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@ -272,27 +272,26 @@ $$
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F_{\theta,1}\!=\!m l_{c g}{\dot{\phi}}^{2}{\hat{u}}_{c g}\sin({\overline{{\theta}}}+\beta)+m l_{c g}w_{0}\phi^{2}\big(\nu^{\prime}\cos({\overline{{\theta}}})\!-\!(u^{\prime}\!+\!l_{p i}^{\prime})\sin({\overline{{\theta}}})\big)
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$$
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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
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其中第一项与 $c g$ 在 $\hat{x}$ 方向上的偏移相关联，第二项与从旋转中心到 $c g$ 的距离相关联。变桨作用的影响由下式描述：
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其中第一项与 $c g$ 在 $\hat{x}$ 方向上的偏移量相关联，而第二项与从旋转中心到 $c g$ 的距离相关联。变桨作用的影响由下式描述：
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$$
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\begin{array}{l}{{F_{\theta,2}=(I_{c g}+m l_{c g}^{2})\ddot{\beta}-\ddot{\beta}m l_{c g}(u\cos(\overline{{{\theta}}})+\nu\sin(\overline{{{\theta}}}))+m l_{c g}\dot{\beta}^{2}\big((u+l_{p i})\sin(\overline{{{\theta}}})-\nu\cos(\overline{{{\theta}}})\big)}}\\ {{\mathrm{}}}\\ {{\mathrm{}\qquad+\,2m l_{c g}\dot{\beta}(\dot{u}\cos(\overline{{{\theta}}})+\dot{\nu}\sin(\overline{{{\theta}}}))}}\end{array}
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$$
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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:
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其中，第一项是与 $(x,y,z)$ 坐标系绕 $z$ 轴的角加速度相关联的虚角加速度。第二项是与 $(x,y,z)$ 坐标系绕 $z$ 轴旋转相关联的虚离心力。最后一项是与 $(x,y,z)$ 坐标系绕 $z$ 轴的旋转以及 $cg$ 在弦向的速度相关联的虚科里奥利力。风轮的加速度导致以下项：
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其中第一项是与 $(x,y,z)$ 坐标系绕 $z$ 轴的角加速度相关的虚拟角加速度。第二项是与 $(x,y,z)$ 坐标系绕 $z$ 轴旋转相关的虚拟离心力。最后一项是与 $(x,y,z)$ 坐标系绕 $z$ 轴的旋转以及 $cg$ 在弦向的速度相关的虚拟科里奥利力。风轮的加速度导致以下项：
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$$
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F_{\theta,3}\,{=}\,{-}m\ddot{\phi}w_{0}l_{c g}\sin(\overline{{\theta}}+\beta)
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$$
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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
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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：
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这是与 $(x,y,z)$ 坐标系绕 $Y$ 轴的角加速度相关的重心 $cg$ 的虚拟角加速度。重力效应由
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$$
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F_{\theta,4}=-l_{c g}\big(\mathrm{sin}\big(\beta+\overline{{\theta}}\big)+\theta\cos\big(\beta+\overline{{\theta}}\big)m g\sin(\phi)+l_{c g}\big(\nu^{\prime}\cos\big(\overline{{\theta}}\big)-(u^{\prime}+l_{p i}^{\prime})\sin(\overline{{\theta}}\big)\big)m g\cos(\phi)
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$$
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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
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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。
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其中第一项是由重力在$\hat{x}$方向的分量以及$cg$和$ea$在${\hat{y}}$方向上的距离引起的扭转力矩。最后一项是由$cg$和$ea$之间的距离以及投影到变形叶片截面上的重力在$z$方向的分量引起的扭转力矩。叶片的弯曲和扭转之间的弹性耦合由下式描述。
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$$
