From 870b0990e40553b3d6c1a3ccae4b36fd4712fe29 Mon Sep 17 00:00:00 2001 From: yize Date: Wed, 13 Aug 2025 09:28:08 +0800 Subject: [PATCH] vault backup: 2025-08-13 09:28:08 --- .obsidian/plugins/copilot/data.json | 2 +- .../auto/CASEstab_theory_manual.md | 86 ++++++++++++------- 2 files changed, 56 insertions(+), 32 deletions(-) diff --git a/.obsidian/plugins/copilot/data.json b/.obsidian/plugins/copilot/data.json index 333e304..6b239b5 100644 --- a/.obsidian/plugins/copilot/data.json +++ b/.obsidian/plugins/copilot/data.json @@ -302,7 +302,7 @@ "name": "Translate to Chinese", "prompt": "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 \n \n\n{copilot-selection}\n\n1. 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，turbine、wind turbine翻译为机组、风电机组，span翻译为展向，deflection翻译为变形，mode翻译为模态，normal mode翻译为简正模态，jacket 翻译为导管架，superelement翻译为超单元，shaft翻译为主轴，azimuth、azimuth angle翻译为方位角，neutral axes 翻译为中性轴\n2. Return only the translated text.\n", "showInContextMenu": true, - "modelKey": "phi4:latest|ollama" + "modelKey": "gemini-2.5-flash|google" }, { "name": "Summarize", 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 0905084..2b9a9d0 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 @@ -510,7 +510,7 @@ For the implementation of the above inertia force components, each substructure 为了实现上述惯性力分量，每个子结构对象必须提供一个函数，计算方程（1.22）、（1.29）、（1.31）、（1.33）和（1.35）中蓝色标记的所有标量、矢量和矩阵。其中一些具有物理意义，例如子结构的总质量 $M$、当前的重心 $\boldsymbol{\mathsf{r}}_{c g}$ 及其对一阶和二阶自由度导数 $(\mathsf{r}_{c g,q_{i}}\ a n d\ r_{c g,q_{i}q_{j}})$，以及由（1.23）给出的当前转动惯量分量矩阵 $I_{b a s e}$。其余标量可以简化为子结构上的两个独特的体积积分，它们是局部质量矩阵和非线性陀螺矩阵的条目： $$ -m_{i j}^{11}=\int_{\mathcal{V}}\rho\,\mathbf{r}_{1,q_{i}}^{T}\mathbf{r}_{1,q_{j}}d\mathcal{V}\;\;\mathbf{a}\mathsf{n}\mathsf{d}\;\;h_{i j k}^{111}=\int_{\mathcal{V}}\rho\,\mathbf{r}_{1,q_{i}}^{T}\mathbf{r}_{1,q_{j}q_{k}}d\mathcal{V} +m_{i j}^{11}=\int_{\mathcal{V}}\rho\,\mathbf{r}_{1,q_{i}}^{T}\mathbf{r}_{1,q_{j}}d\mathcal{V}\;\;\mathbf{a}\mathsf{n}\mathsf{d}\;\;h_{i j k}^{111}=\int_{\mathcal{V}}\rho\,\mathbf{r}_{1,q_{i}}^{T}\mathbf{r}_{1,q_{j}q_{k}}d\mathcal{V}\tag{1.36} $$ for all $i,j,k\,\in\,{\bf d}_{b}$ where ${\mathfrak{d}}_{b}$ is the DOF index vector for the substructure $b$ . The remaining matrices can be reduced to these three unique volume integrals over the substructure: @@ -518,7 +518,7 @@ for all $i,j,k\,\in\,{\bf d}_{b}$ where ${\mathfrak{d}}_{b}$ is the DOF index ve 对于所有 $i,j,k\,\in\,{\bf d}_{b}$，其中 ${\mathfrak{d}}_{b}$ 是子结构 $b$ 的自由度 (DOF) 索引向量。