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.obsidian/plugins/copilot/data.json vendored
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@ -302,7 +302,7 @@
"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 \n </instruction>\n\n<text>{copilot-selection}</text>\n<restrictions>\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</restrictions>",
"showInContextMenu": true,
"modelKey": "gemini-2.5-flash|google"
"modelKey": "phi4:latest|ollama"
},
{
"name": "Summarize",

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书籍/力学书籍/CASEstab_theory_manual/auto/CASEstab_theory_manual.md
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@ -72,7 +72,7 @@ E.3 Miscellaneous formulas 39
This chapter contains the derivations of the nonlinear equations of motion based on the kinematic formulation of articulated substructures to form the entire turbine structure.
本章包含基于铰接子结构运动学公式的非线性运动方程的推导，以形成整个机组结构。
## 1.1 Lagrange equations of articulated substructures
## 1.1 Lagrange equations of articulated substructures铰接子结构的拉格朗日方程
The Lagrangian of a structure ${{L}}=T-V$ is given by its total kinetic energy $T$ and the total potential energy $V$ of the conservative forces acting on the structure, e.g. gravity and elastic forces. Using Lagrange’s equations, the nonlinear equations of motion can be derived as
结构的拉格朗日量 ${{L}}=T-V$ 由其总动能 $T$ 和作用在结构上的保守力（例如重力和弹力）的总势能 $V$ 给出。利用拉格朗日方程，可以推导出非线性运动方程，如下所示：
@ -397,7 +397,7 @@ $$
where $\begin{array}{r}{\mathsf{A}_{b a s e,1,i j}\;\equiv\;\int_{\mathcal{V}}\!\rho\,\mathsf{r}_{1,q_{j}}\mathsf{r}_{1,q_{i}}^{T}d\mathcal{V}}\end{array}$ is introduced for the volume integral over the substructure of the matrix defined by the first derivatives of the local deformation vector with respect to the local DOFs.
其中 $\begin{array}{r}{\mathsf{A}_{base,1,ij}\;\equiv\;\int_{\mathcal{V}}\!\rho\,\mathsf{r}_{1,q_{j}}\mathsf{r}_{1,q_{i}}^{T}d\mathcal{V}}\end{array}$ 是对由局部变形向量相对于局部自由度的第一导数定义的矩阵子结构的体积积分。
# Nonlinear gyroscopic matrix非线性陀螺矩阵
### Nonlinear gyroscopic matrix非线性陀螺矩阵
Inserting (1.12) into (1.8c), expanding and sorting the terms into the two $_{2\times2}$ block matrix forms (1.20) yield the nonlinear gyroscopic matrix elements
将(1.12)代入(1.8c)，展开并按两个$_{2\times2}$块矩阵形式(1.20)对各项进行排序，得到非线性陀螺矩阵元素
@ -455,10 +455,10 @@ h_{ijk}^{111} = \int_{\mathcal{V}} \rho \, \mathbf{r}_{1,q_i}^T \mathbf{r}_{1,q_
$$
where $\begin{array}{r}{\mathsf{A}_{b a s e,2,i j}\;\equiv\;\int_{\mathcal{V}}\rho\,\mathsf{r}_{1,q_{i},q_{j}}\,\mathsf{r}_{1}^{T}d\mathcal{V}}\end{array}$ is introduced for the volume integral over the substructure of the matrix defined by the local deformation vector and its second derivatives with respect to the local DOFs.
