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The structural motion and the aerodynamic forces described in the previous two chapters are now coupled in a closed set of aeroelastic equations of motion. The generalized aerodynamic forces on the structural DOFs are given on a generic form in Equation (2.18) for the integration of the aerodynamic forces and moment distribution over a single substructure. The relative flow at the airfoil section of a blade depend on the structural motion as described on generic form in Section 2.2. The aerodynamic forces and moment at the airfoil section depend on this relative flow, the wind field, the induced velocities, and local dynamic effects of shed vorticity and dynamic stall. The aerodynamic discretization of the blade into aerodynamic calculation points (ACPs) and the piecewise linear functions used to describe the distributions of forces and moment are introduced in Section 2.3. The unsteady airfoil aerodynamic model is described in Section 2.2.2. The wake model of dynamic inflow (the unsteadiness of the induced velocities) depend on the local thrust and torque coefficients and the yaw and upflow angles as described in Section 2.4.
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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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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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叶片在每个结构自由度上的总广义气动力在(2.12)中给出，它等于组成叶片$\beta$的$B_{\beta}$个子结构中每个子结构$b=b_{\beta},\ldots,b_{\beta}+B_{\beta}-1$上的气动力和力矩分布贡献之和。这些贡献由方程(2.18)和(2.19)中每个子结构上的积分给出。在本节中，我们利用子结构上气动力和力矩分布的分段线性函数，推导了CASEStab中不同类型子结构这些积分的表达式。在气动中心点（ACPs）处的截面力和力矩向量分别表示为${f}_{1,\beta,b,j}$和$\mathbf{m}_{1,\beta,b,j}$，其中第一个下标“1”表示该向量已在第三个下标所指的子结构$b$的坐标系中计算或转换，第二个下标$\beta$是叶片编号，第四个也是最后一个下标是叶片上的气动中心点编号$j=1,\dots,N_{a}$。
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We define an index vector $j_{b,k}$ for $k=1,\ldots,N_{a,b}$ containing the indices of the $N_{a,b}$ ACPs which sectional forces and moments define the forces and moment distributions over the substructure $b$ . We collect these forces and moments needed for the linear interpolation over substructure number $b$ in the following vectors:
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我们定义一个索引向量$j_{b,k}$，其中$k=1,\ldots,N_{a,b}$，它包含$N_{a,b}$个ACP的索引，这些ACP的截面力和力矩定义了子结构$b$上的力和力矩分布。我们将子结构$b$上线性插值所需的这些力和力矩收集到以下向量中：
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$$
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\mathbf{f}_{a l l,1,\beta,b}=\left\{\begin{array}{c}{\mathbf{R}_{0,b}^{T}\mathbf{R}_{0,b-1}\mathbf{f}_{1,\beta,b-1,j_{b,1}}}\\ {\mathbf{f}_{1,\beta,b,j_{b,2}}}\\ {\mathbf{f}_{1,\beta,b,j_{b,3}}}\\ {\vdots}\\ {\mathbf{f}_{1,\beta,b,j_{b,N_{a,b}-1}}}\\ {\mathbf{R}_{0,b}^{T}\mathbf{R}_{0,b+1}\mathbf{f}_{1,\beta,b+1,j_{b,N_{a,b}}}}\end{array}\right\}\;\;\mathrm{and}\;\;\mathbf{m}_{a l l,1,\beta,b}=\left\{\begin{array}{c}{\mathbf{R}_{0,b}^{T}\mathbf{R}_{0,b-1}\mathbf{m}_{1,\beta,b-1,j_{b,1}}}\\ {\mathbf{m}_{1,\beta,j_{b,2}}}\\ {\mathbf{m}_{1,\beta,b,j_{b,3}}}\\ {\vdots}\\ {\mathbf{m}_{1,\beta,b,j_{b,N_{a,b}-1}}}\\ {\mathbf{R}_{0,b}^{T}\mathbf{R}_{0,b+1}\mathbf{m}_{1,\beta,b+1,j_{b,N_{a,b}}}}\end{array}\right\}