剩余的矩阵可以简化为子结构上的这三个独特的体积积分： $$ -\mathsf{A}_{b a s e,i}\equiv\int_{\mathcal{V}}\mathsf{r}_{1,q_{i}}\mathsf{r}_{1}^{T}d\mathcal{V}\;,\;\;\mathsf{A}_{b a s e,1,i j}\equiv\int_{\mathcal{V}}\rho\,\mathsf{r}_{1,q_{j}}\mathsf{r}_{1,q_{i}}^{T}d\mathcal{V}\;\;\mathrm{and}\;\;\mathsf{A}_{b a s e,2,i j}\equiv\int_{\mathcal{V}}\rho\,\mathsf{r}_{1,q_{i},q_{j}}\mathsf{r}_{1}^{T}d\mathcal{V} +\mathsf{A}_{b a s e,i}\equiv\int_{\mathcal{V}}\mathsf{r}_{1,q_{i}}\mathsf{r}_{1}^{T}d\mathcal{V}\;,\;\;\mathsf{A}_{b a s e,1,i j}\equiv\int_{\mathcal{V}}\rho\,\mathsf{r}_{1,q_{j}}\mathsf{r}_{1,q_{i}}^{T}d\mathcal{V}\;\;\mathrm{and}\;\;\mathsf{A}_{b a s e,2,i j}\equiv\int_{\mathcal{V}}\rho\,\mathsf{r}_{1,q_{i},q_{j}}\mathsf{r}_{1}^{T}d\mathcal{V}\tag{1.37} $$ for all $i,j\in{\bf d}_{b}$ . Note that the following symmetry rules apply: ${\pmb{\mathsf{A}}}_{b a s e,2,j i}={\pmb{\mathsf{A}}}_{b a s e,2,i j}$ and $\mathbf{A}_{b a s e,1,j i}=\mathbf{A}_{b a s e,1,i j}^{T}$ @@ -563,7 +563,7 @@ The following sections contain The position vector of particles on the rigid body is purely a function of the local spacial coordinates 刚体上质点的位置矢量纯粹是局部空间坐标的函数 $$ -\mathbf{r}_{1,b}=\left\{\begin{array}{l}{x}\\ {y}\\ {z}\end{array}\right\} +\mathbf{r}_{1,b}=\left\{\begin{array}{l}{x}\\ {y}\\ {z}\end{array}\right\}\tag{1.38} $$ where $(x,y,z)\in\mathcal{V}$ @@ -574,93 +574,117 @@ The nonlinear co­rotational finite beam element methodology used for this type 用于此类子结构的非线性同向旋转有限梁单元方法在附录A中详细描述。该方法与Crisfield和Krenk的方法类似，不同之处在于非线性几何公式是显式的，即单元坐标系和节点的局部小旋转被表示为全局节点自由度的显式非线性函数。单元的弹性变形和柔度则改编自Krenk和Couturier提出的平衡单元。 A co­rotational beam substructure consists of finite beam elements with two end-­nodes having six DOFs: three translations and three rotations (using Rodrigues parameters) in the substructure coordinate system. The DOF vector of the substructure 一种共转动梁子结构由有两个端节点的有限梁单元组成，在子结构坐标系中，每个端节点具有六个自由度（DOFs）：三个平移和三个旋转（使用罗德里格斯参数）。该子结构的自由度向量如下： -# Inertia forces +### Inertia forces The volume integral over a co­rotational beam substructure can be computed as the sum of integrals over the volume $\mathcal{V}_{n}$ of element $n$ , here generically written as +共旋梁子结构上的体积积分可以计算为单元 $n$ 的体积 $\mathcal{V}_{n}$ 上的积分之和，这里通常写为 $$ -\int_{\mathcal{V}}\left(\mathbf{\Phi}\right)d\mathcal{V}=\sum_{n=1}^{N_{e,b}}\int_{\mathcal{V}_{n}}\left(\mathbf{\Phi}\right)d\mathcal{V}_{n}=\sum_{n=1}^{N_{e,b}}\left(\frac{l_{n}}{2}\int_{-1}^{1}\int_{\mathcal{A}}\left(\mathbf{\Phi}\right)d\mathcal{A}d\zeta\right) +\int_{\mathcal{V}}\left(\right)d\mathcal{V}=\sum_{n=1}^{N_{e,b}}\int_{\mathcal{V}_{n}}\left(\right)d\mathcal{V}_{n}=\sum_{n=1}^{N_{e,b}}\left(\frac{l_{n}}{2}\int_{-1}^{1}\int_{\mathcal{A}}\left(\right)d\mathcal{A}d\zeta\right)\tag{1.39} $$ where each volume integral over the element of initial length $l_{n}$ are split into an area integral over each cross­ section and a line integral over the non­dimensional element coordinate $\zeta$ from the mid­point to the end nodes at $\zeta=\pm1$ . The position vector of particles on element number $n$ is +其中，初始长度为$l_{n}$的单元上的每个体积积分都被分解为每个横截面上的面积积分，以及从单元中点到$\zeta=\pm1$处端节点的无量纲单元坐标$\zeta$上的线积分。 +单元n上质点的位置向量为 $$ -\mathbf{r}_{1,b,n}=\mathbf{r}_{\mathrm{mid},b,n}\left(\mathbf{q}_{b,n}\right)+\mathbf{E}_{b,n}\left(\mathbf{q}_{b,n}\right)\mathbf{v}_{b,n}\left(\mathbf{q}_{b,n};x,y,\zeta\right) +\mathbf{r}_{1,b,n}=\mathbf{r}_{\mathrm{mid},b,n}\left(\mathbf{q}_{b,n}\right)+\mathbf{E}_{b,n}\left(\mathbf{q}_{b,n}\right)\mathbf{v}_{b,n}\left(\mathbf{q}_{b,n};x,y,\zeta\right)\tag{1.40} $$ -where ${\boldsymbol{\mathsf{r}}}_{{\boldsymbol{\mathsf{m i d}}},{\boldsymbol{b}},n}$ is a vector from the substructure base to the mid­point of the element linearly dependent on the displacement DOFs of element nodes, $\mathsf{E}_{b,n}$ is the element coordinate system dependent on all twelve nodal DOFs $\mathbf{q}_{b,n}$ of element $n$ , and the cross­sectional displacement vector is given by +where ${\boldsymbol{\mathsf{r}}}_{{\boldsymbol{\mathsf{m i d}}},{\boldsymbol{b}},n}$ is a vector from the substructure base to the mid­point of the element linearly dependent on the displacement DOFs of element nodes, $\mathsf{E}_{b,n}$ is the element coordinate system dependent on all twelve nodal DOFs $\mathbf{q}_{b,n}$ of element $n$ , and the cross-­sectional displacement vector is given by +其中， ${\boldsymbol{\mathsf{r}}}_{{\boldsymbol{\mathsf{m i d}}},{\boldsymbol{b}},n}$ 是从子结构基座到单元中点的向量，其线性依赖于单元节点的位移自由度，$\mathsf{E}_{b,n}$ 是依赖于单元 $n$ 的所有十二个节点自由度 $\mathbf{q}_{b,n}$ 的单元坐标系，横截面位移向量由下式给出 $$ -\begin{array}{r}{\mathbf{v}_{b,n}=\left\{\begin{array}{c}{x}\\ {y}\\ {\frac{1}{2}l_{n}\zeta}\end{array}\right\}+\left\{\begin{array}{c}{u_{x,n}(\zeta)}\\ {u_{y,n}(\zeta)}\\ {u_{z,n}(\zeta)}\end{array}\right\}+\left[\begin{array}{c c c}{0}&{-\theta_{z,n}(\zeta)}&{\theta_{y,n}(\zeta)}\\ {\theta_{z,n}(\zeta)}&{0}&{-\theta_{x,n}(\zeta)}\\ {-\theta_{y,n}(\zeta)}&{\theta_{x,n}(\zeta)}&{0}\end{array}\right]\left\{\begin{array}{c}{x}\\ {y}\\ {0}\end{array}\right\}}\end{array} +\begin{array}{r}{\mathbf{v}_{b,n}=\left\{\begin{array}{c}{x}\\ {y}\\ {\frac{1}{2}l_{n}\zeta}\end{array}\right\}+\left\{\begin{array}{c}{u_{x,n}(\zeta)}\\ {u_{y,n}(\zeta)}\\ {u_{z,n}(\zeta)}\end{array}\right\}+\left[\begin{array}{c c c}{0}&{-\theta_{z,n}(\zeta)}&{\theta_{y,n}(\zeta)}\\ {\theta_{z,n}(\zeta)}&{0}&{-\theta_{x,n}(\zeta)}\\ {-\theta_{y,n}(\zeta)}&{\theta_{x,n}(\zeta)}&{0}\end{array}\right]\left\{\begin{array}{c}{x}\\ {y}\\ {0}\end{array}\right\}}\end{array}\tag{1.41} $$ -where $x,y$ are the coordinates of the cross­section and the cross­sectional translations and (small) rotations are given by the shape function polynomials - +where $x,y$ are the coordinates of the cross-­section and the cross-­sectional translations and (small) rotations are given by the shape function polynomials +其中 $x,y$ 是截面坐标，截面平移和（小）转动由形函数多项式给出 $$ -\left\{\begin{array}{l}{u_{x,n}}\\ {u_{y,n}}\\ {u_{z,n}}\\ {\theta_{x,n}}\\ {\theta_{y,n}}\\ {\theta_{z,n}}\end{array}\right\}=\sum_{p=0}^{P+3}\mathbf{N}_{n,p}\,\mathbf{g}\left(\mathbf{q}_{b,n}\right)\zeta^{p}=\sum_{p=0}^{P+3}\left[\begin{array}{l}{\bar{\mathbf{N}}_{n,p}}\\ {\tilde{\mathbf{N}}_{n,p}}\end{array}\right]\mathbf{g}\left(\mathbf{q}_{b,n}\right)\zeta^{p} +\left\{\begin{array}{l}{u_{x,n}}\\ {u_{y,n}}\\ {u_{z,n}}\\ {\theta_{x,n}}\\ {\theta_{y,n}}\\ {\theta_{z,n}}\end{array}\right\}=\sum_{p=0}^{P+3}\mathbf{N}_{n,p}\,\mathbf{g}\left(\mathbf{q}_{b,n}\right)\zeta^{p}=\sum_{p=0}^{P+3}\left[\begin{array}{l}{\bar{\mathbf{N}}_{n,p}}\\ {\tilde{\mathbf{N}}_{n,p}}\end{array}\right]\mathbf{g}\left(\mathbf{q}_{b,n}\right)\zeta^{p}\tag{1.42} $$ -where $\aleph_{n,p}$ are the $6\uptimes7$ coefficient matrices of the local shape functions, $P$ is the polynomial order of the element properties (e.g. for prismatic elements $P=0$ ), and $\mathfrak{g}$ is the nonlinear $7\times1$ vector function of the twelve nodal DOFs. In the following, we split the local shape function coefficient matrix into two, one $3\mathrm{x7}$ matrix for the displacements $\bar{\aleph}_{n,p}$ and another 3x7 matrix for the rotations $\tilde{\mathsf{N}}_{n,p}$ . +where $\mathsf{N}_{n,p}$ are the $6\times7$ coefficient matrices of the local shape functions, $P$ is the polynomial order of the element properties (e.g. for prismatic elements $P=0$ ), and $\mathfrak{g}$ is the nonlinear $7\times1$ vector function of the twelve nodal DOFs. In the following, we split the local shape function coefficient matrix into two, one $3\times7$ matrix for the displacements $\bar{\mathsf{N}}_{n,p}$ and another $3\times7$ matrix for the rotations $\tilde{\mathsf{N}}_{n,p}$ . -This form of the cross­sectional displacement vector is inconvenient for the isolation of the spacial variables $x,y,\zeta$ . We therefore rewrite it as +This form of the cross-­sectional displacement vector is inconvenient for the isolation of the spacial variables $x,y,\zeta$ . We therefore rewrite it as +其中 $\mathsf{N}_{n,p}$ 是局部形状函数的 $6\times7$ 系数矩阵，$P$ 是单元属性的多项式阶数（例如，对于棱柱单元 $P=0$），${g}$ 是十二个节点自由度的非线性 $7\times1$ 向量函数。