diff --git a/.obsidian/plugins/copilot/data.json b/.obsidian/plugins/copilot/data.json
index 8841520..7f611bf 100644
--- a/.obsidian/plugins/copilot/data.json
+++ b/.obsidian/plugins/copilot/data.json
@@ -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    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    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，span翻译为展向\n    Return only the translated text.</instruction>\n\n<text>{copilot-selection}</text>",
       "showInContextMenu": true,
       "modelKey": "gemma3:12b|ollama"
     },
diff --git a/力学书籍/OpenFast/modeling of the uae wind turbine for refinement of fast_ad/auto/modeling of the uae wind turbine for refinement of fast_ad.md b/力学书籍/OpenFast/modeling of the uae wind turbine for refinement of fast_ad/auto/modeling of the uae wind turbine for refinement of fast_ad.md
index 3af48e2..fedfe0f 100644
--- a/力学书籍/OpenFast/modeling of the uae wind turbine for refinement of fast_ad/auto/modeling of the uae wind turbine for refinement of fast_ad.md	
+++ b/力学书籍/OpenFast/modeling of the uae wind turbine for refinement of fast_ad/auto/modeling of the uae wind turbine for refinement of fast_ad.md	
@@ -839,14 +839,15 @@ where $\phi_{1 T}(h)$ and $\phi_{2T}(h)$ are the first and second natural mode s
 其中 $\phi_{1 T}(h)$ 和 $\phi_{2T}(h)$ 分别是塔的第一个和第二个固有振型。 在这些表达式中，塔柔性部分沿高度方向的坐标 $h$ 从零到 $H$ 变化。需要注意的是，$h$ 在相对于地球表面的高度 $H_{S}$ 时等于零。 此外，振型的导数在高度 $h=H$ 处进行评估，如所示。 这些塔的固有振型的推导见第 3.2 节，其中假设塔在纵向和横向方向上独立挠曲；然而，假设每个方向的固有振型在每个方向上是相同的。 方程 (3.4) 中存在负号，因为塔顶底板的正向纵向位移倾向于使底板绕负 ${\pmb a}_{3}$ 轴旋转。 与塔尖挠度相关的广义坐标是时间的函数；塔顶底板的纵向和横向旋转也是时间的函数。
 
 Attached to the tower-top base plate is a yaw bearing (O).  The yaw bearing allows everything atop the tower to rotate $\left(q_{\delta}\right)$ as winds change direction.  The yaw bearing also has the flexibility to allow everything atop the tower to tilt $(q_{\cal S})$ when responding to wind loads.  The nacelle houses the generator and gearbox and supports the rotor.  The center of mass of the nacelle (D) is related to the tower-top base plate by the position vector $r^{O D10}$ :  
-
+塔顶底板上安装有偏航轴承（O）。偏航轴承允许塔顶所有部件随着风向变化而旋转（$q_{6}$）。此外，偏航轴承还具有一定的柔性，使其能够在应对风荷载时允许塔顶所有部件倾斜（$q_{5}$）。机舱内容纳了发电机和齿轮箱，并支撑着风轮。机舱的质心（D）与塔顶底板之间的位置由位置向量 $r^{O D10}$ 描述：
 $$
-\pmb{r}^{O D}=D_{N M I}\pmb{c}_{I}+\left(D_{N M2}+T W R H T O F F S E T\right)\pmb{c}_{2}
+\pmb{r}^{O D}=D_{N M 1}\pmb{c}_{I}+\left(D_{N M2}+T W R H T O F F S E T\right)\pmb{c}_{2}
 $$  
 