其中 $\begin{array}{r}{\mathsf{A}_{base,2,ij}\;\equiv\;\int_{\mathcal{V}}\rho\,\mathsf{r}_{1,q_{i},q_{j}}\,\mathsf{r}_{1}^{T}d\mathcal{V}}\end{array}$ 是对由局部变形向量及其对局部自由度的二阶导数定义的矩阵子结构上的体积积分。
# Centrifugal force vector
### Centrifugal force vector
Inserting (1.12) into (1.8d) and expanding lead to these acceleration (centrifugal) forces on the supporting DOFs and the local substructure DOFs:
将(1.12)代入(1.8d)并展开，得到作用在支撑自由度和局部子结构自由度上的加速度（离心）力：
$$
\begin{array}{l}{{\displaystyle F_{c,i}^{0}=\int_{\mathcal{V}}\rho\left({\bf r}_{0,q_{i}}^{T}\ddot{\bf r}_{0}+\ddot{\bf r}_{0}^{T}{\bf R}_{0,q_{i}}{\bf r}_{1}+{\bf r}_{0,q_{i}}^{T}\ddot{\bf R}_{0}{\bf r}_{1}+{\bf r}_{1}^{T}{\bf R}_{0,q_{i}}^{T}\ddot{\bf R}_{0}{\bf r}_{1}\right)d\mathcal{V}}}\\ {{\displaystyle F_{c,i}^{1}=\int_{\mathcal{V}}\rho\left(\ddot{\bf r}_{0}^{T}{\bf R}_{0}{\bf r}_{1,q_{i}}+{\bf r}_{1}^{T}\ddot{\bf R}_{0}^{T}{\bf R}_{0}{\bf r}_{1,q_{i}}\right)d\mathcal{V}}}\end{array}
$$
@ -466,15 +466,15 @@ $$
which can be rewritten as
$$
\begin{array}{r l}&{F_{c,i}^{0}=\!M\mathbf{r}_{0,q_{i}}^{T}\ddot{\mathbf{r}}_{0}+\left(\ddot{\mathbf{r}}_{0}^{T}\mathbf{R}_{0,q_{i}}+\mathbf{r}_{0,q_{i}}^{T}\ddot{\mathbf{R}}_{0}\right)M\mathbf{r}_{c g}+\left(\mathbf{R}_{0,q_{i}}^{T}\ddot{\mathbf{R}}_{0}\right):\mathbf{l}_{b a s e}}\\ &{F_{c,i}^{1}=\!\ddot{\mathbf{r}}_{0}^{T}\mathbf{R}_{0}M\mathbf{r}_{c g,q_{i}}+\left(\ddot{\mathbf{R}}_{0}^{T}\mathbf{R}_{0}\right):\mathsf{A}_{b a s e,i}}\end{array}
\begin{array}{r l}&{F_{c,i}^{0}=\!M\mathbf{r}_{0,q_{i}}^{T}\ddot{\mathbf{r}}_{0}+\left(\ddot{\mathbf{r}}_{0}^{T}\mathbf{R}_{0,q_{i}}+\mathbf{r}_{0,q_{i}}^{T}\ddot{\mathbf{R}}_{0}\right)M\mathbf{r}_{c g}+\left(\mathbf{R}_{0,q_{i}}^{T}\ddot{\mathbf{R}}_{0}\right):\mathbf{I}_{b a s e}}\\ &{F_{c,i}^{1}=\!\ddot{\mathbf{r}}_{0}^{T}\mathbf{R}_{0}M\mathbf{r}_{c g,q_{i}}+\left(\ddot{\mathbf{R}}_{0}^{T}\mathbf{R}_{0}\right):\mathsf{A}_{b a s e,i}}\end{array}
$$
using (E.7) and the volume integral notations introduced above.
