vault backup: 2025-06-12 17:13:32

.obsidian/plugins/copilot/data.json

@@ -266,7 +266,7 @@
     },
     {
       "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    4. 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，span翻译为展向\n    Return only the translated text.</instruction>\n\n<text>{copilot-selection}</text>",
+      "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    4. 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，span翻译为展向，deflection翻译为变形，normal mode翻译为简正模式，jacket 翻译为导管架，superelement翻译为超单元\n    Return only the translated text.</instruction>\n\n<text>{copilot-selection}</text>",
       "showInContextMenu": true,
       "modelKey": "gemma3:12b|ollama"
     },

书籍/力学书籍/BladedTheoryManual/Structural Dynamics/auto/Structural Dynamics.md

@@ -342,7 +342,7 @@ In order to calculate the internal forces of flexible body components, the defor
 为了计算柔性构件的内部力，因此需要从静态平衡分析计算所有位置的变形，其中施加力被计算为所有外部力的总和，包括惯性载荷。如果一些基本自由度受到约束，则系统将针对一组减少的独立自由度进行求解，并计算与约束相关的拉格朗日乘子。最后，**从梁单元的基本平衡方程计算所有梁单元两端的内部力**。外部载荷对计算出的内部力的二阶效应通过几何刚度模型进行考虑，如 (Przemieniecki, 1968) 中所述。
 
 
-# Equilibrium-Based Method  
+## Equilibrium-Based Method  
 
 The method employs the multibody_approach used by Bladed for modelling of the complete wind turbine structure on a flexible body component level. With this approach described in (Nim et al.,. 2024), a flexible body is modelled as an assembly of rigid or deformable elements, interconnected at $N$ nodes each including six nodal DOFs, i.e., three translation components and three rotation components. According to the multibody approach, the deformation state of a flexible element is described by $N_{\epsilon}^{\epsilon}$ generalised strains that are collected in the vector $\epsilon^{\mathrm{{e}}}$ . The external element loading including inertial loads is represented by the vector $\mathbf{p}_{\mathrm{r}}^{\mathrm{e}}$ conjugate to nodal motion and the vector $\mathbf{p}_{\epsilon}^{\mathrm{e}}$ conjugate to generalised strains, defining the deformation state of a deformable element with elastic properties defined by the stiffness matrix ${\bf K}_{\epsilon\epsilon}^{\mathrm{e}}$ . The kinematical relations between nodal displacement and the generalised strains motion are described by $N_{\mathrm{c}}^{\mathrm{e}}$ geometric constraint relations in terms of the constraint matrices ${\bf C}_{\mathrm{r}}^{\mathrm{e}}$ and ${\bf C}_{\epsilon}^{\mathrm{e}},$ which associate a set of unknown Lagrange multipliers $\lambda^{\mathrm{~e~}}$ . By application of the principle of virtual work, it appears that the resulting equilibrium equations for an element can be written in a simplified form as  
 
@@ -351,46 +351,98 @@ The method employs the multibody_approach used by Bladed for modelling of the co
 
 $$
 \begin{equation}
-\left[\begin{array}{c}
+\left[
-{\bmatrix{C}_{\mathrm{r}}^{\mathrm{e}}}^T \\
+\begin{array}{c}
-{\bmatrix{C}_{\epsilon}^{\mathrm{e}}}^T
+{\mathbf{C}_{\mathrm{r}}^{\mathrm{e}}}^T \\
-\end{array}\right] \bvector{\uplambda}^{\mathrm{e}}=\left[\begin{array}{c}
+{\mathbf{C}_{\epsilon}^{\mathrm{e}}}^T
-\bvector{p}_{\mathrm{r}}^{\mathrm{e}}+\bvector{f}_0^{\mathrm{e}} \\
+\end{array}
-\bvector{p}_{\epsilon}^{\mathrm{e}}-\bmatrix{K}_{\epsilon \epsilon}^{\mathrm{e}} \bvector{\epsilon}^{\mathrm{e}}
+\right]
-\end{array}\right]
+\boldsymbol{\lambda}^{\mathrm{e}}
-\label{eq:motionforaflexelement}
+=
+\left[
+\begin{array}{c}
+\mathbf{p}_{\mathrm{r}}^{\mathrm{e}} + \mathbf{f}_0^{\mathrm{e}} \\
+\mathbf{p}_{\epsilon}^{\mathrm{e}} - \mathbf{K}_{\epsilon \epsilon}^{\mathrm{e}} \boldsymbol{\epsilon}^{\mathrm{e}}
+\end{array}
+\right]
 \end{equation}
 $$  
 