 Blade 1 is at an azimuth angle of $q_{4}$ .  The zero-azimuth reference position can be located by the azimuth offset parameter $z(4)$ .  Blade 2 is naturally $180^{\circ}$ out of phase with blade 1.  Drive train flexibility allows the induction generator to see an angular velocity that is different than $n$ times the angular velocity of the rotor, where $n$ is the gearbox ratio.  The twist of the low-speed shaft is, because of its torsional flexibility, modeled with the $q_{I5}$ parameter.  The azimuth angle $q_{I5}$ , is essentially the sum of the azimuth angle, $q_{4}$ , and the twist of the low-speed shaft.  
-
+叶片 1 的方位角为 $q_{4}$ 。零方位角参考位置可以通过方位角偏移参数 $z(4)$ 确定。叶片 2 与叶片 1 固有相位差为 $180^{\circ}$。传动系统的柔性使得感应发电机看到的角速度与转子角速度的 $n$ 倍不同，其中 $n$ 是齿轮箱比。由于其扭转柔性，低速轴的扭转由参数 $q_{15}$ 进行建模。方位角 $q_{15}$，本质上是方位角，即 $q_{4}$，与低速轴扭转之和。
 If applicable to the wind turbine under consideration, teeter motion of the rotor is about a pin (P) fixed on the low-speed shaft.  The position vector connecting the teeter pin to the tower-top base plate, $r^{O P}$ , is10:  
+如果适用于所考虑的风力发电机，转子倾斜运动是绕固定在低速轴上的一个销（P）。连接倾斜销到塔顶底板的位置向量，$r^{O P}$，为10：
 
 $$
 \pmb{r}^{O P}=D_{N}\pmb{c}_{I}+T W R H T O F F S E T\pmb{c}_{2}
@@ -856,42 +857,53 @@ For an upwind turbine configuration, the distance between the tower-top base pla
 
 The delta-3 angle, $\delta_{3}$ , orients the axis of the unconed rotor blades so it is no longer perpendicular to the axis of the teeter pin.  In this case, the teeter motion has both a flapping and a pitching component.  If the delta-3 angle is zero, teeter motion is purely flapping motion.  
 
-Each blade can be coned a different amount ( $\beta_{I}$ for blade 1 and $\beta_{2}$ for blade 2), though the coning angles are constant, not changing with time.  Coning begins in the very center of the hub at point Q, which is offset of the teeter pin (P) by a distance $R_{U}$ along the central axis of the hub (undersling length).  The position vector connecting the blade axes intersection point and the teeter pin, $\bar{\mathbf{r}}^{P Q}$ , is:  
+对于迎风式风力发电机组配置，塔顶底板与偏摆销在 $c_{I}=e_{I}$ 方向上的距离，$D_{N}$，必须小于零。
 
+delta-3 角，$\delta_{3}$，使非偏摆的叶片旋转轴不再垂直于偏摆销的轴线。在这种情况下，偏摆运动既有摆动分量，也有俯仰分量。如果 delta-3 角为零，偏摆运动纯粹是摆动运动。
+
+Each blade can be coned a different amount ( $\beta_{I}$ for blade 1 and $\beta_{2}$ for blade 2), though the coning angles are constant, not changing with time.  Coning begins in the very center of the hub at point Q, which is offset of the teeter pin (P) by a distance $R_{U}$ along the central axis of the hub (undersling length).  The position vector connecting the blade axes intersection point and the teeter pin, $\bar{\mathbf{r}}^{P Q}$ , is:  
+每个叶片可以被锥入不同的角度（叶片 1 为 $\beta_{I}$，叶片 2 为 $\beta_{2}$），尽管锥入角度是恒定的，不会随时间变化。锥入始于枢轴中心点 Q，该点相对于偏心铰链 (P) 沿枢轴中心轴（悬挂长度）偏移了距离 $R_{U}$。连接叶片轴线交点和偏心铰链的位移矢量 $\bar{\mathbf{r}}^{P Q}$ 为：
 $$
 {\pmb r}^{P Q}=-R_{U}{\pmb g}_{I}
 $$  
 
