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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

@@ -174,7 +174,7 @@ where ${\dot{\beta}}+{\dot{\theta}}$ is the angular velocity of the blade sectio
 # 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：
+非保守力通过描述它们对任何容许变分所做的变分功来加以考虑：
 $$
 \delta Q=T_{g c n}\delta\phi+M_{p i t c h}\delta\beta+\int_{r}^{R}(\mathbf{f}^{\mathrm{{T}}}\cdot\delta\mathbf{r}_{e a}+M\delta(\theta+\beta))\mathrm{d}s
 $$  
@@ -182,7 +182,7 @@ $$
 where $\mathbf{f}=\mathbf{T}_{\phi}^{\mathrm{T}}\mathbf{T}_{\beta}^{\mathrm{T}}[f_{u},f_{\nu},f_{w}]^{\mathrm{T}}$ and  
 
 $$
-\mathbf{r}_{e a}\!=\!\{\mathbf{I},\mathbf{J},\mathbf{K}\}\mathbf{T}_{\phi}^{\mathrm{T}}\mathbf{T}_{\beta}^{\mathrm{T}}[l_{p i}+u,\nu,w_{0}+w_{1}]^{\mathrm{T}}
+\mathbf{r}_{e a}\!=\![\mathbf{I},\mathbf{J},\mathbf{K}]\mathbf{T}_{\phi}^{\mathrm{T}}\mathbf{T}_{\beta}^{\mathrm{T}}[l_{p i}+u,\nu,w_{0}+w_{1}]^{\mathrm{T}}
 $$  
 
 is a position vector describing the elastic axis.  
@@ -191,18 +191,20 @@ is a position vector describing the elastic axis.
 
 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. 
 
-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  
+通过要求作用量积分 $\delta H\equiv\int_{\mathbf{t_{1}}}^{\mathbf{t_{2}}}(\delta T-\delta V_{e l a}-\delta V_{g r a}+\delta Q)d t$ 的任何可容许变分为零，导出了一组运动偏微分方程和一组边界条件方程（广义哈密顿原理11）。$w_{1}$项的变分导致运动方程中出现积分项，而$w_{1}$本身不出现，因为它相对较小。首先，介绍了叶片弯曲和扭转运动的偏微分方程，接着是相应的边界条件。其次，介绍了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  
-
+$x$ 和 $y$ 方向的运动方程变为
 $$
 \begin{array}{r l}&{m\big(\ddot{u}-\ddot{\theta}l_{c g}\sin(\overline{{\theta}})\big)+F_{u,1}\big(\ddot{\beta},\dot{\beta},\dot{\phi},\dot{\nu},\dot{\theta},u^{\prime},u,\nu,\theta,\beta\big)+F_{u,2}\big(\dot{\phi},\dot{u},\dot{\nu},u^{\prime},\nu,\theta,\beta\big)}\\ &{\quad+F_{u,3}(\phi,\beta,\theta,u^{\prime},\nu^{\prime})+F_{u,4}(u^{\prime\prime},\nu^{\prime\prime},\theta)+F_{u,5}\big(\ddot{\phi},\beta\big)=f_{u}+\Big((u^{\prime}+l_{p i}^{\prime})\int_{s}^{R}f_{w}\mathrm{d}\rho\Big)^{\prime}}\\ &{\quad m\big(\ddot{\nu}+\ddot{\theta}l_{c g}\cos(\overline{{\theta}})\big)+F_{\nu,1}\big(\ddot{\beta},\dot{\beta},\dot{\phi},\dot{\theta},\nu^{\prime},u,\nu,\theta,\beta\big)+F_{\nu,2}\big(\dot{\phi},\dot{u},\dot{\nu},u^{\prime},\nu,\theta,\beta\big)}\\ &{\quad+F_{\nu,3}\big(\phi,\beta,\theta,u^{\prime},\nu^{\prime}\big)+F_{\nu,4}\big(u^{\prime\prime},\nu^{\prime\prime},\theta\big)+F_{\nu,5}\big(\ddot{\phi},\beta\big)=f_{\nu}+\Big(\nu^{\prime}\int_{s}^{R}f_{w}\mathrm{d}\rho\Big)^{\prime}}\end{array}
 $$  
 
 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  
 
-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：
+通过改变$\beta$角可以互换$x$轴和$y$轴的方向，因此方程(12a)和(12b)中各项的唯一区别是力的投影方向。在下文中，展示了方程(12)中的各项并讨论了它们的物理意义。由于方程(12a)和(12b)中各项的相似性，将只讨论方程(12a)中的各项。变桨作用的影响由
+
 
 $$
 F_{u,1}\!=\!-\ddot{\beta}m\nu_{c g}-\dot{\beta}^{2}m u_{c g}-2\dot{\beta}m\dot{\nu}_{c g}+\left(T_{1}l_{c g}\cos(\overline{{{\theta}}})\right)^{\prime}+\left((u^{\prime}+l_{p i}^{\prime})\!\int_{s}^{R}T_{1}\!\mathrm{d}\rho\right)^{\prime}
@@ -213,7 +215,7 @@ F_{u,1}\!=\!\ddot{\beta}m u_{c g}-\dot{\beta}^{2}m\nu_{c g}+2\dot{\beta}m\dot{u}
 $$  
 
 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。
+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 $\nu_{c g}=\nu+l_{c g}\mathrm{sin}(\overline{{\theta}})+l_{c g}\theta\mathrm{cos}(\overline{{\theta}})$ 分别是质心在 $(x,y,z)$ 坐标系中的 $x$ 和 $y$ 坐标，$\Gamma_{1}=2m\dot{\beta}\dot{\phi}((u+l_{p i})\mathrm{sin}(\beta)+\nu\mathrm{cos}(\beta)+l_{c g}\mathrm{sin}(\overline{{{\theta}}}+\beta))$ 是在 $z$ 方向上的科里奥利力，它与 $(x,y,z)$ 坐标系绕 $z$ 轴和绕 $\hat{y}$ 轴的角速度相关联。方程 (13a) 中的第一项是与 $(x,\,y,\,z)$ 坐标系绕 $z$ 轴的角加速度相关联的 $x$ 方向上的虚拟力*。第二项是与 $(x,y,z)$ 坐标系绕 $z$ 轴的角速度以及质心在 $x$ 方向上的偏移相关联的虚拟离心力。第三项是与 $(x,y,z)$ 坐标系绕 $z$ 轴的角速度以及质心在 $y$ 方向上的速度相关联的科里奥利力。方程 (13a) 中的第四项是由质心偏移和科里奥利力 $T_{1}$ 引起的力矩的空间导数。最后一项是由科里奥利力 $T_{1}$ 作用在叶片从该点到叶尖的剩余部分上所引起的弯矩。风轮转速的影响由下式描述：
 
 
 $$