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\mathbf{f}_{a l l,1,\beta,b}=\left\{\begin{array}{c}{\mathbf{R}_{0,b}^{T}\mathbf{R}_{0,b-1}\mathbf{f}_{1,\beta,b-1,j_{b,1}}}\\ {\mathbf{f}_{1,\beta,b,j_{b,2}}}\\ {\mathbf{f}_{1,\beta,b,j_{b,3}}}\\ {\vdots}\\ {\mathbf{f}_{1,\beta,b,j_{b,N_{a,b}-1}}}\\ {\mathbf{R}_{0,b}^{T}\mathbf{R}_{0,b+1}\mathbf{f}_{1,\beta,b+1,j_{b,N_{a,b}}}}\end{array}\right\}\;\;\mathrm{and}\;\;\mathbf{m}_{a l l,1,\beta,b}=\left\{\begin{array}{c}{\mathbf{R}_{0,b}^{T}\mathbf{R}_{0,b-1}\mathbf{m}_{1,\beta,b-1,j_{b,1}}}\\ {\mathbf{m}_{1,\beta,j_{b,2}}}\\ {\mathbf{m}_{1,\beta,b,j_{b,3}}}\\ {\vdots}\\ {\mathbf{m}_{1,\beta,b,j_{b,N_{a,b}-1}}}\\ {\mathbf{R}_{0,b}^{T}\mathbf{R}_{0,b+1}\mathbf{m}_{1,\beta,b+1,j_{b,N_{a,b}}}}\end{array}\right\}\tag{3.1}
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$$
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where the first and last vectors defined on adjacent substructures must be transformed into the frame of sub structure $b$ by the orientation matrices $\mathbf{R}_{0,b-1}$ , $\pmb{\mathsf{R}}_{0,b}$ , and $\mathsf{R}_{0,b+1}$ for each involved substructure (1.13). If the blade is described by a single substructure then $j_{b}=1$ and $N_{a,b}=N_{a}$ , and the above vectors reduce to
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其中，在相邻子结构上定义的首末向量必须通过方向矩阵 $\mathbf{R}_{0,b-1}$ 、 $\pmb{\mathsf{R}}_{0,b}$ 和 $\mathsf{R}_{0,b+1}$ 转换为子结构 $b$ 的坐标系，适用于每个相关子结构 (1.13)。如果叶片由单个子结构描述，则 $j_{b}=1$ 且 $N_{a,b}=N_{a}$ ，并且上述向量简化为
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$$
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\mathbf{f}_{a l l,1,\beta,b}=\left\{\begin{array}{l}{\mathbf{f}_{1,\beta,b,j_{b,1}}}\\ {\mathbf{f}_{1,\beta,b,j_{b,2}}}\\ {\qquad\vdots}\\ {\mathbf{f}_{1,\beta,b,j_{b,N_{a}}}}\end{array}\right\}\,\,\,\mathbf{and}\,\,\,\mathbf{m}_{a l l,1,\beta,b}=\left\{\begin{array}{l}{\mathbf{m}_{1,\beta,b,j_{b,1}}}\\ {\mathbf{m}_{1,\beta,b,j_{b,2}}}\\ {\qquad\vdots}\\ {\mathbf{m}_{1,\beta,b,j_{b,N_{a}}}}\end{array}\right\}
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\mathbf{f}_{a l l,1,\beta,b}=\left\{\begin{array}{l}{\mathbf{f}_{1,\beta,b,j_{b,1}}}\\ {\mathbf{f}_{1,\beta,b,j_{b,2}}}\\ {\qquad\vdots}\\ {\mathbf{f}_{1,\beta,b,j_{b,N_{a}}}}\end{array}\right\}\,\,\,\mathbf{and}\,\,\,\mathbf{m}_{a l l,1,\beta,b}=\left\{\begin{array}{l}{\mathbf{m}_{1,\beta,b,j_{b,1}}}\\ {\mathbf{m}_{1,\beta,b,j_{b,2}}}\\ {\qquad\vdots}\\ {\mathbf{m}_{1,\beta,b,j_{b,N_{a}}}}\end{array}\right\}\tag{3.2}
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$$
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as one special case of (3.1). The other two special cases are when $b$ is the first or last of several substructures on the blade.