 Located between point $\mathrm{P}$ and point Q, along the central axis of the hub, is the hub center of mass (C) (not shown in Fig. 3.1).  The position vector connecting the hub center of mass and the teeter pin, $\bar{\pmb{r}}^{P C}$ , is:  
-
+位于点 $\mathrm{P}$ 和点 Q 之间，沿着轮毂的中心轴，是轮毂质心 (C)（如图 3.1 所示未标注）。连接轮毂质心和摇臂销的位移矢量 $\bar{\pmb{r}}^{P C}$ 为：
 $$
 {\pmb r}^{P C}=-R_{\phantom{}_{U M}}{\pmb g}_{I}
 $$  
 
 Similar to the tower, the root of each rotor blade can be considered rigid to some radius $R_{H}$ representing the robustness of the hub (hub radius).  The length of the flexible part of each blade is thus $R-R_{H},$ where $R$ is the total radius of the rotor (also not shown in Fig. 3.1).  The flexible part of each blade is assumed to deflect in the local flapwise (out-of-plane of rotor if pitch and twist distribution equal zero) and local edgewise (in-plane of rotor if pitch and twist distribution equal zero) directions independently.  Local means that the flapwise and edgewise directions are unique to each blade element as defined by the sum of the distributed structural pretwist angle, $\theta_{S}(\bar{r^{\flat}})^{9}$ , and the blade collective pitch angle.  Unlike the tower, the natural mode shapes in each direction are permitted to be different.  The natural mode shapes for each blade are assumed to be identical.  
 
+类似于塔架，每个风轮叶片的根部可以被视为在一定半径 $R_{H}$ 内刚性，该半径代表轮毂的稳固性（轮毂半径）。每个叶片柔性部分的长度因此为 $R-R_{H}$，其中 $R$ 是风轮的总半径（如图 3.1 所示）。每个叶片的柔性部分被假定在局部挥舞（如果变桨角度和扭角分布为零，则为风轮平面外方向）和局部摆振（如果变桨角度和扭角分布为零，则为风轮平面内方向）方向上独立挠曲。局部意味着挥舞和摆振方向是每个叶片单元所特有的，由分布的结构预扭角之和 $\theta_{S}(\bar{r^{\flat}})^{9}$ 和叶片整体变桨角度共同定义。与塔架不同，每个方向的固有振型可以不同。每个叶片的固有振型被假定是相同的。
+
 Because each blade can have some distributed structural pretwist, defining the deflection in two directions is complicated.  The most viable method is to define the total blade curvature as the combination of the local curvature in each local blade element direction (flapwise or edgewise), resolved into in-plane and out-of-plane components by orienting them with the structural pretwist and blade collective pitch angles.  This curvature can then be integrated twice to get the total deflection shape. Assuming that the blade deflections are small, the local curvatures in the flapwise and edgewise directions at a span of $r$ , and time $t$ , $\kappa_{\/F}(r_{\/,}t)$ and $\kappa_{E}(r,t)$ respectively, for blade 1, are11:  
 
+**由于每一叶片都可能存在分布式结构预扭角，因此定义两个方向的挠曲变得复杂**。**最可行的方法是将总叶片曲率定义为每个局部叶片单元方向（挥舞方向或摆振方向）的局部曲率之和，通过与结构预扭角和叶片整体变桨角度对齐，将其分解为平面内和平面外分量。然后，可以将此曲率进行两次积分，以获得总挠曲形状**。假设叶片挠曲较小，在跨度 $r$ ，以及时间 $t$ 时，挥舞方向和摆振方向的局部曲率分别表示为 $\kappa_{F}(r,t)$ 和 $\kappa_{E}(r,t)$，对于叶片 1，为${^{11}}$：
+
 $$
-\kappa_{\scriptscriptstyle F}(r,t)\!=\!q_{\scriptscriptstyle I}\frac{d^{\,2}\phi_{\scriptscriptstyle I B F}(r)}{d r^{2}}\!+\!q_{\scriptscriptstyle I I}\frac{d^{\,2}\phi_{\scriptscriptstyle2B F}(r)}{d r^{2}}
+\kappa_{\scriptscriptstyle F}(r,t)\!=\!q_{\scriptscriptstyle I}\frac{d^{\,2}\phi_{\scriptscriptstyle 1 B F}(r)}{d r^{2}}\!+\!q_{\scriptscriptstyle I I}\frac{d^{\,2}\phi_{\scriptscriptstyle2B F}(r)}{d r^{2}}
 $$  
 
