From 9c12c9c17732b1d526837b5ce48a034a6c68ac88 Mon Sep 17 00:00:00 2001
From: yize <yizeguo1@126.com>
Date: Wed, 20 Aug 2025 10:12:56 +0800
Subject: [PATCH] vault backup: 2025-08-20 10:12:54

---
 .obsidian/plugins/copilot/data.json           |   8 +-
 .../auto/CASEstab_theory_manual.md            | 139 ++++++++++++------
 2 files changed, 97 insertions(+), 50 deletions(-)

diff --git a/.obsidian/plugins/copilot/data.json b/.obsidian/plugins/copilot/data.json
index 6b239b5..8879700 100644
--- a/.obsidian/plugins/copilot/data.json
+++ b/.obsidian/plugins/copilot/data.json
@@ -144,8 +144,8 @@
       "provider": "google",
       "enabled": true,
       "isBuiltIn": false,
-      "baseUrl": "http://60.205.246.14:8000",
-      "apiKey": "gyz",
+      "baseUrl": "https://generativelanguage.googleapis.com",
+      "apiKey": "AIzaSyC9DWwXIbAjfhTTHNwCRAIckuZWRFzqYhA",
       "isEmbeddingModel": false,
       "capabilities": [
         "reasoning",
@@ -153,8 +153,8 @@
         "websearch"
       ],
       "stream": true,
-      "enableCors": true,
-      "displayName": "gemini-2.5-flash"
+      "displayName": "gemini-2.5-flash-gemini",
+      "enableCors": true
     },
     {
       "name": "gemini-2.5-pro",
diff --git a/书籍/力学书籍/CASEstab_theory_manual/auto/CASEstab_theory_manual.md b/书籍/力学书籍/CASEstab_theory_manual/auto/CASEstab_theory_manual.md
index c60ece6..2e44cfc 100644
--- a/书籍/力学书籍/CASEstab_theory_manual/auto/CASEstab_theory_manual.md
+++ b/书籍/力学书籍/CASEstab_theory_manual/auto/CASEstab_theory_manual.md
@@ -1246,129 +1246,176 @@ $$
 $$  
 
 where the first derivatives of the coefficient vectors $\mathbf{r}_{a c,1,b,n,m,p,q_{i}}$ for $m=1,\dots,N_{a,b,n}-1$ are  
-
+其中系数向量 $\mathbf{r}_{a c,1,b,n,m,p,q_{i}}$ 对于 $m=1,\dots,N_{a,b,n}-1$ 的一阶导数为
 $$
-\begin{array}{r}{\mathbf{r}_{a c,1,b,n,m,p,q_{i}}=\left\{\begin{array}{l l}{\mathbf{r}_{o,b,n,p,q_{i}}+c_{a c,x,b,n,m,0}\mathbf{r}_{x,b,n,p,q_{i}}+c_{a c,y,b,n,m,0}\mathbf{r}_{y,b,n,p,q_{i}}\qquad\qquad p=0}\\ {\qquad}&{\qquad}\\ {\mathbf{r}_{o,b,n,p,q_{i}}+c_{a c,x,b,n,m,0}\mathbf{r}_{x,b,n,p,q_{i}}+c_{a c,y,b,n,m,0}\mathbf{r}_{y,b,n,p,q_{i}}}\\ {\qquad+c_{a c,x,b,n,m,1}\mathbf{r}_{x,b,n,p-1,q_{i}}+c_{a c,y,b,n,m,1}\mathbf{r}_{y,b,n,p-1,q_{i}}\qquad\forall p\in[1:P+3]}\\ {\qquad}&{\qquad}\\ {c_{a c,x,b,n,m,1}\mathbf{r}_{x,b,n,p-1,q_{i}}+c_{a c,y,b,n,m,1}\mathbf{r}_{y,b,n,p-1,q_{i}}\qquad\qquad p=P+4}\end{array}\right.}\end{array}
+\begin{array}{r}{\mathbf{r}_{a c,1,b,n,m,p,q_{i}}=\left\{\begin{array}{l l}{\mathbf{r}_{o,b,n,p,q_{i}}+c_{a c,x,b,n,m,0}\mathbf{r}_{x,b,n,p,q_{i}}+c_{a c,y,b,n,m,0}\mathbf{r}_{y,b,n,p,q_{i}}\qquad\qquad p=0}\\ {\qquad}&{\qquad}\\ {\mathbf{r}_{o,b,n,p,q_{i}}+c_{a c,x,b,n,m,0}\mathbf{r}_{x,b,n,p,q_{i}}+c_{a c,y,b,n,m,0}\mathbf{r}_{y,b,n,p,q_{i}}}\\ {\qquad+c_{a c,x,b,n,m,1}\mathbf{r}_{x,b,n,p-1,q_{i}}+c_{a c,y,b,n,m,1}\mathbf{r}_{y,b,n,p-1,q_{i}}\qquad\forall p\in[1:P+3]}\\ {\qquad}&{\qquad}\\ {c_{a c,x,b,n,m,1}\mathbf{r}_{x,b,n,p-1,q_{i}}+c_{a c,y,b,n,m,1}\mathbf{r}_{y,b,n,p-1,q_{i}}\qquad\qquad p=P+4}\end{array}\right.}\end{array}\tag{3.17}
 $$  
 
 where the coefficient vectors are given by (1.57). Although the second DOF derivatives are not needed for the integrations of generalized aerodynamic forces in (2.18) and (2.19), we will need them later for a linearization of these generalized forces. The second DOF derivatives of $\mathbf{r}_{a c,1,b,n}$ are  
-
+其中系数向量由(1.57)给出。尽管在(2.18)和(2.19)中广义气动力的积分不需要二阶DOF导数，但我们稍后将需要它们来对这些广义力进行线性化。$\mathbf{r}_{a c,1,b,n}$的二阶DOF导数为
 $$
-\mathbf{r}_{a c,1,b,n,q_{i},q_{j}}=\left\{\begin{array}{c c}{\sum_{p=0}^{P+4}\mathbf{r}_{a c,1,b,n,1,p,q_{i},q_{j}}\;\zeta^{p}}&{a_{b,n,1}\leq\zeta<b_{b,n,1}}\\ {\sum_{p=0}^{P+4}\mathbf{r}_{a c,1,b,n,2,p,q_{i},q_{j}}\;\zeta^{p}}&{a_{b,n,2}\leq\zeta<b_{b,n,2}}\\ {\vdots}&{\vdots}\\ {\sum_{p=0}^{P+4}\mathbf{r}_{a c,1,b,n,N_{a,b,n}-1,p,q_{i},q_{j}}\;\zeta^{p}}&{a_{b,n,N_{a,b,n}-1}\leq\zeta\leq b_{b,n,N_{a,b,n}-1}}\end{array}\right.
+\mathbf{r}_{a c,1,b,n,q_{i},q_{j}}=\left\{\begin{array}{c c}{\sum_{p=0}^{P+4}\mathbf{r}_{a c,1,b,n,1,p,q_{i},q_{j}}\;\zeta^{p}}&{a_{b,n,1}\leq\zeta<b_{b,n,1}}\\ {\sum_{p=0}^{P+4}\mathbf{r}_{a c,1,b,n,2,p,q_{i},q_{j}}\;\zeta^{p}}&{a_{b,n,2}\leq\zeta<b_{b,n,2}}\\ {\vdots}&{\vdots}\\ {\sum_{p=0}^{P+4}\mathbf{r}_{a c,1,b,n,N_{a,b,n}-1,p,q_{i},q_{j}}\;\zeta^{p}}&{a_{b,n,N_{a,b,n}-1}\leq\zeta\leq b_{b,n,N_{a,b,n}-1}}\end{array}\right.\tag{3.18}
 $$  
 