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作为（3.1）的一个特例。另外两种特例是当$b$是叶片上几个子结构中的第一个或最后一个时。
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The outcome of this section are matrices that transform the $N_{a}$ forces and $N_{a}$ moments of the ACPs on the blade to the total aerodynamic force and moment, as well as the
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本节的结果是矩阵，可将ACPs在叶片上的$N_{a}$个力和$N_{a}$个力矩转换为总气动力和力矩，以及
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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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To be derived ...
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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 endbreak 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 endbreakpoints 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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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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$$
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\zeta_{b,n,1}=-1-\frac{l_{n-1}}{l_{n}}\left(1-\zeta_{b,n-1,N_{a,b,n-1}-1}\right)\quad\mathrm{~and~}\quad\zeta_{b,n,N_{a,b,n}}=1+\frac{l_{n+1}}{l_{n}}\left(1+\zeta_{b,n+1,2}\right)
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\zeta_{b,n,1}=-1-\frac{l_{n-1}}{l_{n}}\left(1-\zeta_{b,n-1,N_{a,b,n-1}-1}\right)\quad\mathrm{~and~}\quad\zeta_{b,n,N_{a,b,n}}=1+\frac{l_{n+1}}{l_{n}}\left(1+\zeta_{b,n+1,2}\right)\tag{3.3}
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$$
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where $l_{n}$ is the initial lengths of elements $n\,=\,1,\ldots,N_{e,b}$ . Note that special rules apply to the first and last elements of a substructure, where the adjacent element may lie on an adjacent substructure.
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其中 $l_{n}$ 是单元 $n\,=\,1,\ldots,N_{e,b}$ 的初始长度。请注意，特殊规则适用于子结构的第一个和最后一个单元，其中相邻单元可能位于相邻子结构上。
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A curve integral over the corotational beam substructure $b$ is the sum of integrals over each element $n$ as
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共旋梁子结构$b$上的曲线积分是每个单元$n$上的积分之和，表示为
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$$
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\int_{0}^{L_{b}}\left(\mathbf{\Phi}\right)d\sigma=\sum_{n=1}^{N_{e,b}}\frac{l_{n}}{2}\int_{-1}^{1}\left(\mathbf{\Phi}\right)d\zeta=\sum_{n=1}^{N_{e,b}}\frac{l_{n}}{2}\left(\sum_{m=1}^{N_{a,b,n}-1}\int_{a_{b,n,m}}^{b_{b,n,m}}\left(\mathbf{\Phi}\right)d\zeta\right)
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\int_{0}^{L_{b}}\left(\mathbf{}\right)d\sigma=\sum_{n=1}^{N_{e,b}}\frac{l_{n}}{2}\int_{-1}^{1}\left(\mathbf{}\right)d\zeta=\sum_{n=1}^{N_{e,b}}\frac{l_{n}}{2}\left(\sum_{m=1}^{N_{a,b,n}-1}\int_{a_{b,n,m}}^{b_{b,n,m}}\left(\mathbf{}\right)d\zeta\right)\tag{3.4}
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$$