 and  
 
 $$
-\kappa_{\scriptscriptstyle E}(r,t)\!=\!q_{\scriptscriptstyle I3}\frac{d^{2}\phi_{\scriptscriptstyle I B E}(r)}{d r^{2}}
+\kappa_{\scriptscriptstyle E}(r,t)\!=\!q_{\scriptscriptstyle 13}\frac{d^{2}\phi_{\scriptscriptstyle 1 B E}(r)}{d r^{2}}
 $$  
 
-where $\phi_{I B F}(r)$ and $\phi_{2B F}(r)$ are the first and second natural mode shapes, respectively, of the blades in the flapwise direction and $\phi_{I B E}(r)$ is first natural mode shape of the blades in the edgewise direction.  The dependency on these curvatures with time is inherent in the generalized coordinates $q_{I},\,q_{I I}$ , and $q_{I3}$ .  In these expressions, the radius along the flexible part of the blade,  
+where $\phi_{1 B F}(r)$ and $\phi_{2B F}(r)$ are the first and second natural mode shapes, respectively, of the blades in the flapwise direction and $\phi_{1 B E}(r)$ is first natural mode shape of the blades in the edgewise direction.  The dependency on these curvatures with time is inherent in the generalized coordinates $q_{1},\,q_{11}$ , and $q_{13}$ .  In these expressions, the radius along the flexible part of the blade, $r$ , ranges from zero to $R-R_{H}$ .  Note that $r$ equals zero at a span of $R_{H}$ relative to the axis of the hub.  The derivation of these natural mode shapes of the blades is presented in section 3.2.   
+其中，$\phi_{1 B F}(r)$ 和 $\phi_{2B F}(r)$ 分别是叶片在挥舞方向上的第一和第二自然振型，而 $\phi_{1 B E}(r)$ 是叶片在摆振方向上的第一自然振型。 这些曲率随时间的变化内嵌在广义坐标 $q_{1},\,q_{11}$ 和 $q_{13}$ 中。 在这些表达式中，叶片柔性部分的半径 $r$ 的范围从零到 $R-R_{H}$。 注意，$r$ 在相对于轴的轴承跨度为 $R_{H}$ 时等于零。 这些叶片自然振型的推导见 3.2 节。
+
+11 The curvature of a curve $y(x)$, $\kappa(x)$, is
 
 $$
-\kappa(x)\!=\!\frac{\displaystyle{\bigg|\frac{d^{\,2}y(x)}{d x^{\,2}}\bigg|}}{\displaystyle{\bigg\{\!I\!+\!\!\left[\frac{d y(x)}{d x}\right]^{2}\!\bigg\}^{\,3/2}}}
+\kappa(x)\!=\!\frac{\displaystyle{\bigg|\frac{d^{\,2}y(x)}{d x^{\,2}}\bigg|}}{\displaystyle{\bigg\{\!1\!+\!\!\left[\frac{d y(x)}{d x}\right]^{2}\!\bigg\}^{\,3/2}}}
 $$  
 
 If the curve is composed only of small deflections, then:  
 
 $$
-\frac{\mathit{d y}(x)}{\mathit{d x}}<<I
+\frac{\mathit{d y}(x)}{\mathit{d x}}<<1
 $$  
 
 and the curvature simplifies to the following form:  
@@ -900,16 +912,19 @@ $$
 \kappa(x)\!=\!\left|\!\frac{d^{2}y(x)}{d x^{2}}\!\right|
 $$  
 