-where the coefficient vectors rac,1,b,n,m,p,qi,qj for $m=1,\dots,N_{a,b,n}-1$ are  
-
+where the coefficient vectors $\mathbf{r}_{a c,1,b,n,m,p,q_{i},q_{j}}$ for $m=1,\dots,N_{a,b,n}-1$ are  
+其中系数向量$\mathbf{r}_{a c,1,b,n,m,p,q_{i},q_{j}}$对于$m=1,\dots,N_{a,b,n}-1$为
 $$
-\begin{array}{r}{\cdot_{1,b,n,m,p,q_{i},q_{j}}=\left\{\begin{array}{l l}{\mathbf{r}_{o,b,n,p,q_{i},q_{j}}+c_{a c,x,b,n,m,0}\mathbf{r}_{x,b,n,p,q_{i},q_{j}}+c_{a c,y,b,n,m,0}\mathbf{r}_{y,b,n,p,q_{i},q_{j}}}&{p=0}\\ {\mathbf{r}_{o,b,n,p,q_{i},q_{j}}+c_{a c,x,b,n,m,0}\mathbf{r}_{x,b,n,p,q_{i},q_{j}}+c_{a c,y,b,n,m,0}\mathbf{r}_{y,b,n,p,q_{i},q_{j}}}&{}\\ {\quad+c_{a c,x,b,n,m,1}\mathbf{r}_{x,b,n,p-1,q_{i},q_{j}}+c_{a c,y,b,n,m,1}\mathbf{r}_{y,b,n,p-1,q_{i},q_{j}}}&{\forall p\in[1:P+3]}\\ {\quad c_{a c,x,b,n,m,1}\mathbf{r}_{x,b,n,p-1,q_{i},q_{j}}+c_{a c,y,b,n,m,1}\mathbf{r}_{y,b,n,p-1,q_{i},q_{j}}}&{p=P+4}\end{array}\right.}\end{array}
+\begin{array}{r}{\mathbf{r}_{a c,1,b,n,m,p,q_{i},q_{j}}=\left\{\begin{array}{l l}{\mathbf{r}_{o,b,n,p,q_{i},q_{j}}+c_{a c,x,b,n,m,0}\mathbf{r}_{x,b,n,p,q_{i},q_{j}}+c_{a c,y,b,n,m,0}\mathbf{r}_{y,b,n,p,q_{i},q_{j}}}&{p=0}\\ {\mathbf{r}_{o,b,n,p,q_{i},q_{j}}+c_{a c,x,b,n,m,0}\mathbf{r}_{x,b,n,p,q_{i},q_{j}}+c_{a c,y,b,n,m,0}\mathbf{r}_{y,b,n,p,q_{i},q_{j}}}&{}\\ {\quad+c_{a c,x,b,n,m,1}\mathbf{r}_{x,b,n,p-1,q_{i},q_{j}}+c_{a c,y,b,n,m,1}\mathbf{r}_{y,b,n,p-1,q_{i},q_{j}}}&{\forall p\in[1:P+3]}\\ {\quad c_{a c,x,b,n,m,1}\mathbf{r}_{x,b,n,p-1,q_{i},q_{j}}+c_{a c,y,b,n,m,1}\mathbf{r}_{y,b,n,p-1,q_{i},q_{j}}}&{p=P+4}\end{array}\right.}\end{array}\tag{3.19}
 $$  
 
 where the second derivatives of the coefficient vectors are given by (1.58).  
-
-For the integrations of generalized aerodynamic forces in (2.18) and (2.19), we also need function of the vari­ ation of the first unit­vector $\mathfrak{e}_{1,c,1,b,n}$ of the chord coordinate system (CCS) over the element number $n$ , and its DOF derivatives. The variation has two components: a constant component due to the geometrical variation of the blade shape and a dynamic component due to the small rotational elastic deformations of the element:  
-
+其中系数向量的二阶导数由(1.58)给出。
+For the integrations of generalized aerodynamic forces in (2.18) and (2.19), we also need function of the vari­ation of the first unit-­vector $\mathfrak{e}_{1,c,1,b,n}$ of the chord coordinate system (CCS) over the element number $n$ , and its DOF derivatives. The variation has two components: a constant component due to the geometrical variation of the blade shape and a dynamic component due to the small rotational elastic deformations of the element:  
+对于（2.18）和（2.19）中广义气动力的积分，我们还需要弦坐标系（CCS）的第一个单位向量${e}_{1,c,1,b,n}$随单元编号$n$的变化函数及其自由度导数。这种变化包含两个分量：一个是由叶片形状的几何变化引起的常数分量，另一个是由单元的小旋转弹性变形引起的动态分量：
 $$
-\begin{array}{r}{\mathbf{e}_{1,c,1,b,n}(\mathbf{q}_{b},\zeta)=\left\{\begin{array}{c c}{\mathbf{E}_{b,n}(\mathbf{q}_{b})\,\mathbf{R}_{1,b,n}(\mathbf{q}_{b},\zeta)\,\mathbf{e}_{1,c,e,b,n,1}(\zeta)}&{a_{b,n,1}\leq\zeta<b_{b,n,1}}\\ {\mathbf{E}_{b,n}(\mathbf{q}_{b})\,\mathbf{R}_{1,b,n}(\mathbf{q}_{b},\zeta)\,\mathbf{e}_{1,c,e,b,n,2}(\zeta)}&{a_{b,n,2}\leq\zeta<b_{b,n,2}}\\ &{\vdots}&{\vdots}\\ {\mathbf{E}_{b,n}(\mathbf{q}_{b})\,\mathbf{R}_{1,b,n}(\mathbf{q}_{b},\zeta)\,\mathbf{e}_{1,c,e,b,n,N_{a,b,n}-1}(\zeta)}&{a_{b,n,N_{a,b,n}-1}\leq\zeta\leq b_{b,n,N_{a,b,n}-1}}\end{array}\right.}\end{array}
+\begin{array}{r}{\mathbf{e}_{1,c,1,b,n}(\mathbf{q}_{b},\zeta)=\left\{\begin{array}{c c}{\mathbf{E}_{b,n}(\mathbf{q}_{b})\,\mathbf{R}_{1,b,n}(\mathbf{q}_{b},\zeta)\,\mathbf{e}_{1,c,e,b,n,1}(\zeta)}&{a_{b,n,1}\leq\zeta<b_{b,n,1}}\\ {\mathbf{E}_{b,n}(\mathbf{q}_{b})\,\mathbf{R}_{1,b,n}(\mathbf{q}_{b},\zeta)\,\mathbf{e}_{1,c,e,b,n,2}(\zeta)}&{a_{b,n,2}\leq\zeta<b_{b,n,2}}\\ {\vdots}&{\vdots}\\ {\mathbf{E}_{b,n}(\mathbf{q}_{b})\,\mathbf{R}_{1,b,n}(\mathbf{q}_{b},\zeta)\,\mathbf{e}_{1,c,e,b,n,N_{a,b,n}-1}(\zeta)}&{a_{b,n,N_{a,b,n}-1}\leq\zeta\leq b_{b,n,N_{a,b,n}-1}}\end{array}\right.}\end{array}\tag{3.20}
 $$  
 