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where each curve integral over the element length is a line integral over the nondimensional element coordinate $\zeta$ between element its nodes at $\zeta\,=\,\pm1$ . To ensure that the integral kernels will be single polynomials (not piecewise), the line integral over each element is further subdivided into line integrals between the nodes and the $N_{a,b,n}-2$ internal breakpoints on the element. The limits of these subdivided integrals are
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其中，单元长度上的每个曲线积分是单元在无量纲单元坐标 $\zeta$ 介于其节点 $\zeta\,=\,\pm1$ 之间的线积分。为了确保积分核是单一多项式（而非分段多项式），每个单元上的线积分进一步细分为节点与单元上 $N_{a,b,n}-2$ 个内部断点之间的线积分。这些细分积分的限值是
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$$
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\begin{array}{r c l}{{(a_{b,n,1},b_{b,n,1})}}&{{=}}&{{(-1,\zeta_{b,n,2})}}\\ {{(a_{b,n,2},b_{b,n,2})}}&{{=}}&{{(\zeta_{b,n,2},\zeta_{b,n,3})}}\\ {{}}&{{\vdots}}&{{}}\\ {{(a_{b,n,N_{a,b,n}-1},b_{b,n,N_{a,b,n}-1})}}&{{=}}&{{(\zeta_{b,N_{a,b,n}-1},1)}}\end{array}
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\begin{array}{r c l}{{(a_{b,n,1},b_{b,n,1})}}&{{=}}&{{(-1,\zeta_{b,n,2})}}\\ {{(a_{b,n,2},b_{b,n,2})}}&{{=}}&{{(\zeta_{b,n,2},\zeta_{b,n,3})}}\\ {{}}&{{\vdots}}&{{}}\\ {{(a_{b,n,N_{a,b,n}-1},b_{b,n,N_{a,b,n}-1})}}&{{=}}&{{(\zeta_{b,N_{a,b,n}-1},1)}}\end{array}\tag{3.5}
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$$
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Note that in case there are no ACPs on an element, when $N_{a,b,n}=2$ , only a single integration over the element is needed $\left(a_{b,n,1},b_{b,n,1}\right)\,=\,\left(-1,1\right)$ . The piecewise linear distributions of the aerodynamic force and moment vectors over the element can be written on generic forms as
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请注意，如果某个单元上没有ACP，当$N_{a,b,n}=2$时，只需对该单元进行一次积分$\left(a_{b,n,1},b_{b,n,1}\right)\,=\,\left(-1,1\right)$。单元上气动力和力矩矢量的分段线性分布可以写成通用形式，如下所示：
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$$
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\mathbf{f}_{1,b,n}\left(\zeta\right)=\left\{\begin{array}{c c}{\sum_{r=0}^{1}\mathbf{f}_{1,b,n,0,r}\zeta^{r}}&{a_{b,n,1}\leq\zeta<b_{b,n,1}}\\ {\sum_{r=0}^{1}\mathbf{f}_{1,b,n,1,r}\zeta^{r}}&{a_{b,n,2}\leq\zeta<b_{b,n,2}}\\ {\vdots}&{\vdots}\\ {\sum_{r=0}^{1}\mathbf{f}_{1,b,n,N_{a,b,n},r}\zeta^{r}}&{a_{b,n,N_{a,b,n}}\leq\zeta<b_{b,n,N_{a,b,n}}}\end{array}\right.
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\mathbf{f}_{1,b,n}\left(\zeta\right)=\left\{\begin{array}{c c}{\sum_{r=0}^{1}\mathbf{f}_{1,b,n,0,r}\zeta^{r}}&{a_{b,n,1}\leq\zeta<b_{b,n,1}}\\ {\sum_{r=0}^{1}\mathbf{f}_{1,b,n,1,r}\zeta^{r}}&{a_{b,n,2}\leq\zeta<b_{b,n,2}}\\ {\vdots}&{\vdots}\\ {\sum_{r=0}^{1}\mathbf{f}_{1,b,n,N_{a,b,n},r}\zeta^{r}}&{a_{b,n,N_{a,b,n}}\leq\zeta<b_{b,n,N_{a,b,n}}}\end{array}\right.\tag{3.6}
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$$