-$r$ , ranges from zero to $R-R_{H}$ .  Note that $r$ equals zero at a span of $R_{H}$ relative to the axis of the hub.  The derivation of these natural mode shapes of the blades is presented in section 3.2.  
+
 
 The curvatures in the out-of-plane and in-plane directions at a span of $r$ , and time $t$ , $\kappa_{O}(r_{,t})$ and $\kappa_{I}(r,t)$ respectively, for blade 1, are12:  
-
+在 $r$ 径向位置和 $t$ 时刻，叶片1的平面外弯曲度和平面内弯曲度分别为 $\kappa_{O}(r,t)$ 和 $\kappa_{I}(r,t)$，为12：
 $$
-\begin{array}{l}{{\kappa_{o}(r,t)\!=\!\!\left[q_{I}\frac{d^{\,2}\phi_{_{I B F}}(r)}{d r^{\,2}}\!+\!q_{I I}\frac{d^{\,2}\phi_{_{2B F}}(r)}{d r^{\,2}}\right]\!c o s[\theta_{s}(r)\!+\!\theta_{P}]+}}\\ {{\hphantom{\frac{(r_{0})^{2}\phi_{_{I B F}}(r)}{d r^{\,2}}}\left[q_{I b}\frac{d^{\,2}\phi_{_{I B E}}(r)}{d r^{\,2}}\right]\!s i n[\theta_{s}(r)\!+\!\theta_{P}]}}\end{array}
+\begin{array}{l}{{\kappa_{o}(r,t)\!=\!\!\left[q_{1}\frac{d^{\,2}\phi_{_{1 B F}}(r)}{d r^{\,2}}\!+\!q_{11}\frac{d^{\,2}\phi_{_{2B F}}(r)}{d r^{\,2}}\right]\!c o s[\theta_{s}(r)\!+\!\theta_{P}]+}}\\ {{\hphantom{\frac{(r_{0})^{2}\phi_{_{1 B F}}(r)}{d r^{\,2}}}\left[q_{1 3}\frac{d^{\,2}\phi_{_{1 B E}}(r)}{d r^{\,2}}\right]\!s i n[\theta_{s}(r)\!+\!\theta_{P}]}}\end{array}
 $$  
 
 $$
-\begin{array}{l}{{\kappa_{I}}(r,t)\!=\!-\!\Bigg[q_{I}\frac{d^{2}\phi_{I B F}(r)}{\;d r^{2}}\!+\!q_{I I}\frac{d^{2}\phi_{2B F}(r)}{d r^{2}}\!\Bigg]s i n\!\big[\theta_{s}(r)\!+\!\theta_{P}\big]+}}\\ {{\Bigg[q_{I3}\frac{d^{2}\phi_{I B E}(r)}{\;d r^{2}}\!\Bigg]c o s\!\big[\theta_{s}(r)\!+\!\theta_{P}\big]\!}}\end{array}
+\begin{align}
+    \kappa_{I}(r,t) = -\Bigg[q_{1}\frac{d^{2}\phi_{1BF}(r)}{\,dr^{2}} + q_{11}\frac{d^{2}\phi_{2BF}(r)}{dr^{2}}\Bigg]\sin\big[\theta_{s}(r) + \theta_{P}\big] \\
+    + \Bigg[q_{13}\frac{d^{2}\phi_{1BE}(r)}{\,dr^{2}}\Bigg]\cos\big[\theta_{s}(r) + \theta_{P}\big]
+\end{align}
 $$  
 
 or equivalently:  
@@ -922,7 +937,7 @@ $$
 \kappa_{\scriptscriptstyle I}{(r,t)}\!=\!q_{\scriptscriptstyle I}\frac{d^{2}\psi_{\scriptscriptstyle I}{(r)}}{d r^{2}}\!+\!q_{\scriptscriptstyle I I}\frac{d^{2}\psi_{\scriptscriptstyle2}{(r)}}{d r^{2}}\!+\!q_{\scriptscriptstyle I3}\frac{d^{2}\psi_{\scriptscriptstyle3}{(r)}}{d r^{2}}
 $$  
 