在下文中，我们将局部形状函数系数矩阵分为两个，一个用于位移的 $3\times7$ 矩阵 $\bar{\mathsf{N}}_{n,p}$ 和另一个用于旋转的 $3\times7$ 矩阵 $\tilde{\mathsf{N}}_{n,p}$。 + +这种形式的横截面位移向量不便于分离空间变量 $x,y,\zeta$。因此，我们将其改写为 $$ -\begin{array}{r}{\mathbf{v}_{b,n}=\left\{\begin{array}{c}{x}\\ {y}\\ {\frac{1}{2}l_{n}\zeta}\end{array}\right\}+\displaystyle\sum_{p=0}^{P+3}\left\{\begin{array}{c}{u_{x,n,p}}\\ {u_{y,n,p}}\\ {u_{z,n,p}}\end{array}\right\}\zeta^{p}+\displaystyle\sum_{p=0}^{P+3}\left(\left\{\begin{array}{c}{0}\\ {\theta_{z,n,p}}\\ {-\theta_{y,n,p}}\end{array}\right\}x+\left\{\begin{array}{c}{-\theta_{z,n,p}}\\ {0}\\ {\theta_{x,n,p}}\end{array}\right\}y\right)\zeta^{p}}\end{array} +\begin{array}{r}{\mathbf{V}_{b,n}=\left\{\begin{array}{c}{x}\\ {y}\\ {\frac{1}{2}l_{n}\zeta}\end{array}\right\}+\displaystyle\sum_{p=0}^{P+3}\left\{\begin{array}{c}{u_{x,n,p}}\\ {u_{y,n,p}}\\ {u_{z,n,p}}\end{array}\right\}\zeta^{p}+\displaystyle\sum_{p=0}^{P+3}\left(\left\{\begin{array}{c}{0}\\ {\theta_{z,n,p}}\\ {-\theta_{y,n,p}}\end{array}\right\}x+\left\{\begin{array}{c}{-\theta_{z,n,p}}\\ {0}\\ {\theta_{x,n,p}}\end{array}\right\}y\right)\zeta^{p}}\end{array}\tag{1.43} $$ -where the polynomial coefficients of the cross­sectional displacements and rotations are given by - +where the polynomial coefficients of the cross­-sectional displacements and rotations are given by +其中截面位移和转角的多项式系数由下式给出 $$ -\left\{\begin{array}{c}{u_{x,n,p}}\\ {u_{y,n,p}}\\ {u_{z,n,p}}\end{array}\right\}=\bar{\aleph}_{n,p}\,\mathbf{g}\left(\mathbf{q}_{b,n}\right)\quad\mathsf{a n d}\quad\left\{\begin{array}{c}{\theta_{x,n,p}}\\ {\theta_{y,n,p}}\\ {\theta_{z,n,p}}\end{array}\right\}=\tilde{\aleph}_{n,p}\mathbf{g}\left(\mathbf{q}_{b,n}\right) +\left\{\begin{array}{c}{u_{x,n,p}}\\ {u_{y,n,p}}\\ {u_{z,n,p}}\end{array}\right\}=\bar{\mathsf{N}}_{n,p}\,\mathbf{g}\left(\mathbf{q}_{b,n}\right)\quad\mathsf{a n d}\quad\left\{\begin{array}{c}{\theta_{x,n,p}}\\ {\theta_{y,n,p}}\\ {\theta_{z,n,p}}\end{array}\right\}=\tilde{\mathsf{N}}_{n,p}\mathbf{g}\left(\mathbf{q}_{b,n}\right)\tag{1.44} $$ that depend on the nodal DOFs of the element. - +取决于单元的节点自由度。 