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.obsidian/core-plugins.json
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"file-recovery": true,
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"publish": false,
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"sync": false,
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"webviewer": false
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"webviewer": false,
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"footnotes": false,
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"bases": true
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}
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@ -1103,7 +1103,7 @@ The chapter is structured as follows: Section 3.1
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前两章中描述的结构运动和气动力现在耦合在一组封闭的气动弹性运动方程中。结构自由度上的广义气动力以通用形式在方程（2.18）中给出，用于在单个子结构上积分气动力和力矩分布。叶片翼型截面处的相对流取决于结构运动，如第2.2节中以通用形式描述的。翼型截面处的气动力和力矩取决于这种相对流、风场、诱导速度以及脱落涡量和动态失速的局部动态效应。叶片的气动离散化为气动计算点（ACPs）以及用于描述力和力矩分布的分段线性函数在第2.3节中介绍。非定常翼型气动模型在第2.2.2节中描述。动态入流的尾流模型（诱导速度的非定常性）取决于局部推力和扭矩系数以及偏航和上洗角，如第2.4节所述。
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本章结构如下：第3.1节
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# 3.1 Generalized aerodynamic forces for different substructure types
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## 3.1 Generalized aerodynamic forces for different substructure types
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The total generalized aerodynamic force from a blade on each structural DOF is given in (2.12) by the sum of the contributions from aerodynamic forces and moment distributions on each substructure $b=b_{\beta},\ldots,b_{\beta}+B_{\beta}-1$ of the $B_{\beta}$ substructures comprising blade number $\beta$ . These contributions are given by the integrals over each substructure in Equations (2.18) and (2.19). In this section, we derive expressions for these integrals for the different types of substructures in CASEStab using piecewise linear functions of the aerodynamic forces and moment distributions over the substructure. The sectional force and moment vectors at the ACPs are denoted $\mathfrak{f}_{1,\beta,b,j}$ and $\mathbf{m}_{1,\beta,b,j}$ , where the first subscript “1” refers to the fact that the vector has been computed in or transformed to the frame of the substructure number $b$ being the third subscript, the second subscript $\beta$ is the blade number, and the fourth and last subscript is the ACP number $j=1,\dots,N_{a}$ on the blade.
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@ -1130,11 +1130,11 @@ The outcome of this section are matrices that transform the $N_{a}$ forces and $
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Note that we omit the subscript $\beta$ for the blade number in the following sections because we are referring to a single blade.
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请注意，在以下章节中，我们省略了表示叶片编号的下标 $\beta$，因为我们指的是单个叶片。
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# 3.1.1 Rigid body substructure
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### 3.1.1 Rigid body substructure
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To be derived ...
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# 3.1.2 Corotational beam substructure
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### 3.1.2 Corotational beam substructure