# Centrifugal stiffness matrix
使用(E.7)和上述引入的体积分符号。
### Centrifugal stiffness matrix离心刚度矩阵
Inserting (1.12) into (1.9), expanding and sorting the terms into the $_{2\times2}$ block matrix form (1.19) yield the centrifugal stiffness matrix elements
将（1.12）代入（1.9），展开并按${2\times2}$块矩阵形式（1.19）对各项进行排序，得到离心刚度矩阵元素
$$
k_{c,ij}^{00} = \int_{\mathcal{V}} \rho \left( \ddot{\mathbf{r}}_{0,q_i}^T \mathbf{r}_{0,q_j} + \mathbf{r}_{0,q_i}^T \ddot{\mathbf{r}}_{0,q_j} + \ddot{\mathbf{r}}_{0,q_i,q_j}^T \mathbf{R}_{0} \mathbf{r}_{1} + \mathbf{r}_{1}^T \mathbf{R}_{0,q_i}^T \ddot{\mathbf{R}}_{0,q_j} \mathbf{r}_{1} + \mathbf{r}_{0,q_i}^T \ddot{\mathbf{R}}_{0,q_j} \mathbf{r}_{1} + \mathbf{r}_{0,q_j}^T \ddot{\mathbf{R}}_{0,q_i} \mathbf{r}_{1} + \mathbf{r}_{1}^T \mathbf{R}_{0,q_i}^T \ddot{\mathbf{R}}_{0} \mathbf{r}_{1} + \mathbf{r}_{1}^T \ddot{\mathbf{R}}_{0}^T \mathbf{R}_{0,q_i} \mathbf{r}_{1} \right) d \mathcal{V} \tag{1.34a}
$$
@ -503,9 +503,11 @@ $$
using (E.7) and the volume integral notations introduced above.
# Generic implementation of inertia forces from a substructure
### Generic implementation of inertia forces from a substructure 子结构惯性力的通用实现
For the implementation of the above inertia force components, each substructure object must provide a function computing all scalers, vectors and matrices marked in blue in Equations (1.22), (1.29), (1.31), (1.33) and (1.35). Some have physical meaning, e.g. the total mass of the substructure $M$ , the current center of gravity $\boldsymbol{\mathsf{r}}_{c g}$ and its DOF derivatives to the first and second order $(\mathsf{r}_{c g,q_{i}}\ a n d\ r_{c g,q_{i}q_{j}})$ , and the current matrix of rotational moments of inertia components $\mid_{b a s e}$ as given by (1.23). The remaining scalars can be reduced to these two unique volume integrals over the substructure that are the entries of the local mass and nonlinear gyroscopic matrices:
For the implementation of the above inertia force components, each substructure object must provide a function computing all scalers, vectors and matrices marked in blue in Equations (1.22), (1.29), (1.31), (1.33) and (1.35). Some have physical meaning, e.g. the total mass of the substructure $M$ , the current center of gravity $\boldsymbol{\mathsf{r}}_{c g}$ and its DOF derivatives to the first and second order $(\mathsf{r}_{c g,q_{i}}\ a n d\ r_{c g,q_{i}q_{j}})$ , and the current matrix of rotational moments of inertia components $I_{b a s e}$ as given by (1.23). The remaining scalars can be reduced to these two unique volume integrals over the substructure that are the entries of the local mass and nonlinear gyroscopic matrices:
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：
$$
m_{i j}^{11}=\int_{\mathcal{V}}\rho\,\mathbf{r}_{1,q_{i}}^{T}\mathbf{r}_{1,q_{j}}d\mathcal{V}\;\;\mathbf{a}\mathsf{n}\mathsf{d}\;\;h_{i j k}^{111}=\int_{\mathcal{V}}\rho\,\mathbf{r}_{1,q_{i}}^{T}\mathbf{r}_{1,q_{j}q_{k}}d\mathcal{V}
@ -513,6 +515,8 @@ $$
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:
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：
$$
\mathsf{A}_{b a s e,i}\equiv\int_{\mathcal{V}}\mathsf{r}_{1,q_{i}}\mathsf{r}_{1}^{T}d\mathcal{V}\;,\;\;\mathsf{A}_{b a s e,1,i j}\equiv\int_{\mathcal{V}}\rho\,\mathsf{r}_{1,q_{j}}\mathsf{r}_{1,q_{i}}^{T}d\mathcal{V}\;\;\mathrm{and}\;\;\mathsf{A}_{b a s e,2,i j}\equiv\int_{\mathcal{V}}\rho\,\mathsf{r}_{1,q_{i},q_{j}}\mathsf{r}_{1}^{T}d\mathcal{V}
$$
@ -520,49 +524,56 @@ $$
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}$
All scalars, vectors, and matrices for each substructure must be computed in every time step for all combinations of substructure DOFs. The combinations leads to a large number of computations, e.g. the nonlinear gyroscopic elements combines over three DOF indices, with symmetry for two of them, resulting in $\left(N_{b}^{3}+N_{b}^{2}\right)/2$ scalar values (where $N_{b}$ is the number of DOFs in structure $b$ ) to be computed for the nonlinear gyroscopic matrix. The computations of (1.23), (1.36), (1.37), and the derivatives of $\pmb{r}_{c g}$ in the object function of the substructure must therefore be optimized for speed.