-where $\mathsf{E}_{b,n}$ is the element coordinate system (ECS) matrix defined in the substructure frame, $\mathsf{R}_{1,b,n}$ is the rotation matrix function for the elastic deformation, and the unit­vector functions $\mathsf{e}_{1,c,e,b,n,m}(\zeta)$ with $m=1,\ldots,N_{a,b,n}-1$ describe the geometrical variation of the CCS matrix over the interval $\zeta\,\in\,[a_{b,n,m},b_{b,n,m}]$ in the frame of the ECS. Assuming small rotations inside the elements, the elastic rotation matrix can be approximated as  
-
+where $\mathsf{E}_{b,n}$ is the element coordinate system (ECS) matrix defined in the substructure frame, $\mathsf{R}_{1,b,n}$ is the rotation matrix function for the elastic deformation, and the unit-­vector functions $\mathsf{e}_{1,c,e,b,n,m}(\zeta)$ with $m=1,\ldots,N_{a,b,n}-1$ describe the geometrical variation of the CCS matrix over the interval $\zeta\,\in\,[a_{b,n,m},b_{b,n,m}]$ in the frame of the ECS. Assuming small rotations inside the elements, the elastic rotation matrix can be approximated as  
+其中 $\mathsf{E}_{b,n}$ 是定义在子结构坐标系中的单元坐标系 (ECS) 矩阵，$\mathsf{R}_{1,b,n}$ 是用于弹性变形的旋转矩阵函数，并且单位向量函数 $\mathsf{e}_{1,c,e,b,n,m}(\zeta)$ (其中 $m=1,\ldots,N_{a,b,n}-1$) 描述了 CCS 矩阵在 ECS 坐标系中，于区间 $\zeta\,\in\,[a_{b,n,m},b_{b,n,m}]$ 内的几何变化。假设单元内部存在小旋转，则弹性旋转矩阵可以近似为
 $$
-\mathbf{R}_{1,b,n}(\mathbf{q}_{b},\zeta)=\left[\begin{array}{c c c}{1}&{-\theta_{z,n}(\zeta)}&{\theta_{y,n}(\zeta)}\\ {\theta_{z,n}(\zeta)}&{1}&{-\theta_{x,n}(\zeta)}\\ {-\theta_{y,n}(\zeta)}&{\theta_{x,n}(\zeta)}&{1}\end{array}\right]=\mathbf{I}+\sum_{p=0}^{P+3}\mathbf{S}\left\{\theta_{n,p}\right\}\zeta^{p}
+\mathbf{R}_{1,b,n}(\mathbf{q}_{b},\zeta)=\left[\begin{array}{c c c}{1}&{-\theta_{z,n}(\zeta)}&{\theta_{y,n}(\zeta)}\\ {\theta_{z,n}(\zeta)}&{1}&{-\theta_{x,n}(\zeta)}\\ {-\theta_{y,n}(\zeta)}&{\theta_{x,n}(\zeta)}&{1}\end{array}\right]=\mathbf{I}+\sum_{p=0}^{P+3}\mathbf{S}\left\{\theta_{n,p}\right\}\zeta^{p}\tag{3.21}
 $$  
 
 where the skew symmetric parts (cf. (E.1)) are  
-
+其中反对称部分 (参见 (E.1)) 为
 $$
-\mathbf{S}\left\{\theta_{n,p}\right\}=\left[\begin{array}{c c c}{0}&{-\theta_{z,n,p}}&{\theta_{y,n,p}}\\ {\theta_{z,n,p}}&{0}&{-\theta_{x,n,p}}\\ {-\theta_{y,n,p}}&{\theta_{x,n,p}}&{0}\end{array}\right]
+\mathbf{S}\left\{\theta_{n,p}\right\}=\left[\begin{array}{c c c}{0}&{-\theta_{z,n,p}}&{\theta_{y,n,p}}\\ {\theta_{z,n,p}}&{0}&{-\theta_{x,n,p}}\\ {-\theta_{y,n,p}}&{\theta_{x,n,p}}&{0}\end{array}\right]\tag{3.22}
 $$  
 