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Figure 3.1 shows a 2D illustration of the aerodynamic force distribution over a blade and an element of a corotational beam element substructure of the blade. The blade consists of two substructures $b-1$ and $b$ described by several beam elements with their end nodes are marked by $\bullet$ . The force distribution over the blade is a piecewise linear function with the $N_{a}$ ACPs (marked by ◦) as breakpoints. The piecewise linear function over the element $n$ of substructure $b$ is here defined by four ACPs, written as $N_{a,b,n}\,=\,4$ , i.e., the force (and moment) distribution over this element is described by two internal breakpoints and two end-break points on the adjacent elements. We define a vector $\zeta_{b,n,k}$ for $k=1,\ldots,N_{a,b,n}$ containing the nondimensional element coordinate of these breakpoints for each element $n$ . Note that the end-break-points will always lie at the element nodes $\left\langle\zeta\right.=\pm1)$ ) or outside the element, i.e., $\zeta_{b,n,1}\,\leq\,-1$ and $\zeta_{b,n,N_{a,b,n}}\,\geq\,1$ . In case that these points lie on the adjacent elements, their element coordinates can be computed as
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图3.1展示了叶片上的气动力分布以及叶片共旋梁单元子结构的一个单元的二维示意图。叶片由两个子结构$b-1$和$b$组成，它们由若干梁单元描述，其端节点用$\bullet$标记。叶片上的力分布是一个分段线性函数，以$N_{a}$个ACPs（用◦标记）作为分段点。子结构$b$的单元$n$上的分段线性函数由四个ACPs定义，记作$N_{a,b,n}\,=\,4$，即该单元上的力（和力矩）分布由两个内部断点和相邻单元上的两个端部断点描述。我们定义一个向量$\zeta_{b,n,k}$，其中$k=1,\ldots,N_{a,b,n}$，包含每个单元$n$的这些断点的无量纲单元坐标。注意，端部断点将始终位于单元节点（$\left\langle\zeta\right.=\pm1)$）或单元外部，即$\zeta_{b,n,1}\,\leq\,-1$和$\zeta_{b,n,N_{a,b,n}}\,\geq\,1$。如果这些点位于相邻单元上，它们的单元坐标可以计算为
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@ -1187,49 +1187,65 @@ $$
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where $\mathsf{f}_{1,b,j}$ and $\mathbf{m}_{1,b,j}$ are the force and moment vectors at $j$ ’th ACP described in the frame of substructure $b$ , and the index vector $j_{b,n,k}$ with $k=1,\ldots,N_{a,b,n}$ contains the indices $j$ of the ACPs involved in the piecewise linear function over element $n$ on substructure $b$ , cf. Figure 3.1. We rewrite these coefficients as
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其中，$\mathsf{f}_{1,b,j}$ 和 $\mathbf{m}_{1,b,j}$ 是在子结构 $b$ 坐标系中描述的第 $j$ 个 ACP 上的力和力矩向量，索引向量 $j_{b,n,k}$（其中 $k=1,\ldots,N_{a,b,n}$）包含子结构 $b$ 上单元 $n$ 的分段线性函数所涉及的 ACP 的索引 $j$，参见图 3.1。我们将这些系数重写为
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$$
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\begin{array}{r l}&{\mathbf{f}_{1,b,n,m,r}=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}}}\\ &{\mathbf{m}_{1,b,n,m,r}=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}}}\end{array}
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$$
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{\mathbf{f}_{1,b,n,m,r}=w_{b,n,m,r}\mathbf{f}_{1,b,j_{b,n,m}}+w_{b,n,m+1,r}\mathbf{f}_{1,b,j_{b,n,m+1}}}\tag{3.10a}
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$$
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$$
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{\mathbf{m}_{1,b,n,m,r}=w_{b,n,m,r}\mathbf{m}_{1,b,j_{b,n,m}}+w_{b,n,m+1,r}\mathbf{m}_{1,b,j_{b,n,m+1}}}\tag{3.10b}
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$$
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where $r=0,1$ and the scalar weights are given by the break points
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其中 $r=0,1$ 且标量权重由拐点给出
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$$
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\begin{array}{r c l}{w_{b,n,m,0}=\frac{\zeta_{b,n,m+1}}{\zeta_{b,n,m+1}-\zeta_{b,n,m}}\;,}&{w_{b,n,m+1,0}=\frac{-\zeta_{b,n,m}}{\zeta_{b,n,m+1}-\zeta_{b,n,m}}\;,}\\ {w_{b,n,m,1}=\frac{-1}{\zeta_{b,n,m+1}-\zeta_{b,n,m}}\;,}&{\mathsf{a n d}\;\;w_{b,n,m+1,1}=\frac{1}{\zeta_{b,n,m+1}-\zeta_{b,n,m}}}\end{array}
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\begin{array}{r c l}{w_{b,n,m,0}=\frac{\zeta_{b,n,m+1}}{\zeta_{b,n,m+1}-\zeta_{b,n,m}}\;,}&{w_{b,n,m+1,0}=\frac{-\zeta_{b,n,m}}{\zeta_{b,n,m+1}-\zeta_{b,n,m}}\;,}\\ {w_{b,n,m,1}=\frac{-1}{\zeta_{b,n,m+1}-\zeta_{b,n,m}}\;,}&{\mathsf{a n d}\;\;w_{b,n,m+1,1}=\frac{1}{\zeta_{b,n,m+1}-\zeta_{b,n,m}}}\end{array}\tag{3.11}
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$$
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which are precomputed in the initialization of the coupling of the substructure with the aerodynamic blade.