Using (1.43) with (1.42), the position vector for element $n$ on substructure $b$ (1.40) can be rewritten as $$ -\mathbf{r}_{1,b,n}=\sum_{p=0}^{P+3}\left(\mathbf{r}_{o,b,n,p}+x\,\mathbf{r}_{x,b,n,p}+y\,\mathbf{r}_{y,b,n,p}\right)\zeta^{p} +\mathbf{r}_{1,b,n}=\sum_{p=0}^{P+3}\left(\mathbf{r}_{o,b,n,p}+x\,\mathbf{r}_{x,b,n,p}+y\,\mathbf{r}_{y,b,n,p}\right)\zeta^{p}\tag{1.45} $$ -where the sub­vectors are +where the sub-­vectors are $$ -\begin{array}{r l}&{\mathbf{r}_{o,b,n,p}\left(\mathbf{q}_{b,n}\right)=\pmb{\Sigma}_{b,n}\left(\mathbf{q}_{b,n}\right)\,\bar{\mathbf{N}}_{n,p}\,\mathbf{g}\left(\mathbf{q}_{b,n}\right)+\delta_{0p}\,\mathbf{r}_{\mathrm{mid},b,n}\left(\mathbf{q}_{b,n}\right)+\delta_{1p}\,\frac{l_{n}}{2}\mathbf{e}_{3,b,n}\left(\mathbf{q}_{b,n}\right)}\\ &{\mathbf{r}_{x,b,n,p}\left(\mathbf{q}_{b,n}\right)=\pmb{\Sigma}_{b,n}\left(\mathbf{q}_{b,n}\right)\,\mathbf{P}_{x}\,\tilde{\mathbf{N}}_{n,p}\,\mathbf{g}\left(\mathbf{q}_{b,n}\right)+\delta_{0p}\,\mathbf{e}_{1,b,n}\left(\mathbf{q}_{b,n}\right)}\\ &{\mathbf{r}_{y,b,n,p}\left(\mathbf{q}_{b,n}\right)=\pmb{\Sigma}_{b,n}\left(\mathbf{q}_{b,n}\right)\,\mathbf{P}_{y}\,\tilde{\mathbf{N}}_{n,p}\,\mathbf{g}\left(\mathbf{q}_{b,n}\right)+\delta_{0p}\,\mathbf{e}_{2,b,n}\left(\mathbf{q}_{b,n}\right)}\end{array} -$$ +{\mathbf{r}_{o,b,n,p}\left(\mathbf{q}_{b,n}\right)={\mathbf{E}}_{b,n}\left(\mathbf{q}_{b,n}\right)\,\bar{\mathbf{N}}_{n,p}\,\mathbf{g}\left(\mathbf{q}_{b,n}\right)+\delta_{0p}\,\mathbf{r}_{\mathrm{mid},b,n}\left(\mathbf{q}_{b,n}\right)+\delta_{1p}\,\frac{l_{n}}{2}\mathbf{e}_{3,b,n}\left(\mathbf{q}_{b,n}\right)}\tag{1.46a} +$$ + +$$ +{\mathbf{r}_{x,b,n,p}\left(\mathbf{q}_{b,n}\right)=\mathbf{E}_{b,n}\left(\mathbf{q}_{b,n}\right)\,\mathbf{P}_{x}\,\tilde{\mathbf{N}}_{n,p}\,\mathbf{g}\left(\mathbf{q}_{b,n}\right)+\delta_{0p}\,\mathbf{e}_{1,b,n}\left(\mathbf{q}_{b,n}\right)}\tag{1.46b} +$$ + +$$ +{\mathbf{r}_{y,b,n,p}\left(\mathbf{q}_{b,n}\right)=\mathbf{E}_{b,n}\left(\mathbf{q}_{b,n}\right)\,\mathbf{P}_{y}\,\tilde{\mathbf{N}}_{n,p}\,\mathbf{g}\left(\mathbf{q}_{b,n}\right)+\delta_{0p}\,\mathbf{e}_{2,b,n}\left(\mathbf{q}_{b,n}\right)}\tag{1.46c} +$$ + where $\delta_{i j}$ is Kronecker’s delta which is 1 for $i=j$ , and otherwise 0. Matrices $\mathsf{P}_{x}$ and $\mathsf{P}_{y}$ are constant permutation matrices: - +其中 $\delta_{i j}$ 是克罗内克函数，当 $i=j$ 时为1，否则为0。矩阵 $\mathsf{P}_{x}$ 和 $\mathsf{P}_{y}$ 是常数置换矩阵： $$ -\begin{array}{r}{\mathsf{P}_{x}=\left[\begin{array}{c c c}{0}&{0}&{0}\\ {0}&{0}&{1}\\ {0}&{-1}&{0}\end{array}\right]\quad\mathsf{a n d}\quad\mathsf{P}_{y}=\left[\begin{array}{c c c}{0}&{0}&{-1}\\ {0}&{0}&{0}\\ {1}&{0}&{0}\end{array}\right]}\end{array} +\begin{array}{r}{\mathsf{P}_{x}=\left[\begin{array}{c c c}{0}&{0}&{0}\\ {0}&{0}&{1}\\ {0}&{-1}&{0}\end{array}\right]\quad\mathsf{a n d}\quad\mathsf{P}_{y}=\left[\begin{array}{c c c}{0}&{0}&{-1}\\ {0}&{0}&{0}\\ {1}&{0}&{0}\end{array}\right]}\end{array}\tag{1.47} $$ -Note that $\mathsf{P}_{x}$ and $\mathsf{P}_{y}$ are equal to the matrix $\boldsymbol{\mathsf{N}}$ in Eq. (E.3) related to the angular derivative of a rotation matrix with a constant unit­vectors in the $x$ ­ and $y$ ­directions, respectively. +Note that $\mathsf{P}_{x}$ and $\mathsf{P}_{y}$ are equal to the matrix $\boldsymbol{\mathsf{N}}$ in Eq. (E.3) related to the angular derivative of a rotation matrix with a constant unit-­vectors in the $x$ ­ and $y$ ­directions, respectively. -In the following, the element­index $n$ and the substructure­index $b$ are omitted for brevity. +In the following, the element-­index $n$ and the substructure-­index $b$ are omitted for brevity. -Integration over the cross­sectional area spanned by $x$ and $y$ of the element coordinate system are defined by the mass per unit­length $m$ , the center of gravity coordinates $x_{c g}$ and $y_{c g}$ , and the moments of inertia about the $x$ ­ and $y$ ­axes and their cross­coupling. These cross­sectional properties are given by polynomials to the order of $P$ along the element lengthwise coordinate $\zeta\in[-1,1]$ as +Integration over the cross­-sectional area spanned by $x$ and $y$ of the element coordinate system are defined by the mass per unit­­-length $m$ , the center of gravity coordinates $x_{c g}$ and $y_{c g}$ , and the moments of inertia about the $x$ ­ and $y$ ­axes and their cross­­-coupling. These cross­­-sectional properties are given by polynomials to the order of $P$ along the element lengthwise coordinate $\zeta\in[-1,1]$ as +请注意，$\mathsf{P}_{x}$ 和 $\mathsf{P}_{y}$ 等于方程 (E.3) 中的矩阵 $\boldsymbol{\mathsf{N}}$，该矩阵分别与旋转矩阵在 $x$ 和 $y$ 方向上具有常量单位向量的角导数相关。 + +在下文中，为了简洁，省略了单元索引 $n$ 和子结构索引 $b$。 + +单元坐标系中由 $x$ 和 $y$ 构成的截面面积上的积分由单位长度质量 $m$、重心坐标 $x_{c g}$ 和 $y_{c g}$、以及绕 $x$ 轴和 $y$ 轴的转动惯量及其交叉耦合定义。这些截面特性由沿着单元长度坐标 $\zeta\in[-1,1]$ 的 $P$ 阶多项式给出，如下所示： $$ -\begin{array}{l}{\displaystyle\int_{A}\rho\,d A=\sum_{r=0}^{P}a_{m,r}\,\zeta^{r}\;,\;\;\int_{A}\rho x\,d A=\sum_{r=0}^{P}a_{m x_{c g},r}\,\zeta^{r}\;,\;\;\int_{A}\rho y\,d A=\sum_{r=0}^{P}a_{m y_{c g},r}\,\zeta^{r}\;,}\\ {\displaystyle\int_{A}\rho x^{2}\,d A=\sum_{r=0}^{P}a_{I_{x x},r}\,\zeta^{r}\;,\;\;\displaystyle\int_{A}\rho y^{2}\,d A=\sum_{r=0}^{P}a_{I_{y y},r}\,\zeta^{r}\;,\;\;\displaystyle\int_{A}\rho x y\,d A=\sum_{r=0}^{P}a_{I_{x y},r}\,\zeta^{r}}\end{array} +\begin{array}{l}{\displaystyle\int_{A}\rho\,d A=\sum_{r=0}^{P}a_{m,r}\,\zeta^{r}\;,\;\;\int_{A}\rho x\,d A=\sum_{r=0}^{P}a_{m x_{c g},r}\,\zeta^{r}\;,\;\;\int_{A}\rho y\,d A=\sum_{r=0}^{P}a_{m y_{c g},r}\,\zeta^{r}\;,}\\ {\displaystyle\int_{A}\rho x^{2}\,d A=\sum_{r=0}^{P}a_{I_{x x},r}\,\zeta^{r}\;,\;\;\displaystyle\int_{A}\rho y^{2}\,d A=\sum_{r=0}^{P}a_{I_{y y},r}\,\zeta^{r}\;,\;\;\displaystyle\int_{A}\rho x y\,d A=\sum_{r=0}^{P}a_{I_{x y},r}\,\zeta^{r}}\end{array}\tag{1.48} $$ -where the polynomial coefficients are generated from the model input in a pre­simulation processing step. The coefficients $a_{m x_{c g},r}$ and $a_{m y_{c g},r}$ represent a polynomial fit to the product of the mass per unit­length $m$ and the individual center of gravity coordinates $x_{c g}$ and $y_{c g}$ in the element coordinate system. +where the polynomial coefficients are generated from the model input in a pre­-simulation processing step. The coefficients $a_{m x_{c g},r}$ and $a_{m y_{c g},r}$ represent a polynomial fit to the product of the mass per unit-­length $m$ and the individual center of gravity coordinates $x_{c g}$ and $y_{c g}$ in the element coordinate system. Integration over an element with the local position vector (1.45) and using (1.48), the mass of the element can be computed as +其中多项式系数是在预仿真处理步骤中从模型输入生成的。系数 $a_{m x_{c g},r}$ 和 $a_{m y_{c g},r}$ 表示对单位长度质量 $m$ 与单元坐标系中单个重心坐标 $x_{c g}$ 和 $y_{c g}$ 乘积的多项式拟合。 +对具有局部位置矢量 (1.45) 的单元进行积分并使用 (1.48)，可以计算出单元的质量为 $$ -M=\frac{l}{2}\sum_{r=0}^{P}c(r)a_{m,r} +M=\frac{l}{2}\sum_{r=0}^{P}c(r)a_{m,r}\tag{1.49} $$ where the coefficient function is given by $$ -c\left(r\right)={\frac{\left(-1\right)^{r}+1}{1+r}}={\left\{\begin{array}{l l}{2/(1+r)}&{r\;\;{\mathsf{e v e n}}}\\ {\;\;\;\;0}&{r\;\;\;{\mathsf{o d d}}}\end{array}\right.} +c\left(r\right)={\frac{\left(-1\right)^{r}+1}{1+r}}={\left\{\begin{array}{l l}{2/(1+r)}&{r\;\;{\mathsf{e v e n}}}\\ {\;\;\;\;0}&{r\;\;\;{\mathsf{o d d}}}\end{array}\right.}\tag{1.50} $$ The element center of gravity position times the element mass and the nodal derivatives can be computed as +单元重心位置乘以单元质量和节点导数可以计算为 $$ \begin{array}{l l}{{\displaystyle M\,{\bf r}_{c g}=\frac{l}{2}\sum_{p=0}^{P+3}\left(\sum_{r=0}^{P}c(p+r)\left(a_{m,r}\,{\bf r}_{o,p}+a_{m x_{c g},r}\,{\bf r}_{x,p}+a_{m y_{c g},r}\,{\bf r}_{y,p}\right)\right)}}\\ {{\displaystyle M\,{\bf r}_{c g,q_{i}}=\frac{l}{2}\sum_{p=0}^{P+3}\left(\sum_{r=0}^{P}c(p+r)\left(a_{m,r}\,{\bf r}_{o,p,q_{i}}+a_{m x_{c g},r}\,{\bf r}_{x,p,q_{i}}+a_{m y_{c g},r}\,{\bf r}_{y,p,q_{i}}\right)\right)}}\\ {{\displaystyle M\,{\bf r}_{c g,q_{i},q_{j}}=\frac{l}{2}\sum_{p=0}^{P+3}\left(\sum_{r=0}^{P}c(p+r)\left(a_{m,r}\,{\bf r}_{o,p,q_{i},q_{j}}+a_{m x_{c g},r}\,{\bf r}_{x,p,q_{i},q_{j}}+a_{m y_{c g},r}\,{\bf r}_{y,p,q_{i},q_{j}}\right)\right)}}\end{array}