每个子结构的全部标量、向量和矩阵都必须在每个时间步中针对子结构自由度的所有组合进行计算。这些组合导致大量的计算，例如，非线性陀螺元件组合了三个自由度指标，其中两个具有对称性，从而导致非线性陀螺矩阵需要计算 $\left(N_{b}^{3}+N_{b}^{2}\right)/2$ 个标量值（其中 $N_{b}$ 是结构 $b$ 中的自由度数量）。因此，子结构目标函数中 (1.23)、(1.36)、(1.37) 的计算以及 $\pmb{r}_{c g}$ 的导数必须进行速度优化。
Each substructure object must also provide a function computing its contributions to the positions and orienta­ tions of any connection points to subsequent substructures.
Each substructure object must also provide a function computing its contributions to the positions and orienta­tions of any connection points to subsequent substructures.
每个子结构对象还必须提供一个函数，用于计算其对后续子结构的任何连接点的位置和方向的贡献。
### 1.1.3 Gravitational forces on and from a substructure
# 1.1.3 Gravitational forces on and from a substructure
# 1.1.4 Linear elastic forces inside a substructure
### 1.1.4 Linear elastic forces inside a substructure
The flexible substructures will have internal forces that can be modeled as linear elastic forces. These forces can be included in the Lagrange equation through their potential energy $V_{e,b}=V_{e,b}(\mathbf{q}_{b})$ which is only a function of the substructure DOFs. Thus, the internal forces and the associated tangent stiffness matrix do not lead to coupling to the other substructures as the inertia forces.
# 1.1.5 Linear purely dissipative forces inside a substructure
柔性子结构将具有内力，这些内力可以建模为线性弹性力。这些力可以通过其势能 $V_{e,b}=V_{e,b}(\mathbf{q}_{b})$ 包含在拉格朗日方程中，该势能仅是子结构自由度的函数。因此，内力及相关的切向刚度矩阵不会像惯性力那样导致与其他子结构的耦合。
### 1.1.5 Linear purely dissipative forces inside a substructure
The dissipation of vibrational energy in a substructure ...
# 1.1.6 Generalized forces from external forces and moments
### 1.1.6 Generalized forces from external forces and moments
... aerodynamic forces ...
# 1.2 Substructure models
## 1.2 Substructure models
The following model types can be used to model substructures:
• Rigid body
• Flexible co­rotational finite beam element body • Flexible linear Craig­Bampton super­element body
- Rigid body
- Flexible co­rotational finite beam element body
- Flexible linear Craig­-Bampton super-­element body
The following sections contain
以下模型类型可用于对子结构建模：
# 1.2.1 Rigid body substructure
- 刚体
- 柔性同向旋转有限梁单元体
- 柔性线性Craig-Bampton超单元体
以下章节包含
### 1.2.1 Rigid body substructure
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\}
$$
where $(x,y,z)\in\mathcal{V}$
# 1.2.2 Co­rotational beam substructure
### 1.2.2 Co­rotational beam substructure
The nonlinear co­rotational finite beam element methodology used for this type of substructures is described in details in Appendix A. The methodology is similar to Crisfield [2] and Krenk [3] except that the nonlinear geometric formulation is explicit, i.e., the element coordinate systems and the local small rotations of the nodes are given as explicit nonlinear functions of the global nodal DOFs. The elastic deformation and compliance of the element is adapted from the equilibrium element proposed by Krenk and Couturier [4].
A co­rotational beam substructure consists of finite beam elements with two end­nodes having six DOFs: three translations and three rotations (using Rodrigues parameters) in the substructure coordinate system. The DOF vector of the substructure
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
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