-using the coefficients for the angles $\pmb{\theta}_{n,p}=\{\theta_{x,n,p},\theta_{y,n,p},\theta_{z,n,p}\}^{T}=\tilde{\aleph}_{n,p}\pmb{\mathfrak{g}}\left(\pmb{\mathfrak{q}}_{b,n}\right)$ as defined in (1.44).  
-
-The geometrical variation of the first unit­vector of the CCS matrix over the interval $\zeta\,\in\,[a_{b,n,m},b_{b,n,m}]$ in the frame of the ECS is approximated by a linear interpolation function:  
+using the coefficients for the angles $\pmb{\theta}_{n,p}=\{\theta_{x,n,p},\theta_{y,n,p},\theta_{z,n,p}\}^{T}=\tilde{N}_{n,p}\pmb{{g}}\left(\pmb{{q}}_{b,n}\right)$ as defined in (1.44).  
+使用角度 $\pmb{\theta}_{n,p}=\{\theta_{x,n,p},\theta_{y,n,p},\theta_{z,n,p}\}^{T}=\tilde{N}_{n,p}\pmb{{g}}\left(\pmb{{q}}_{b,n}\right)$ 的系数，其定义如(1.44)所示。
 
+The geometrical variation of the first unit-­vector of the CCS matrix over the interval $\zeta\,\in\,[a_{b,n,m},b_{b,n,m}]$ in the frame of the ECS is approximated by a linear interpolation function:  
+CCS矩阵的第一个单位向量在ECS坐标系中，在区间 $\zeta\,\in\,[a_{b,n,m},b_{b,n,m}]$ 上的几何变化通过线性插值函数近似：
 $$
-\begin{array}{r}{\mathbf{e}_{1,c,e,b,n,m}(\zeta)=(\mathbf{e}_{1,c,e,b,n,m,a}(b_{m,n}-\zeta)+\mathbf{e}_{1,c,e,b,n,m,b}(\zeta-a_{m,n}))/(b_{m,n}-a_{m,n})}\end{array}
+\begin{array}{r}{\mathbf{e}_{1,c,e,b,n,m}(\zeta)=(\mathbf{e}_{1,c,e,b,n,m,a}(b_{m,n}-\zeta)+\mathbf{e}_{1,c,e,b,n,m,b}(\zeta-a_{m,n}))/(b_{m,n}-a_{m,n})}\end{array}\tag{3.23}
 $$  
-
-where $\mathbf{e}_{1,c,e,b,n,m,a}$ and $\mathtt{e}_{1,c,e,b,n,m,b}$ are the first unit­vectors of the CCS in the ECS frame on the lower and upper limits of the integration range, respectively. We rewrite this approximation as  
-
+where $\mathbf{e}_{1,c,e,b,n,m,a}$ and $\mathtt{e}_{1,c,e,b,n,m,b}$ are the first unit­-vectors of the CCS in the ECS frame on the lower and upper limits of the integration range, respectively. We rewrite this approximation as  
+其中 $\mathbf{e}_{1,c,e,b,n,m,a}$ 和 $\mathtt{e}_{1,c,e,b,n,m,b}$ 分别是积分范围的下限和上限处，CCS在ECS参考系中的第一个单位向量。我们将这个近似重写为
 $$
-\mathbf{e}_{1,c,e,b,n,m}(\zeta)=\sum_{r=0}^{1}\mathbf{e}_{1,c,e,b,n,m,r}\,\zeta^{r}
+\mathbf{e}_{1,c,e,b,n,m}(\zeta)=\sum_{r=0}^{1}\mathbf{e}_{1,c,e,b,n,m,r}\,\zeta^{r}\tag{3.24}
 $$  
 
 where coefficient vectors are  
-
+其中系数向量为
 $$
-\begin{array}{r l}&{\pmb{\mathrm{e}}_{1,c,e,b,n,m,0}=\frac{\pmb{\mathrm{e}}_{1,c,e,b,n,m,a}\,b_{b,n,m}-\pmb{\mathrm{e}}_{1,c,e,b,n,m,b}\,a_{b,n,m}}{b_{b,n,m}-a_{b,n,m}}}\\ &{\pmb{\mathrm{e}}_{1,c,e,b,n,m,1}=\frac{\pmb{\mathrm{e}}_{1,c,e,b,n,m,b}-\pmb{\mathrm{e}}_{1,c,e,b,n,m,a}}{b_{b,n,m}-a_{b,n,m}}}\end{array}
+\begin{array}{r l}&{\pmb{\mathrm{e}}_{1,c,e,b,n,m,0}=\frac{\pmb{\mathrm{e}}_{1,c,e,b,n,m,a}\,b_{b,n,m}-\pmb{\mathrm{e}}_{1,c,e,b,n,m,b}\,a_{b,n,m}}{b_{b,n,m}-a_{b,n,m}}}\\ &{\pmb{\mathrm{e}}_{1,c,e,b,n,m,1}=\frac{\pmb{\mathrm{e}}_{1,c,e,b,n,m,b}-\pmb{\mathrm{e}}_{1,c,e,b,n,m,a}}{b_{b,n,m}-a_{b,n,m}}}\end{array}\tag{3.25}
 $$  
 
 Note that the approximation (3.23) does not ensure that the length of the first axis vector of CCS in the ECS frame is unity, but the error will decrease with increased number of ACPs and can therefore be quantified as part of the aerodynamic discretization error in a convergence study.  
+请注意，近似式 (3.23) 并不能确保CCS在ECS坐标系中第一个轴向量的长度为单位一，但误差会随着ACPs数量的增加而减小，因此可以在收敛性研究中作为气动离散误差的一部分进行量化。
 
-Using (3.21) and (3.25), the variation of the first unit­vector $\mathbf{e}_{1,c,1,b,n}$ of the CCS over the element number $n$ (3.20) can be written as  
-
+Using (3.21) and (3.25), the variation of the first unit-­vector $\mathbf{e}_{1,c,1,b,n}$ of the CCS over the element number $n$ (3.20) can be written as  
+使用(3.21)和(3.25)，CCS的第一个单位向量$\mathbf{e}_{1,c,1,b,n}$随单元编号$n$ (3.20)的变化可以写成
 $$
-\begin{array}{r}{\mathbf{e}_{1,c,1,b,n}(\mathbf{q}_{b},\zeta)=\left\{\begin{array}{c c}{\sum_{p=0}^{P+4}\mathbf{e}_{1,c,1,b,n,1,p}\,\zeta^{p}}&{a_{b,n,1}\leq\zeta<b_{b,n,1}}\\ {\sum_{p=0}^{P+4}\mathbf{e}_{1,c,1,b,n,2,p}\,\zeta^{p}}&{a_{b,n,2}\leq\zeta<b_{b,n,2}}\\ {\vdots}&{\vdots}\\ {\sum_{p=0}^{P+4}\mathbf{e}_{1,c,1,b,n,N_{a,b,n}-1,p}\,\zeta^{p}}&{a_{b,n,N_{a,b,n}-1}\leq\zeta\leq b_{b,n,N_{a,b,n}-1}}\end{array}\right.}\end{array}
+\begin{array}{r}{\mathbf{e}_{1,c,1,b,n}(\mathbf{q}_{b},\zeta)=\left\{\begin{array}{c c}{\sum_{p=0}^{P+4}\mathbf{e}_{1,c,1,b,n,1,p}\,\zeta^{p}}&{a_{b,n,1}\leq\zeta<b_{b,n,1}}\\ {\sum_{p=0}^{P+4}\mathbf{e}_{1,c,1,b,n,2,p}\,\zeta^{p}}&{a_{b,n,2}\leq\zeta<b_{b,n,2}}\\ {\vdots}&{\vdots}\\ {\sum_{p=0}^{P+4}\mathbf{e}_{1,c,1,b,n,N_{a,b,n}-1,p}\,\zeta^{p}}&{a_{b,n,N_{a,b,n}-1}\leq\zeta\leq b_{b,n,N_{a,b,n}-1}}\end{array}\right.}\end{array}\tag{3.26}
 $$  
 