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and
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$$
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\begin{array}{r}{\mathfrak{m}_{1,b,n}\left(\zeta\right)=\left\{\begin{array}{c c}{\sum_{r=0}^{1}\mathbf{m}_{1,b,n,0,r}\,\zeta^{r}}&{a_{b,n,1}\leq\zeta<b_{b,n,1}}\\ {\sum_{r=0}^{1}\mathbf{m}_{1,b,n,1,r}\,\zeta^{r}}&{a_{b,n,2}\leq\zeta<b_{b,n,2}}\\ {\vdots}&{\vdots}\\ {\sum_{r=0}^{1}\mathbf{m}_{1,b,n,N_{a,b,n},r}\,\zeta^{r}}&{a_{b,n,N_{a,b,n}}\leq\zeta<b_{b,n,N_{a,b,n}}}\end{array}\right.}\end{array}
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\begin{array}{r}{\mathfrak{m}_{1,b,n}\left(\zeta\right)=\left\{\begin{array}{c c}{\sum_{r=0}^{1}\mathbf{m}_{1,b,n,0,r}\,\zeta^{r}}&{a_{b,n,1}\leq\zeta<b_{b,n,1}}\\ {\sum_{r=0}^{1}\mathbf{m}_{1,b,n,1,r}\,\zeta^{r}}&{a_{b,n,2}\leq\zeta<b_{b,n,2}}\\ {\vdots}&{\vdots}\\ {\sum_{r=0}^{1}\mathbf{m}_{1,b,n,N_{a,b,n},r}\,\zeta^{r}}&{a_{b,n,N_{a,b,n}}\leq\zeta<b_{b,n,N_{a,b,n}}}\end{array}\right.}\end{array}\tag{3.7}
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$$
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Figure 3.1: Illustration in 2D of the aerodynamic force distribution over an element of a substructure of a blade. The blade consists of two substructures $b-1$ and $b$ described by corotational beam elements with end nodes $(\bullet)$ . The force distribution over the blade is a piecewise linear function with the $N_{a}$ aerodynamic calculation points $(\circ)$ as break points. The force distribution over element number $n$ of substructure $b$ has two internal break points $(N_{a,b,n}=4)$ and its end break points on the adjacent elements.
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图3.1：叶片子结构单元上气动力分布的二维示意图。叶片由两个子结构$b-1$和$b$组成，它们由具有端节点$(\bullet)$的同向旋转梁单元描述。叶片上的力分布是一个分段线性函数，以$N_{a}$个气动计算点$(\circ)$作为断点。子结构$b$的第$n$个单元上的力分布有四个内部断点$(N_{a,b,n}=4)$，其端点位于相邻单元上。
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where the coefficient vectors $\mathsf{f}_{1,b,n,m,r}$ and $\mathbf{m}_{1,b,n,m,r}$ for each integration interval $m\,=\,1,\dots,N_{a,b,n}\,-\,1$ and order $r=0,1$ of the linear function can be computed as
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其中，对于每个积分区间 $m\,=\,1,\dots,N_{a,b,n}\,-\,1$ 和线性函数阶次 $r=0,1$，系数向量 $\mathsf{f}_{1,b,n,m,r}$ 和 $\mathbf{m}_{1,b,n,m,r}$ 可以计算为
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$$
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\mathbf{f}_{1,b,n,m,0}=\frac{\mathbf{f}_{1,b,j_{b,n,m}}\zeta_{b,n,m+1}-\mathbf{f}_{1,b,j_{b,n,m+1}}\zeta_{b,n,m}}{\zeta_{b,n,m+1}-\zeta_{b,n,m}}\;\;\mathrm{and}\;\;\mathbf{f}_{1,b,n,m,1}=\frac{\mathbf{f}_{1,b,j_{b,n,m+1}}-\mathbf{f}_{1,b,j_{b,n,m}}}{\zeta_{b,n,m+1}-\zeta_{b,n,m}}
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\mathbf{f}_{1,b,n,m,0}=\frac{\mathbf{f}_{1,b,j_{b,n,m}}\zeta_{b,n,m+1}-\mathbf{f}_{1,b,j_{b,n,m+1}}\zeta_{b,n,m}}{\zeta_{b,n,m+1}-\zeta_{b,n,m}}\;\;\mathrm{and}\;\;\mathbf{f}_{1,b,n,m,1}=\frac{\mathbf{f}_{1,b,j_{b,n,m+1}}-\mathbf{f}_{1,b,j_{b,n,m}}}{\zeta_{b,n,m+1}-\zeta_{b,n,m}}\tag{3.8}