-where the twisted shape functions [the $\phi_{i}(r)$ ’s and $\psi_{i}(r)^{:}$ ’s for $i=1,2$ , and 3] are defined as:  
+where the twisted shape functions (the $\phi_{i}(r)$ ’s and $\psi_{i}(r)^{:}$ ’s for $i=1,2$ , and 3) are defined as:  
 
 $$
 \frac{d^{2}\phi_{l}(r)}{d r^{2}}\!=\!\!\left[\frac{d^{2}\phi_{l B F}(r)}{d r^{2}}\right]\!c o s\!\left[\theta_{s}(r)\!+\!\theta_{P}\right]
@@ -949,7 +964,7 @@ $$
 $$  
 
 The curvatures can be integrated over $r$ to obtain the deflections of blade 1 in the out-of-plane $(i_{I})$ and in-plane $(i_{2})$ directions at a span of $r$ , and time t, $u(r,t)$ and $\nu(r,t)$ respectively.  Since the deflections at the root of each blade (a span $R_{H}$ relative to the hub axis or $r=0$ ) must be zero, the deflections of blade 1 in the out-of-plane and in-plane directions become:  
-
+叶片曲率可以对 $r$ 进行积分，以获得在展向 $r$ 和时间 $t$ 时，叶片 1 在平面外方向 $(i_{I})$ 和平面内方向 $(i_{2})$ 的挠度，分别表示为 $u(r,t)$ 和 $\nu(r,t)$。由于每个叶片在根部（相对于轮毂轴的展向 $R_{H}$ 或 $r=0$）处的挠度必须为零，因此叶片 1 在平面外和平面内的挠度变为：
 $$
 u(\boldsymbol{r},t)\!=\!\int_{0}^{R-R_{H}}\!\Bigg[\int_{0}^{r}\kappa_{O}\big(\boldsymbol{r}^{\prime},t\big)d\boldsymbol{r}^{\prime}\Bigg]d r
 $$  
@@ -970,12 +985,13 @@ $$
 
 where $\acute{r}$ is a dummy variable representing the span along the flexible part of the blade.  
 
-As discussed in section 3.2, an axial (radial or span-wise) deflection of the blades will directly result from any lateral deflection if the flexible blades are assumed to remain fixed in length.  By  
+As discussed in section 3.2, an axial (radial or span-wise) deflection of the blades will directly result from any lateral deflection if the flexible blades are assumed to remain fixed in length.  By  an extension of the derivations of section 3.2, this axial deflection for blade 1 at a span of $r$ , and time t, $w(r,t)$ , is:  
 
-an extension of the derivations of section 3.2, this axial deflection for blade 1 at a span of $r$ , and time t, $w(r,t)$ , is:  
+其中 $\acute{r}$ 是代表风轮叶片柔性部分展向的虚拟变量。
 
+如3.2节所述，如果假设柔性叶片长度不变，任何横向挠度将直接导致轴向挠度。通过扩展3.2节的推导，叶片1在展向r，时间t时的轴向挠度 $w(r,t)$ 为：
 $$
-w(r,t)\!=\!\frac{I}{2}\!\int_{0}^{r}\!\left\{\!\!\left[\frac{\partial u(r^{\prime},t)}{\partial r^{\prime}}\right]^{2}+\!\!\left[\frac{\partial\nu(r^{\prime},t)}{\partial r^{\prime}}\right]^{2}\!\!\right\}\!d r^{\prime}
+w(r,t)\!=\!\frac{1}{2}\!\int_{0}^{r}\!\left\{\!\!\left[\frac{\partial u(r^{\prime},t)}{\partial r^{\prime}}\right]^{2}+\!\!\left[\frac{\partial\nu(r^{\prime},t)}{\partial r^{\prime}}\right]^{2}\!\!\right\}\!d r^{\prime}
 $$  
 