 where  
 
 $$
-\begin{array}{r}{\mathbf{e}_{1,c,1,b,n,m,p}=\left\{\begin{array}{c l}{\mathbf{E}_{b,n}\left(\mathbf{1}+\mathbf{S}\left\{\tilde{\mathbf{N}}_{n,0}\mathbf{g}\right\}\right)\mathbf{e}_{1,c,e,b,n,m,0}}&{p=0}\\ {\mathbf{E}_{b,n}\left(\mathbf{S}\left\{\tilde{\mathbf{N}}_{n,p}\mathbf{g}\right\}\mathbf{e}_{1,c,e,b,n,m,0}+\mathbf{S}\left\{\tilde{\mathbf{N}}_{n,p-1}\mathbf{g}\right\}\mathbf{e}_{1,c,e,b,n,m,1}\right)}&{\forall p\in[1:P+3]}\\ {\mathbf{E}_{b,n}\mathbf{S}\left\{\tilde{\mathbf{N}}_{n,p-1}\mathbf{g}\right\}\mathbf{e}_{1,c,e,b,n,m,1}}&{p=P+4}\end{array}\right.}\end{array}
+\begin{array}{r}{\mathbf{e}_{1,c,1,b,n,m,p}=\left\{\begin{array}{c l}{\mathbf{E}_{b,n}\left(\mathbf{I}+\mathbf{S}\left\{\tilde{\mathbf{N}}_{n,0}\mathbf{g}\right\}\right)\mathbf{e}_{1,c,e,b,n,m,0}}&{p=0}\\ {\mathbf{E}_{b,n}\left(\mathbf{S}\left\{\tilde{\mathbf{N}}_{n,p}\mathbf{g}\right\}\mathbf{e}_{1,c,e,b,n,m,0}+\mathbf{S}\left\{\tilde{\mathbf{N}}_{n,p-1}\mathbf{g}\right\}\mathbf{e}_{1,c,e,b,n,m,1}\right)}&{\forall p\in[1:P+3]}\\ {\mathbf{E}_{b,n}\mathbf{S}\left\{\tilde{\mathbf{N}}_{n,p-1}\mathbf{g}\right\}\mathbf{e}_{1,c,e,b,n,m,1}}&{p=P+4}\end{array}\right.}\end{array}\tag{3.27}
 $$  
 
-The first derivative of this uni­vector with respect to a DOF $q_{i}$ on the substructure becomes  
-
+The first derivative of this unit-­vector with respect to a DOF $q_{i}$ on the substructure becomes  
+该单位向量关于子结构上一个自由度 $q_{i}$ 的一阶导数变为
 $$
-\mathbf{e}_{1,c,1,b,n,q_{i}}(\mathbf{q}_{b},\zeta)=\left\{\begin{array}{c c}{\sum_{p=0}^{P+4}\mathbf{e}_{1,c,1,b,n,1,p,q_{i}}\,\zeta^{p}}&{a_{b,n,1}\leq\zeta<b_{b,n,1}}\\ {\sum_{p=0}^{P+4}\mathbf{e}_{1,c,1,b,n,2,p,q_{i}}\,\zeta^{p}}&{a_{b,n,2}\leq\zeta<b_{b,n,2}}\\ {\vdots}&{\vdots}\\ {\sum_{p=0}^{P+4}\mathbf{e}_{1,c,1,b,n,N_{a,b,n}-1,p,q_{i}}\,\zeta^{p}}&{a_{b,n,N_{a,b,n}-1}\leq\zeta\leq b_{b,n,N_{a,b,n}-1}}\end{array}\right.
+\mathbf{e}_{1,c,1,b,n,q_{i}}(\mathbf{q}_{b},\zeta)=\left\{\begin{array}{c c}{\sum_{p=0}^{P+4}\mathbf{e}_{1,c,1,b,n,1,p,q_{i}}\,\zeta^{p}}&{a_{b,n,1}\leq\zeta<b_{b,n,1}}\\ {\sum_{p=0}^{P+4}\mathbf{e}_{1,c,1,b,n,2,p,q_{i}}\,\zeta^{p}}&{a_{b,n,2}\leq\zeta<b_{b,n,2}}\\ {\vdots}&{\vdots}\\ {\sum_{p=0}^{P+4}\mathbf{e}_{1,c,1,b,n,N_{a,b,n}-1,p,q_{i}}\,\zeta^{p}}&{a_{b,n,N_{a,b,n}-1}\leq\zeta\leq b_{b,n,N_{a,b,n}-1}}\end{array}\right.\tag{3.28}
 $$  
 