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$$
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and
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$$
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\begin{array}{r}{\mathfrak{m}_{1,b,n,m,0}=\frac{\mathfrak{m}_{1,b,j_{b,n,m}}\zeta_{b,n,m+1}-\mathfrak{m}_{1,b,j_{b,n,m+1}}\zeta_{b,n,m}}{\zeta_{b,n,m+1}-\zeta_{b,n,m}}\mathrm{~and~}\,\mathfrak{m}_{1,b,n,m,1}=\frac{\mathfrak{m}_{1,b,j_{b,n,m+1}}-\mathfrak{m}_{1,b,j_{b,n,m}}}{\zeta_{b,n,m+1}-\zeta_{b,n,m}}}\end{array}
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\begin{array}{r}{\mathfrak{m}_{1,b,n,m,0}=\frac{\mathfrak{m}_{1,b,j_{b,n,m}}\zeta_{b,n,m+1}-\mathfrak{m}_{1,b,j_{b,n,m+1}}\zeta_{b,n,m}}{\zeta_{b,n,m+1}-\zeta_{b,n,m}}\mathrm{~and~}\,\mathfrak{m}_{1,b,n,m,1}=\frac{\mathfrak{m}_{1,b,j_{b,n,m+1}}-\mathfrak{m}_{1,b,j_{b,n,m}}}{\zeta_{b,n,m+1}-\zeta_{b,n,m}}}\end{array}\tag{3.9}
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$$
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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
|
||||
$$
|
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\begin{array}{r l}&{\mathbf{f}_{1,b,n,m,r}=w_{b,n,m,r}\mathbf{f}_{1,b,j_{b,n,m}}+w_{b,n,m+1,r}\mathbf{f}_{1,b,j_{b,n,m+1}}}\\ &{\mathbf{m}_{1,b,n,m,r}=w_{b,n,m,r}\mathbf{m}_{1,b,j_{b,n,m}}+w_{b,n,m+1,r}\mathbf{m}_{1,b,j_{b,n,m+1}}}\end{array}
|
||||
$$
|
||||
|
||||
@ -5,7 +5,7 @@
|
||||
{"id":"52c483d4870680c3","type":"text","text":"# 推进计划\nRag系统、agent系统调研\n新闻 公众号\n小说推广系统 可能容易实现一些\n","x":-490,"y":-186,"width":456,"height":347},
|
||||
{"id":"0b25ceb1c28f6da1","type":"text","text":"# 六月已完成\n\nP1 海龟系统测试\n- 代理测试，5分钟间隔，全天监控成功1次\n- 代理池增加，但是没用\n- yiy.one.config会变 检测是否变化，重新修改以及重新加载 done\n- 邮件服务器连接使用代理 done\n","x":-482,"y":240,"width":440,"height":340},
|
||||
{"id":"b79f3a0f35402ec1","type":"text","text":"# 七月已完成\n\nP1 短剧使用AI生成解说视频功能跑通\n\nP1 短剧使用AI生成解说视频\n- 提示词更新，流程缩减 done\n- 考虑增加解说词长度 完成\n\nP1 交易系统 pass 也要增加冷却时间 done\n\nP1 公众号系统工作流 done\n- 每天的文章改名字\n- 控制一批处理新闻的数量加个loop，加一个wait\n- 加个ai控制主题别重复\n- 做一个封面图，以后就用一个\n\n","x":34,"y":240,"width":440,"height":340},
|
||||
{"id":"5aac58c184e57887","type":"text","text":"# 计划\n这周要做的3~5件重要的事情，这些事情能有效推进实现OKR。\n\nP1 必须做。P2 应该做\n\n\nP1 suno youtube视频\n- 生成音频\n- 创建场景形象 - 听音乐观海\n- logo形象 done\n- 名字\n- 横幅\n- 动画\n\nP1 海龟交易法 选股部分\n\n\n","x":-490,"y":-573,"width":450,"height":347}
|
||||
{"id":"5aac58c184e57887","type":"text","text":"# 计划\n这周要做的3~5件重要的事情，这些事情能有效推进实现OKR。\n\nP1 必须做。P2 应该做\n\n\nP1 suno youtube视频\n- 生成音频\n- 创建场景形象 - 听音乐观海 done\n- logo形象 done\n- 修复手\n- 名字\n- 横幅\n- 动画\n\nP1 海龟交易法 \n- position size设置etf最小1000，stock最小100\n- 选股部分\n\n\n","x":-490,"y":-573,"width":450,"height":347}
|
||||
],
|
||||
"edges":[]
|
||||
}
|
||||
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Reference in New Issue
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