 A positive axial deflection is directed along the negative $i_{3}$ -axis.  
@@ -983,7 +999,7 @@ A positive axial deflection is directed along the negative $i_{3}$ -axis.
 Substituting Eqs. (3.24) and (3.25) into Eq. (3.26), results in:  
 
 $$
-w\!\left(r,t\right)\!=\!\frac{I}{2}\!\left(q_{I}^{\,2}S_{I I}+q_{I I}^{\,2}S_{\,22}+q_{I3}^{\,2}S_{\,33}+2q_{I}q_{I I}S_{\,I2}+2q_{I I}q_{I3}S_{\,23}+2q_{I}q_{I3}S_{\,I3}\right)
+w\!\left(r,t\right)\!=\!\frac{1}{2}\!\left(q_{1}^{\,2}S_{11}+q_{1 1}^{\,2}S_{\,22}+q_{13}^{\,2}S_{\,33}+2q_{1}q_{11}S_{\,12}+2q_{1 1}q_{13}S_{\,23}+2q_{1}q_{13}S_{\,13}\right)
 $$  
 
 where the symmetric, three by three, matrix $[S]$ is:  
@@ -992,19 +1008,20 @@ $$
 S_{i j}=\int_{0}^{r}\left\{\left[\frac{d\phi_{i}\left(r^{\prime}\right)}{d r^{\prime}}\right]\left[\frac{d\phi_{j}\left(r^{\prime}\right)}{d r^{\prime}}\right]+\left[\frac{d\psi_{i}\left(r^{\prime}\right)}{d r^{\prime}}\right]\left[\frac{d\psi_{j}\left(r^{\prime}\right)}{d r^{\prime}}\right]\right\}d r^{\prime}
 $$  
 
-The position vector connecting any point S on the deflected blade 1 to the blade axes intersection point (O), $r^{\mathcal{Q}^{S}}$ , is:  
+The position vector connecting any point S on the deflected blade 1 to the blade axes intersection point (O), $r^{{QS}}$ , is:  
+连接到变形叶片1上任意点S到叶片轴线交点(O)的位置向量，表示为 $r^{{QS}}$ ，为：
 
 $$
 \pmb{r}^{Q S}=u\big(r,t\big)\pmb{i}_{I}+\nu\big(r,t\big)\pmb{i}_{2}+\big[r+R_{H}-w\big(r,t\big)\big]\pmb{i}_{3}
 $$  
 
-The position vector connecting any point $\mathbf{S}^{\bullet}$ on the undeflected blade 1 to the blade axis intersection point (O), $r^{\mathcal{Q}^{S^{\prime}}}$ , is:  
+The position vector connecting any point $\mathbf{S}^{'}$ on the undeflected blade 1 to the blade axis intersection point (O), $r^{{QS^{\prime}}}$ , is:  
 
 $$
 {\pmb r}^{Q S^{\prime}}=({\pmb r}+{\cal R}_{H}\,){\pmb i}_{3}
 $$  
 
-For blade 2, only $q_{2}$ needs to be substituted in for $q_{I},\,q_{I}_{2}$ in for $q_{I I}$ , and $q_{I4}$ in for $q_{I3}$ .  The twisted shape functions [the $\phi_{i}(r)$ ’s and $\psi_{i}(r)$ ’s for $i=1,2$ , and 3] are not dependent on the blade considered.  
+For blade 2, only $q_{2}$ needs to be substituted in for $q_{1},,q_{12}$ in for $q_{I I}$ , and $q_{14}$ in for $q_{13}$ .  The twisted shape functions (the $\phi_{i}(r)$ ’s and $\psi_{i}(r)$ ’s for $i=1,2$ , and 3) are not dependent on the blade considered.  
 
 Fixed quantities (not dependent on time) can be translated to any coordinate system by simple coordinate transformations: