先前结果存在差异但其扭转分量的数值相对于摆振和挥舞分量较小因此该误差是可以接受的。第六模态第一扭转模态的摆振和挥舞分量与先前结果存在较大差异。这是因为摆振和挥舞分量主要由高阶摆振和挥舞运动主导而该低阶模型无法捕捉到这些运动。然而摆振和挥舞分量的数值相对于扭转分量较小因此该误差是可以接受的。
# 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.
变桨角度模型通过数值模拟进行说明和测试,其中风轮以恒定角速度 $\dot{\Phi}$ rad $\mathrm{s}^{-1}$ 旋转,并在 $70\,\mathrm{s}$ 时施加了 2° 的变桨角度变化。该变桨角度变化具有 0·2 s 的上升时间和 $1\!\cdot\!5\%$ 的超调量。此示例中未包含任何空气动力学力。通过将规定的变桨动作和计算出的叶片运动输入到(方程 (40))中来计算俯仰力矩。模拟结果与 HAWC2\*,5,6 的结果进行比较,显示出良好的吻合度。
图 3 显示了本模型和 HAWC2 的叶片尖端变形和俯仰力矩。摆振和挥舞运动主要受重力影响,这在 5s 尺度上的振荡中可见(对应于风轮速度为 $0.79\,\mathrm{rad\s^{-1}}$ )。在 $70\,\mathrm{s}$ 时的变桨动作处观察到挥舞运动的小幅度激发。叶片的扭转运动在 70s 时的变桨动作处受到强烈激发。变桨动作期间俯仰力矩较高,此后受到叶片扭转运动的强烈影响。两个模型的挥舞运动非常吻合。在 5s 尺度上的挥舞运动幅度略小于本模型,而 HAWC2 的 $70\,\mathrm{s}$ 处的激发略有增强,但总体而言,两个模型仍然吻合良好。扭转运动的幅度吻合得非常好,但频率存在轻微差异。两个模型的俯仰力矩之间存在良好的吻合度。
![](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, $\cdot\cdot^{\prime}H A W C2^{5,6}$
@ -457,13 +505,17 @@ Figure 3. Tip deflection and pitch moment of a blade rotating with a constant sp
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.
有限差分离散模型的结果表明本模型能够捕捉到叶片的根本特性与HAWCstab.13一致。假设模态模型的结果表明,即使仅使用三个常微分方程,就可以描述叶片的重要基本特性,并且可以很好地模拟变桨叶片相互作用。
运动方程相对简单的结构(方程(39))使其适用于定性分析变桨动作与叶片运动之间的相互作用和/或快速模拟。方程(39)的结构类似于二维叶片截面模型如Chaviaropoulos et al.1和Block and Strganac2所用的运动方程结构因此本模型具有与二维叶片截面模型相同的优势但与真实机组叶片具有明确的关联。风轮位置模型方程(41))可用于分析一个叶片的运动如何影响风轮转速,但更重要的是,它可以很容易地扩展到更多叶片,从而实现各个叶片运动之间的耦合。通过为每个参与叶片添加方程(52)中的每一项,可以扩展风轮位置模型以包含更多叶片。如果使用更多的模态或耦合模态,可以实现对叶片运动的改进描述。但缺点是系统会变得更加复杂,使得解析分析和解释更加困难。
# 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.
本工作扩展了最初由 Hodges 和 Dowell 推导出的非线性偏微分运动方程,考虑了变桨角度作用、风轮转速变化和重力影响。针对变桨角度作用和风轮转速,导出了新的方程。计算了叶片的固有振动频率和形状,并与 HAWCstab 的结果进行了比较,结果吻合良好。通过假设模态离散化,将偏微分运动方程转化为近似的常微分运动方程(方程 (39))。该模型适用于变桨角度作用和叶片运动之间的基本相互作用分析。利用近似的常微分运动方程,模拟了风电机组叶片在快速 2° 变桨角度作用下的响应和俯仰力矩。模拟结果与 HAWC2 的结果进行了比较,结果吻合良好。
本工作是关于变桨叶片相互作用项目的一部分,该模型将进一步扩展,纳入气动模型,并用于分析变桨叶片相互作用的基本特性。
# 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.