 where  
 
 $$
-\begin{array}{r}{\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! 
+\mathbf{e}_{1,c,1,b,n,m,p,q_i} = 
+\begin{cases}
+    \left( \mathbf{E}_{b,n,q_i} \left( \mathbf{I} + \mathbf{S}\{\tilde{\mathbf{N}}_{n,0}\mathbf{g}\} \right) + \mathbf{E}_{b,n} \mathbf{S}\{\tilde{\mathbf{N}}_{n,0}\mathbf{g}_{,q_i}\} \right) \mathbf{e}_{1,c,e,b,n,m,0} & p=0 \\
+    \begin{aligned}
+        &\left( \mathbf{E}_{b,n,q_i} \mathbf{S}\{\tilde{\mathbf{N}}_{n,p}\mathbf{g}\} + \mathbf{E}_{b,n} \mathbf{S}\{\tilde{\mathbf{N}}_{n,p}\mathbf{g}_{,q_i}\} \right) \mathbf{e}_{1,c,e,b,n,m,0} \\
+        &+ \left( \mathbf{E}_{b,n,q_i} \mathbf{S}\{\tilde{\mathbf{N}}_{n,p-1}\mathbf{g}\} + \mathbf{E}_{b,n} \mathbf{S}\{\tilde{\mathbf{N}}_{n,p-1}\mathbf{g}_{,q_i}\} \right) \mathbf{e}_{1,c,e,b,n,m,1}
+    \end{aligned}
+    & \forall p \in [1:P+3] \\
+    \left( \mathbf{E}_{b,n,q_i} \mathbf{S}\{\tilde{\mathbf{N}}_{n,p-1}\mathbf{g}\} + \mathbf{E}_{b,n} \mathbf{S}\{\tilde{\mathbf{N}}_{n,p-1}\mathbf{g}_{,q_i}\} \right) \mathbf{e}_{1,c,e,b,n,m,1} & p=P+4
+\end{cases}
+\tag{3.29}
 $$  
 
 The second derivative of this uni­vector with respect to DOFs $q_{i}$ and $q_{j}$ on the substructure becomes  
-
+这个单位向量关于子结构上自由度$q_{i}$和$q_{j}$的二阶导数是
 $$
-\begin{array}{r}{\mathbf{e}_{1,c,1,b,n,q_{i},q_{j}}(\mathbf{q}_{b},\zeta)=\left\{\begin{array}{c c}{\sum_{p=0}^{P+4}\mathbf{e}_{1,c,1,b,n,1,p,q_{i},q_{j}}\zeta^{p}}&{a_{b,n,1}\leq\zeta<b_{b,n,1}}\\ {\sum_{p=0}^{P+4}\mathbf{e}_{1,c,1,b,n,2,p,q_{i},q_{j}}\zeta^{p}}&{a_{b,n,2}\leq\zeta<b_{b,n,2}}\\ {\vdots}&{\vdots}\\ {\sum_{p=0}^{P+4}\mathbf{e}_{1,c,1,b,n,N_{a,b,n}-1,p,q_{i},q_{j}}\zeta^{p}}&{a_{b,n,N_{a,b,n}-1}\leq\zeta\leq b_{b,n,N_{a,b,n}-1}}\end{array}\right.}\end{array}
+\begin{array}{r}{\mathbf{e}_{1,c,1,b,n,q_{i},q_{j}}(\mathbf{q}_{b},\zeta)=\left\{\begin{array}{c c}{\sum_{p=0}^{P+4}\mathbf{e}_{1,c,1,b,n,1,p,q_{i},q_{j}}\zeta^{p}}&{a_{b,n,1}\leq\zeta<b_{b,n,1}}\\ {\sum_{p=0}^{P+4}\mathbf{e}_{1,c,1,b,n,2,p,q_{i},q_{j}}\zeta^{p}}&{a_{b,n,2}\leq\zeta<b_{b,n,2}}\\ {\vdots}&{\vdots}\\ {\sum_{p=0}^{P+4}\mathbf{e}_{1,c,1,b,n,N_{a,b,n}-1,p,q_{i},q_{j}}\zeta^{p}}&{a_{b,n,N_{a,b,n}-1}\leq\zeta\leq b_{b,n,N_{a,b,n}-1}}\end{array}\right.}\end{array}\tag{3.30}
 $$  
 
 where  
 
 $$
-\begin{array}{r}{\mathfrak{H}_{1,c,1,b,m,m,p,q,i,q}=\left\{\begin{array}{r l}{\left(\begin{array}{l}{\mathbf{E}_{b,n,q,q_{i}}\left(|1\bullet\left\{\mathbf{\tilde{N}}_{1,0}\mathbf{g}\right\}\right\}\right)+\mathbf{E}_{b,n,q_{i}}\mathbf{S}\left\{\mathbf{\tilde{N}}_{n,0}\mathbf{g}_{i,q}\right\}}\\ {+\mathbf{E}_{b,n,q_{i}}\mathbf{S}\left\{\mathbf{\tilde{N}}_{n,0}\mathbf{g}_{i}\right\}+\mathbf{E}_{b,n}\mathbf{S}\left\{\mathbf{\tilde{N}}_{n,0}\mathbf{g}_{i,q,i,q}\right\}\right)\mathbf{e}_{1,c,e,b,n,m,0}}&{p=0}\\ {\left(\mathbf{E}_{b,n,q_{i}}\mathbf{\tilde{Q}}_{1,0}\right)\left\{\mathbf{\tilde{N}}_{n,p}\mathbf{g}_{i}\right\}+\mathbf{E}_{b,n,q_{i}}\mathbf{S}\left\{\mathbf{\tilde{N}}_{n,p}\mathbf{g}_{i}\right\}}\\ {+\mathbf{E}_{b,n,q_{i}}\mathbf{S}\left\{\mathbf{\tilde{N}}_{n,p}\mathbf{g}_{i}\right\}+\mathbf{E}_{b,n}\mathbf{S}\left\{\mathbf{\tilde{N}}_{n,p}\mathbf{g}_{i,q_{i},q_{j}}\right\}\right)\mathbf{e}_{1,c,e,b,n,m,0}}\\ {+\left(\mathbf{E}_{b,n,q_{i},q_{i}}\mathbf{S}\left\{\mathbf{\tilde{N}}_{n,p-1}\mathbf{g}\right\}+\mathbf{E}_{b,n,q_{i}}\mathbf{S}\left\{\mathbf{\tilde{N}}_{n,p-1}\mathbf{g}_{i,q_{i}}\right\}\right)\mathbf{e}_{1,c,e,b,n,m,0}}\\ {+\mathbf{E}_{b,n,q_{i}}\mathbf{S}\left\{\mathbf{\tilde{N}}_{n,p-1}\mathbf{g}_{i}\right\}+\mathbf{E}_{b,n}\mathbf{S}\left\{\mathbf{\tilde{N}}_{n,p-1}\mathbf{g}_{i,q_{i},j}\right\}}\\ {\left(\mathbf{E}_{b,n,q_{i}}\mathbf{S}\left\{\mathbf{\tilde{N}}_{n,p-1}\mathbf{g}_{i}\right\}+\mathbf{E}_{b,n,q_{i}}\mathbf{S}\left\
+\mathbf{e}_{1,c,1,b,n,m,p,q_i,q_j} = 
+\begin{cases}
+    \begin{aligned}
+        &\Bigg\{ \Big( \mathbf{E}_{b,n,q_i,q_j} \left( \mathbf{I} + \mathbf{S}\{\tilde{\mathbf{N}}_{n,0}\mathbf{g}\} \right) \Big) + \mathbf{E}_{b,n,q_i} \mathbf{S}\{\tilde{\mathbf{N}}_{n,0}\mathbf{g}_{,q_j}\} \\
+        &+ \mathbf{E}_{b,n,q_j} \mathbf{S}\{\tilde{\mathbf{N}}_{n,0}\mathbf{g}_{,q_i}\} + \mathbf{E}_{b,n} \mathbf{S}\{\tilde{\mathbf{N}}_{n,0}\mathbf{g}_{,q_i,q_j}\} \Bigg\} \mathbf{e}_{1,c,e,b,n,m,0}
+    \end{aligned}
+    & p=0 \\
+    \begin{aligned}
+        &\Bigg\{ \Big( \mathbf{E}_{b,n,q_i,q_j} \mathbf{S}\{\tilde{\mathbf{N}}_{n,p}\mathbf{g}\} + \mathbf{E}_{b,n,q_i} \mathbf{S}\{\tilde{\mathbf{N}}_{n,p}\mathbf{g}_{,q_j}\} \Big) \\
+        &+ \mathbf{E}_{b,n,q_j} \mathbf{S}\{\tilde{\mathbf{N}}_{n,p}\mathbf{g}_{,q_i}\} + \mathbf{E}_{b,n} \mathbf{S}\{\tilde{\mathbf{N}}_{n,p}\mathbf{g}_{,q_i,q_j}\} \Bigg\} \mathbf{e}_{1,c,e,b,n,m,0} \\
+        &+ \Bigg\{ \Big( \mathbf{E}_{b,n,q_i,q_j} \mathbf{S}\{\tilde{\mathbf{N}}_{n,p-1}\mathbf{g}\} + \mathbf{E}_{b,n,q_i} \mathbf{S}\{\tilde{\mathbf{N}}_{n,p-1}\mathbf{g}_{,q_j}\} \Big) \\
+        &+ \mathbf{E}_{b,n,q_j} \mathbf{S}\{\tilde{\mathbf{N}}_{n,p-1}\mathbf{g}_{,q_i}\} + \mathbf{E}_{b,n} \mathbf{S}\{\tilde{\mathbf{N}}_{n,p-1}\mathbf{g}_{,q_i,q_j}\} \Bigg\} \mathbf{e}_{1,c,e,b,n,m,1}
+    \end{aligned}
+    & \forall p \in [1:P+3] \\
+    \begin{aligned}
+        &\Bigg\{ \Big( \mathbf{E}_{b,n,q_i,q_j} \mathbf{S}\{\tilde{\mathbf{N}}_{n,p-1}\mathbf{g}\} + \mathbf{E}_{b,n,q_i} \mathbf{S}\{\tilde{\mathbf{N}}_{n,p-1}\mathbf{g}_{,q_j}\} \Big) \\
+        &+ \mathbf{E}_{b,n,q_j} \mathbf{S}\{\tilde{\mathbf{N}}_{n,p-1}\mathbf{g}_{,q_i}\} + \mathbf{E}_{b,n} \mathbf{S}\{\tilde{\mathbf{N}}_{n,p-1}\mathbf{g}_{,q_i,q_j}\} \Bigg\} \mathbf{e}_{1,c,e,b,n,m,1}
+    \end{aligned}
+    & p=P+4
+\end{cases}
+\tag{3.31}
 $$  
 
 which is needed for the linearization of the generalized aerodynamic forces.  
-
+其是广义气动力线性化所必需的。
 Using the piecewise linear force and moment functions (3.6) and (3.7) with their coefficients (3.35), the variations of the AC position (3.14), and the chordwise unit­vector of the CCS (3.26), we can compute contribution from element $n$ to the total aerodynamic force vector and moment matrix (2.19) for the substructure $b$ as  
-
+使用分段线性的力和力矩函数(3.6)和(3.7)及其系数(3.35)，AC位置(3.14)的变化量，以及CCS的弦向单位向量(3.26)，我们可以计算出单元$n$对子结构$b$的总气动力向量和力矩矩阵(2.19)的贡献。
 $$
-\mathbf{f}_{b,n}=\frac{l_{n}}{2}\binom{N_{a,b,n}-1}{m=1}\Biggl(\sum_{r=0}^{1}\frac{b_{b,n,m}^{r+1}-a_{b,n,m}^{r+1}}{r+1}\;\left(w_{b,n,m,r}\mathbf{f}_{1,b,j_{b,n,m}}+w_{b,n,m+1,r}\mathbf{f}_{1,b,j_{b,n,m+1}}\right)\Biggr)\Biggr)
+\mathbf{f}_{b,n} = \frac{l_n}{2} \left( \sum_{m=1}^{N_{a,b,n}-1} \left( \sum_{r=0}^{1} \frac{b_{b,n,m}^{r+1} - a_{b,n,m}^{r+1}}{r+1} \left( w_{b,n,m,r} \mathbf{f}_{1,b,j_b,n,m} + w_{b,n,m+1,r} \mathbf{f}_{1,b,j_b,n,m+1} \right) \right) \right)
+\tag{3.32}
 $$  
 
 and  
 
 $$
-\begin{array}{r l}{\lefteqn{\mathbb{M}_{b,n}=\frac{l_{n}}{2}\left(\sum_{m=1}^{N_{a,b,n}-1}\left(\sum_{p=0}^{P+4}\left(\sum_{r=0}^{1}\frac{b_{b,n,m}^{p+r+1}-a_{b,n,m}^{p+r+1}}{p+r+1}\right.\right.}\\ &{\left.\left.\qquad\left(\left(w_{b,n,m,r}\mathbf{f}_{1,b,j_{b,n,m}}+w_{b,n,m+1,r}\mathbf{f}_{1,b,j_{b,n,m+1}}\right)\mathbf{r}_{a c,1,b,n,m,p}^{T}\right.\right.\right.}\\ &{\left.\left.\left.\left.+\left(w_{b,n,m,r}\mathbf{m}_{1,b,j_{b,n,m}}+w_{b,n,m+1,r}\mathbf{m}_{1,b,j_{b,n,m+1}}\right)\mathbf{e}_{1,c,1,b,n,m,p}^{T}\right)\right)\right)\right)}\end{array}
+\mathbf{M}_{b,n} = \frac{l_n}{2} \left( \sum_{m=1}^{N_{a,b,n}-1} \left( \sum_{p=0}^{P+4} \left( \sum_{r=0}^{1} \frac{b_{b,n,m}^{p+r+1} - a_{b,n,m}^{p+r+1}}{p+r+1} \right. \right. \right. \\
+\left. \left( \mathbf{r}_{ac,1,b,n,m,p, q_{i}}^T \left( w_{b,n,m,r} \mathbf{f}_{1,b,j_b,n,m} + w_{b,n,m+1,r} \mathbf{f}_{1,b,j_b,n,m+1} \right)  \right. \right. \\
+\left. \left. \left. + \mathbf{e}_{1,c,1,b,n,m,p, q_{i}}^T  \left( w_{b,n,m,r} \mathbf{m}_{1,b,j_b,n,m} + w_{b,n,m+1,r} \mathbf{m}_{1,b,j_b,n,m+1} \right) \right) \right) \right) 
+\tag{3.33}
 $$  
 
 Similar, the contribution from element $n$ to the generalized aerodynamic force on DOF $q_{i}$ of the substructure $b$ (2.18b) can be written as  
-
+类似地，单元 $n$ 对子结构 $b$ 在自由度 $q_{i}$ 上的广义气动力 (2.18b) 的贡献可以写成
 $$
-\begin{array}{r l}{\lefteqn{Q_{i,b,n}=\frac{l_{n}}{2}\left(\sum_{m=1}^{N_{a,b,n}-1}\left(\sum_{p=0}^{P+4}\left(\sum_{r=0}^{1}\frac{b_{b,n,m}^{p+r+1}-a_{b,n,m}^{p+r+1}}{p+r+1}\right.\right.}\\ &{\left.\left.\qquad\left(\mathbf{r}_{a c,1,b,n,m,p,q_{i}}^{T}\left(w_{b,n,m,r}\mathbf{f}_{1,b,j_{b,n,m}}+w_{b,n,m+1,r}\mathbf{f}_{1,b,j_{b,n,m+1}}\right)\right.\right.\right.}\\ &{\left.\left.\left.\left.+\mathbf{e}_{1,c,1,b,n,m,p,q_{i}}^{T}\left(w_{b,n,m,r}\mathbf{m}_{1,b,j_{b,n,m}}+w_{b,n,m+1,r}\mathbf{m}_{1,b,j_{b,n,m+1}}\right)\right)\right)\right)}\right)}\end{array}
+Q_{i,b,n} = \frac{l_n}{2} \left( \sum_{m=1}^{N_{a,b,n}-1} \left( \sum_{p=0}^{P+4} \left( \sum_{r=0}^{1} \frac{b_{b,n,m}^{p+r+1} - a_{b,n,m}^{p+r+1}}{p+r+1} \right. \right. \right. \\
+\left. \left( \mathbf{r}_{ac,1,b,n,m,p,q_i}^T \left( w_{b,n,m,r} \mathbf{f}_{1,b,j_b,n,m} + w_{b,n,m+1,r} \mathbf{f}_{1,b,j_b,n,m+1} \right) \right. \right. \\
+\left. \left. \left. + \mathbf{e}_{1,c,1,b,n,m,p,q_i}^T \left( w_{b,n,m,r} \mathbf{m}_{1,b,j_b,n,m} + w_{b,n,m+1,r} \mathbf{m}_{1,b,j_b,n,m+1} \right) \right) \right) \right)
+\tag{3.34}
 $$  
 
 We now define the element transformation matrices $\boldsymbol{\mathsf{T}}_{f,b,n}$ for the total force, $\boldsymbol{\mathsf{T}}_{{M f,b,n}}$ and $\mathsf{T}_{M m,b,n}$ for the total moment, and ${\sf T}_{Q f,b,n}$ and $\mathsf{T}_{Q m,b,n}$ for the generalized force as  
-
+我们现在定义用于总力的单元变换矩阵$\boldsymbol{\mathsf{T}}_{f,b,n}$，用于总力矩的$\boldsymbol{\mathsf{T}}_{{M f,b,n}}$和$\mathsf{T}_{M m,b,n}$，以及用于广义力的${\sf T}_{Q f,b,n}$和$\mathsf{T}_{Q m,b,n}$。
 $$
-\begin{array}{r}{\mathbf{f}_{b,n}=\mathsf{T}_{f,b,n}\mathbf{f}_{a l l,1,b,n}\,}\\ {\mathsf{M}_{b,n}=\mathsf{T}_{M f,b,n}\mathbf{f}_{a l l,1,b,n}+\mathsf{T}_{M m,b,n}\mathbf{m}_{a l l,1,b,n}}\\ {\mathbf{Q}_{a l l,b,n}=\mathsf{T}_{Q f,b,n}\mathbf{f}_{a l l,1,b,n}+\mathsf{T}_{Q m,b,n}\mathbf{m}_{a l l,1,b,n}}\end{array}
+\mathbf{f}_{b,n} = \mathbf{T}_{f,b,n} \mathbf{f}_{all,1,b,n}
+\tag{3.35a}
+$$
+$$
+\mathbf{M}_{b,n} = \mathbf{T}_{Mf,b,n} \mathbf{f}_{all,1,b,n} + \mathbf{T}_{Mm,b,n} \mathbf{m}_{all,1,b,n}
+\tag{3.35b}
+$$
+$$
+\mathbf{Q}_{all,b,n} = \mathbf{T}_{Qf,b,n} \mathbf{f}_{all,1,b,n} + \mathbf{T}_{Qm,b,n} \mathbf{m}_{all,1,b,n}
+\tag{3.35c}
 $$  
 
 where $\mathsf{f}_{a l l,1,b,n}$ and $\mathbf{m}_{a l l,1,b,n}$ are vectors of length $3N_{a,b,n}$ containing all forces and moments affecting the aero­ dynamic force and moment distributions over element $n$ on substructure $b$ , and $\mathbf{Q}_{a l l,b,n}$ is a vector of length 12 containing the generalized forces on the 12 DOFs that describe the deformation of element $n$ .  
-
+其中，$\mathsf{f}_{a l l,1,b,n}$ 和 $\mathbf{m}_{a l l,1,b,n}$ 是长度为 $3N_{a,b,n}$ 的向量，包含影响子结构 $b$ 上单元 $n$ 气动力和力矩分布的所有力与力矩；而 $\mathbf{Q}_{a l l,b,n}$ 是一个长度为 12 的向量，包含作用在描述单元 $n$ 变形的 12 个自由度上的广义力。
 # Bibliography  
 
 [1] Meirovitch L. Methods of Analytical Dynamics. New York: McGraw­Hill; 1970.