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其在子结构与气动叶片的耦合初始化中预先计算。
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For the integrations of generalized aerodynamic forces in (2.18) and (2.19), we also need the position vector of the aerodynamic center (AC) in the substructure frame $\boldsymbol{\mathsf{r}}_{a c,1,b}$ and its first DOF derivative $\boldsymbol{\mathsf{r}}_{a c,1,b,q_{i}}$ . Using the position vector formulation (1.45), we can write it as functions of the element coordinate of each element $n$ :
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对于（2.18）和（2.19）中广义气动力的积分，我们还需要气动中心（AC）在子结构坐标系中的位置矢量$\boldsymbol{\mathsf{r}}_{a c,1,b}$及其一阶自由度导数$\boldsymbol{\mathsf{r}}_{a c,1,b,q_{i}}$。使用位置矢量公式（1.45），我们可以将其写成每个单元$n$的单元坐标的函数：
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$$
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\mathsf{r}_{a c,1,b,n}=\sum_{p=0}^{P+3}\left(\mathsf{r}_{o,b,n,p}+x_{a c,b,n}(\zeta)\,\mathsf{r}_{x,b,n,p}+y_{a c,b,n}(\zeta)\,\mathsf{r}_{y,b,n,p}\right)\zeta^{p}
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\mathsf{r}_{a c,1,b,n}=\sum_{p=0}^{P+3}\left(\mathsf{r}_{o,b,n,p}+x_{a c,b,n}(\zeta)\,\mathsf{r}_{x,b,n,p}+y_{a c,b,n}(\zeta)\,\mathsf{r}_{y,b,n,p}\right)\zeta^{p}\tag{3.12}
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$$
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where $\mathbf{r}_{o,b,n,p},\ \mathbf{r}_{x,b,n,p}$ , and ${\sf r}_{y,b,n,p}$ are the coefficient vectors given by (1.46), and functions $x_{a c,b,n}(\zeta)$ and $y_{a c,b,n}(\zeta)$ describe the varying position of the aerodynamic center in the element coordinate system. In CAS EStab, we assume that this variation can be approximated by the piecewise linear function:
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where $\mathbf{r}_{o,b,n,p},\ \mathbf{r}_{x,b,n,p}$ , and ${\sf r}_{y,b,n,p}$ are the coefficient vectors given by (1.46), and functions $x_{a c,b,n}(\zeta)$ and $y_{a c,b,n}(\zeta)$ describe the varying position of the aerodynamic center in the element coordinate system. In CASEStab, we assume that this variation can be approximated by the piecewise linear function:
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其中 $\mathbf{r}_{o,b,n,p},\ \mathbf{r}_{x,b,n,p}$ 和 ${\sf r}_{y,b,n,p}$ 是由 (1.46) 给出的系数向量，函数 $x_{a c,b,n}(\zeta)$ 和 $y_{a c,b,n}(\zeta)$ 描述了气动中心在单元坐标系中变化的位置。在 CASEStab 中，我们假设这种变化可以通过分段线性函数来近似：
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$$
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\begin{array}{r l}&{x_{a c,b,n}(\zeta)=\left\{\begin{array}{c c}{\sum_{r=0}^{1}c_{a c,x,b,n,n,r}\zeta^{r}}&{a_{b,n,1}\leq\zeta<b_{b,n,1}}\\ {\sum_{r=0}^{1}c_{a c,x,b,n,2,r}\zeta^{r}}&{a_{b,n,2}\leq\zeta<b_{b,n,2}}\\ {\vdots}&{\vdots}\\ {\sum_{r=0}^{1}c_{a c,x,b,n,N,n,b,n-1,r}\zeta^{r}}&{a_{b,n,N,a,b,n-1}\leq\zeta\leq b_{b,n,N_{a,b,n}-1}}\end{array}\right.}\\ &{y_{a c,b,n}(\zeta)=\left\{\begin{array}{c c}{\sum_{r=0}^{1}c_{a c,y,b,n,n,r}\zeta^{r}}&{a_{b,n,1}\leq\zeta<b_{b,n,1}}\\ {\sum_{r=0}^{1}c_{a c,y,b,n,1,r}\zeta^{r}}&{a_{b,n,2}\leq\zeta<b_{b,n,2}}\\ {\vdots}&{\vdots}\\ {\sum_{r=0}^{1}c_{a c,y,b,n,N_{a,b,n-1},r}\zeta^{r}}&{a_{b,n,N_{a,b,n-1}}\leq\zeta\leq b_{b,n,N_{a,b,n-1}}}\end{array}\right.}\end{array}
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\begin{array}{r l}&{x_{a c,b,n}(\zeta)=\left\{\begin{array}{c c}{\sum_{r=0}^{1}c_{a c,x,b,n,1,r}\zeta^{r}}&{a_{b,n,1}\leq\zeta<b_{b,n,1}}\\ {\sum_{r=0}^{1}c_{a c,x,b,n,2,r}\zeta^{r}}&{a_{b,n,2}\leq\zeta<b_{b,n,2}}\\ {\vdots}&{\vdots}\\ {\sum_{r=0}^{1}c_{a c,x,b,n,N_{a,b,n}-1,r}\zeta^{r}}&{a_{b,n,N_{a,b,n}-1}\leq\zeta\leq b_{b,n,N_{a,b,n}-1}}\end{array}\right.}\\ &{y_{a c,b,n}(\zeta)=\left\{\begin{array}{c c}{\sum_{r=0}^{1}c_{a c,y,b,n,0,r}\zeta^{r}}&{a_{b,n,1}\leq\zeta<b_{b,n,1}}\\ {\sum_{r=0}^{1}c_{a c,y,b,n,1,r}\zeta^{r}}&{a_{b,n,2}\leq\zeta<b_{b,n,2}}\\ {\vdots}&{\vdots}\\ {\sum_{r=0}^{1}c_{a c,y,b,n,N_{a,b,n}-1,r}\zeta^{r}}&{a_{b,n,N_{a,b,n}-1}\leq\zeta\leq b_{b,n,N_{a,b,n}-1}}\end{array}\right.}\end{array}\tag{3.13}
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$$
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where the two coefficients of each linear functions $c_{a c,x,b,n,m,r}$ and $c_{a c,y,b,n,m,r}$ for $m\,=\,1,\dots,N_{a,b,n}\,-\,1$ are constants computed during the initial model assembly as described in Appendix B. Combining (3.12) and (3.13), the position vector of the aerodynamic center over the element is given by the piecewise polynomial function
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其中，每个线性函数$c_{a c,x,b,n,m,r}$和$c_{a c,y,b,n,m,r}$的两个系数，对于$m\,=\,1,\dots,N_{a,b,n}-1$，是在初始模型组装期间计算的常数，如附录B所述。结合(3.12)和(3.13)，气动中心在单元上的位置向量由分段多项式函数给出
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$$
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\begin{array}{r}{\mathbf{r}_{a c,1,b,n}=\left\{\begin{array}{c c}{\sum_{p=0}^{P+4}\mathbf{r}_{a c,1,b,n,1,p}\,\zeta^{p}}&{a_{b,n,1}\leq\zeta<b_{b,n,1}}\\ {\sum_{p=0}^{P+4}\mathbf{r}_{a c,1,b,n,2,p}\,\zeta^{p}}&{a_{b,n,2}\leq\zeta<b_{b,n,2}}\\ {\vdots}&{\vdots}\\ {\sum_{p=0}^{P+4}\mathbf{r}_{a c,1,b,n,N_{a,b,n}-1,p}\,\zeta^{p}}&{a_{b,n,N_{a,b,n}-1}\leq\zeta\leq b_{b,n,N_{a,b,n}-1}}\end{array}\right.}\end{array}
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\begin{array}{r}{\mathbf{r}_{a c,1,b,n}=\left\{\begin{array}{c c}{\sum_{p=0}^{P+4}\mathbf{r}_{a c,1,b,n,1,p}\,\zeta^{p}}&{a_{b,n,1}\leq\zeta<b_{b,n,1}}\\ {\sum_{p=0}^{P+4}\mathbf{r}_{a c,1,b,n,2,p}\,\zeta^{p}}&{a_{b,n,2}\leq\zeta<b_{b,n,2}}\\ {\vdots}&{\vdots}\\ {\sum_{p=0}^{P+4}\mathbf{r}_{a c,1,b,n,N_{a,b,n}-1,p}\,\zeta^{p}}&{a_{b,n,N_{a,b,n}-1}\leq\zeta\leq b_{b,n,N_{a,b,n}-1}}\end{array}\right.}\end{array}\tag{3.14}
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$$
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where the coefficient vectors $\mathbf{r}_{a c,1,b,n,m,p}$ for $m=1,\dots,N_{a,b,n}-1$ are derived from (1.46) as
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其中，$m=1,\dots,N_{a,b,n}-1$ 的系数向量 $\mathbf{r}_{a c,1,b,n,m,p}$ 由 (1.46) 导出，如下所示：
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$$
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\begin{array}{r}{\mathbf{r}_{a c,1,b,n,m,p}=\left\{\begin{array}{l l}{\mathbf{r}_{o,b,n,p}+c_{a c,x,b,n,m,0}\mathbf{r}_{x,b,n,p}+c_{a c,y,b,n,m,0}\mathbf{r}_{y,b,n,p}\quad\quad}&{p=0}\\ {\;}&{\mathbf{r}_{o,b,n,p}+c_{a c,x,b,n,m,0}\mathbf{r}_{x,b,n,p}+c_{a c,y,b,n,m,0}\mathbf{r}_{y,b,n,p}\quad\quad}\\ {\;}&{+c_{a c,x,b,n,m,1}\mathbf{r}_{x,b,n,p-1}+c_{a c,y,b,n,m,1}\mathbf{r}_{y,b,n,p-1}\quad\quad}\\ {\;}&{c_{a c,x,b,n,m,1}\mathbf{r}_{x,b,n,p-1}+c_{a c,y,b,n,m,1}\mathbf{r}_{y,b,n,p-1}\quad\quad}&{p=P+4}\end{array}\right.}\end{array}
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\mathbf{r}_{ac,1,b,n,m,p} =
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\begin{cases}
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\mathbf{r}_{o,b,n,p} + c_{ac,x,b,n,m,0} \mathbf{r}_{x,b,n,p} + c_{ac,y,b,n,m,0} \mathbf{r}_{y,b,n,p} & p = 0 \\
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\begin{aligned}
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&\mathbf{r}_{o,b,n,p} + c_{ac,x,b,n,m,0} \mathbf{r}_{x,b,n,p} + c_{ac,y,b,n,m,0} \mathbf{r}_{y,b,n,p} \\
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&+ c_{ac,x,b,n,m,1} \mathbf{r}_{x,b,n,p-1} + c_{ac,y,b,n,m,1} \mathbf{r}_{y,b,n,p-1}
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\end{aligned}
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& \forall p \in [1:P+3] \\
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c_{ac,x,b,n,m,1} \mathbf{r}_{x,b,n,p-1} + c_{ac,y,b,n,m,1} \mathbf{r}_{y,b,n,p-1} & p = P+4
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\end{cases}
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\tag{3.15}
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$$
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where $P$ is the order of the structural element. Note that the coefficient vectors in (1.46) are functions of the substructure DOFs and their first and second DOF derivatives are given by (1.57) and (1.58). The first DOF derivatives of $\mathbf{r}_{a c,1,b,n}$ are therefore
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其中 $P$ 是结构单元的阶数。请注意，(1.46) 中的系数向量是子结构自由度及其一阶和二阶自由度导数的函数，由 (1.57) 和 (1.58) 给出。因此，$\mathbf{r}_{a c,1,b,n}$ 的一阶自由度导数为
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$$
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\begin{array}{r}{\mathbf{r}_{a c,1,b,n,q_{i}}=\left\{\begin{array}{c c}{\sum_{p=0}^{P+4}\mathbf{r}_{a c,1,b,n,1,p,q_{i}}\;\zeta^{p}}&{a_{b,n,1}\leq\zeta<b_{b,n,1}}\\ {\sum_{p=0}^{P+4}\mathbf{r}_{a c,1,b,n,2,p,q_{i}}\;\zeta^{p}}&{a_{b,n,2}\leq\zeta<b_{b,n,2}}\\ {\vdots}&{\vdots}\\ {\sum_{p=0}^{P+4}\mathbf{r}_{a c,1,b,n,N_{a,b,n}-1,p,q_{i}}\;\zeta^{p}}&{a_{b,n,N_{a,b,n}-1}\leq\zeta\leq b_{b,n,N_{a,b,n}-1}}\end{array}\right.}\end{array}
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\begin{array}{r}{\mathbf{r}_{a c,1,b,n,q_{i}}=\left\{\begin{array}{c c}{\sum_{p=0}^{P+4}\mathbf{r}_{a c,1,b,n,1,p,q_{i}}\;\zeta^{p}}&{a_{b,n,1}\leq\zeta<b_{b,n,1}}\\ {\sum_{p=0}^{P+4}\mathbf{r}_{a c,1,b,n,2,p,q_{i}}\;\zeta^{p}}&{a_{b,n,2}\leq\zeta<b_{b,n,2}}\\ {\vdots}&{\vdots}\\ {\sum_{p=0}^{P+4}\mathbf{r}_{a c,1,b,n,N_{a,b,n}-1,p,q_{i}}\;\zeta^{p}}&{a_{b,n,N_{a,b,n}-1}\leq\zeta\leq b_{b,n,N_{a,b,n}-1}}\end{array}\right.}\end{array}\tag{3.16}
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$$
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where the first derivatives of the coefficient vectors $\pmb{\Gamma}_{a c,1,b,n,m,p,q_{i}}$ for $m=1,\dots,N_{a,b,n}-1$ are
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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
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$$
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\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}
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