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\begin{array}{r l}&{F_{\theta,5}\!=\!-\!\bigg(E I_{\eta\eta\xi}\big(\tilde{\theta}+\theta\big)^{\prime}\big(u^{\prime\prime}\sin(\overline{{\theta}})\!-\!\nu^{\prime\prime}\cos(\theta)\big)\bigg)^{\prime}\!+\!\bigg(E I_{\eta\xi\xi}\big(\tilde{\theta}+\theta\big)^{\prime}\big(u^{\prime\prime}\cos(\tilde{\theta})\!+\nu^{\prime\prime}\sin(\tilde{\theta})\big)\bigg)^{\prime}}\\ &{\qquad-\left(E I_{\xi}-E I_{\eta}\right)\!\big((u^{\prime\prime2}-\nu^{\prime\prime2})\cos(\tilde{\theta})\sin(\tilde{\theta})\!-u^{\prime\prime}\nu^{\prime\prime}\cos(2\tilde{\theta})\big)}\\ &{\qquad-\left.E I_{\xi}l_{p i}^{\prime\prime}\big(u^{\prime\prime}\cos(\tilde{\theta})\!+\nu^{\prime\prime}\sin(\tilde{\theta})\big)\sin(\tilde{\theta})\!+E I_{\eta}l_{p i}^{\prime\prime}\big(u^{\prime\prime}\sin(\tilde{\theta})\!-\!\nu^{\prime\prime}\cos(\tilde{\theta})\big)\right)\!\cos(\tilde{\theta})}\end{array}
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$$
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@ -305,12 +304,12 @@ F_{\theta,6}\,{=}\,{-}(G J(\theta^{\prime}\,{+}\,\nu^{\prime}(u^{\prime\prime}\,
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$$
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where the polar moment of inertia is $J=\int\int_{\cal A}{\left(\eta^{2}+\xi^{2}\right)}d\eta d\xi$ . The right hand side describes the external moment on the blade $M$ .
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其中极惯性矩为 $J=\int\int_{\cal A}{\left(\eta^{2}+\xi^{2}\right)}d\eta d\xi$ 。右侧描述了作用在叶片上的外力矩 $M$ 。
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其中，极惯性矩为 $J=\int\int_{\cal A}{\left(\eta^{2}+\xi^{2}\right)}d\eta d\xi$ 。等式右侧描述了作用在叶片上的外部力矩 $M$ 。
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Boundary Conditions
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## Boundary Conditions
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The boundary conditions for the root of the blade are given by the geometric constraints:
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叶片根部的边界条件由几何约束给出：
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$$
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u(0,t)\!=\!u^{\prime}(0,t)\!=\!\nu(0,t)\!=\!\nu^{\prime}(0,t)\!=\!\theta(0,t)\!=\!0
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$$
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@ -320,19 +319,19 @@ because the coordinate frame used to describe the blade follows the root of the
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The boundary conditions for the tip of the blade are determined by the boundary condition equations derived by demanding any admissible variation of the action integral to be zero. The boundary conditions become
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因为用于描述叶片的坐标系跟随叶片根部。
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叶片尖部的边界条件由要求作用量积分的任何可行变分均为零所推导出的边界条件方程决定。边界条件变为
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叶尖的边界条件由要求作用量积分的任何容许变分为零所推导出的边界条件方程确定。边界条件为：
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$$
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\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}
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$$
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If $l_{c g}(R)\neq0$ , the boundary conditions for the tip are functions of rotor speed $\dot{\phi}$ and rotor position $\phi$ and therefore time-varying. This is because an offset of the center of gravity from the elastic axis at the blade tip leads to a bending moment at the tip, caused by the gravity and centrifugal force. Most modern wind turbine blades, however, are tapered at the tip, leading to $l_{c g}(R)/R<<\varepsilon_{*}$ , making the time variation of the boundary conditions negligible.
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如果 $l_{c g}(R)\neq0$ ，则叶片尖部的边界条件是风轮转速 $\dot{\phi}$ 和风轮位置 $\phi$ 的函数，因此是随时间变化的。这是因为重力中心相对于叶片尖部的弹性轴存在偏移，导致尖部产生弯矩，由重力和离心力共同作用造成。然而，大多数现代风电机组叶片在尖部采用锥度设计，导致 $l_{c g}(R)/R<<\varepsilon_{*}$ ，使得边界条件的随时间变化可以忽略不计。
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# Pitch Action
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如果 $l_{c g}(R)\neq0$ ，尖端边界条件是风轮转速 $\dot{\phi}$ 和风轮位置 $\phi$ 的函数，因此是时变的。这是因为叶片尖端处重心偏离弹性轴会导致尖端产生由重力和离心力引起的弯矩。然而，大多数现代风电机组叶片在尖端处是锥形的，导致 $l_{c g}(R)/R<<\varepsilon_{*}$ ，使得边界条件的时变性可以忽略不计。
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## Pitch Action
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The equation of motion for the pitch angle is
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$$
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\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}
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\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(l_{p i}^{2}+2l_{c g}l_{p i}\cos(\overline{{\theta}})\big)\ddot{\beta}-m\ddot{u}l_{c g}\sin(\overline{{\theta}})+m\ddot{\nu}\big(l_{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+l_{p i}\big)-f_{u}\nu\big)\mathrm{d}s}}\end{array}
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$$
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where
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@ -348,7 +347,7 @@ F_{\beta,3}\big(\dot{\phi},u,\nu,\beta\big)\!=\dot{\phi}^{2}\int_{r}^{R}m\hat{u}
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$$
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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
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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：
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其中 $\hat{\nu}_{c g}=(u+l_{p i})\mathrm{sin}(\beta)+\nu\mathrm{cos}(\beta)+l_{c g}\mathrm{sin}(\overline{{{\theta}}}+\beta)$ 是重心在 (ˆx,ˆy,ˆz) 坐标系中的 $\hat{y}$ 坐标。 $(x,y,z)$ 坐标系绕 $\hat{y}$ 轴的角加速度的影响由下式描述。
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$$
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F_{\beta,4}\big(\ddot{\phi},u,\nu,\beta\big)\!=-\ddot{\phi}\!\!\int_{r}^{R}m w_{0}\hat{\nu}_{c g}\mathrm{d}s
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$$
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@ -366,19 +365,21 @@ F_{\beta,6}(\vec{\beta},\ddot{u},\ddot{\nu},u,\nu)=\ddot{\beta}\int_{r}^{R}m(u^{
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$$
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is nonlinear inertia.
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是非线性惯性。
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If the pitch angle is prescribed or given by an external model, equation (27) can be used to compute the pitch moment, by solving for $M_{p i t c h}$ and feed in the blade motion and pitch action.
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# Rotor Position
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如果变桨角度是预设的或由外部模型给定的，则方程 (27) 可用于计算变桨力矩，方法是求解 $M_{p i t c h}$ 并输入叶片运动和变桨动作。
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## Rotor Position
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Assuming a rigid drive train and no gearing, the rotor position is described by
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假设传动链为刚性且无齿轮传动，风轮位置由
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$$
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\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}
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$$
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The effect of the fictitious centrifugal force associated with rotation of the $(x,\,y,\,z)$ -frame about the $z$ -axis is described by
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与 $(x,\,y,\,z)$ 坐标系绕 $z$ 轴旋转相关的虚构离心力的影响由下式描述：
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$$
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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
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$$
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@ -390,7 +391,7 @@ F_{\phi,2}\big(\dot{\beta},\dot{u},\dot{\nu},\dot{\theta},\beta\big)\!=-2\dot{\b
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$$
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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
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描述了由 $(x,y,z)$ 坐标系绕 $z$ 轴的旋转以及叶片的相对速度所产生的虚构科里奥利力。重力效应由
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$$
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F_{\phi,3}(u,\phi)=g\sin(\phi)\!\!\int_{r}^{R}m w_{0}\mathrm{d}s+g\cos(\phi)\!\!\int_{r}^{R}m{\hat{u}}_{c g}\mathrm{d}s
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$$
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@ -402,11 +403,15 @@ F_{\phi,4}\big(\ddot{\beta},u,\nu,\beta\big)\!=-\ddot{\beta}\!\!\int_{r}^{R}m w_
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$$
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describes the fictitious acceleration associated with an angular acceleration of the $\left(x,\,y,\,z\right)$ -frame about the $z$ -axis.
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描述了与 $\left(x,\,y,\,z\right)$ 坐标系绕 $z$ 轴的角加速度相关的虚拟加速度。
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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).
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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.
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叶片上力的影响以及叶片运动对风轮转速的影响由方程 (33) 中的两个积分项以及方程 (34) 至 (37) 描述。
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风轮转速方程 (33) 仅包含一个叶片的影响，但可以通过为每个额外的叶片在方程 (33) 中额外增加两个积分项，并增加一套方程 (34) 至 (37) 来扩展以包含更多叶片的影响。
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# Discussion
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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
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