 Thevector $\mathbf{f}_{0}^{\mathrm{{e}}}$ contains the resulting element internal forces, which originate from the connection to neighbouring elements. This vector cancels out in the assembly process, which means that the resulting equilibrium equations for the flexible body take the form  
+矢量 $\mathbf{f}_{0}^{\mathrm{{e}}}$ 包含来自与相邻单元连接产生的单元内部力。此矢量在组装过程中相互抵消，这意味着柔性体的最终平衡方程将呈现如下形式：
+
 
 $$
-\left[\mathbf{C}_{\mathrm{r}}^{\phantom{\dagger}}T\right]\boldsymbol{\lambda}=\left[\begin{array}{c}{\mathbf{p}_{\mathrm{r}}}\\ {\mathbf{p}_{\epsilon}-\mathbf{K}_{\epsilon\epsilon}\epsilon}\end{array}\right]
+\begin{equation}
+\left[\begin{array}{c}
+{\mathbf{C}_{\mathrm{r}}}^T \\
+{\mathbf{C}_{\epsilon}}^T
+\end{array}\right] \boldsymbol{\lambda} = 
+\left[\begin{array}{c}
+\mathbf{p}_{\mathrm{r}} \\
+\mathbf{p}_{\epsilon} - \mathbf{K}_{\epsilon \epsilon} \boldsymbol{\epsilon}
+\end{array}\right]
+\end{equation}
 $$  
 
-For statically determinate flexible bodies, the total number of constraints $N_{\mathrm{c}}=\Sigma_{\mathrm{e}}N_{\mathrm{c}}^{\mathrm{e}}$ equals the number of nodal DOFs $6N$ , i.e., $N_{\mathrm{c}}=6N$ implying that $\mathbf{C}_{\mathrm{r}}$ is a square matrix. In this case, the Lagrange multipliers $\lambda$ for the flexible body are calculated from Equation (2) by the linear system $\mathbf{C}_{\mathrm{r}}{}^{T}\hat{\pmb{\lambda}}=\mathbf{p}_{\mathrm{r}}$ with the assumption that the constraint matrix $\mathbf{C}_{\mathrm{r}}$ is invertible. The extracted element Lagrange multipliers $\lambda^{\mathrm{{e}}}$ are then used for calculating the resulting internal forces at the element end nodes by the relationship $\mathbf{f}_{0}^{\mathrm{e}}=\mathbf{C}_{\mathrm{r}}^{\mathrm{e}T}\lambda^{\mathrm{e}}-\mathbf{p}_{\mathrm{r}}^{\mathrm{e}}$ derived by Equation (1). With this approach, the internal forces are determined in terms of the Lagrange multipliers, which represent the internal constraint forces in the element.  
+For statically determinate flexible bodies, the total number of constraints $N_{\mathrm{c}}=\Sigma_{\mathrm{e}}N_{\mathrm{c}}^{\mathrm{e}}$ equals the number of nodal DOFs $6N$ , i.e., $N_{\mathrm{c}}=6N$ implying that $\mathbf{C}_{\mathrm{r}}$ is a square matrix. In this case, the Lagrange multipliers $\lambda$ for the flexible body are calculated from Equation (2) by the linear system $\mathbf{C}_{\mathrm{r}}{}^{T}{\pmb{\lambda}}=\mathbf{p}_{\mathrm{r}}$ with the assumption that the constraint matrix $\mathbf{C}_{\mathrm{r}}$ is invertible. The extracted element Lagrange multipliers $\lambda^{\mathrm{{e}}}$ are then used for calculating the resulting internal forces at the element end nodes by the relationship $\mathbf{f}_{0}^{\mathrm{e}}=\mathbf{C}_{\mathrm{r}}^{\mathrm{e}T}\lambda^{\mathrm{e}}-\mathbf{p}_{\mathrm{r}}^{\mathrm{e}}$ derived by Equation (1). With this approach, the internal forces are determined in terms of the Lagrange multipliers, which represent the internal constraint forces in the element.  
 
-For statically indeterminate flexible bodies, the number of constraints is larger than the number of nodal DOFs, i.e., $N_{\mathrm{c}}>6N$ implying that $\mathbf{C_{r}}$ is a rectangular matrix. Conventionally, the degree of static indeterminacy $D_{\mathrm{s}}$ is used for quantifying the number of redundant forces, which in the present context means that $\ensuremath{D_{\mathrm{s}}}=\ensuremath{N_{\mathrm{c}}}-6\ensuremath{N_{\mathrm{}}}$ In order to calculate the Lagrange multipliers 入 for the flexible body, it is therefore necessary to include both relations in Equation (2). To this end, the geometric constraints are subdivided into a set of $6N$ determinate constraints and the remaining $D_{\mathrm{s}}$ indeterminate constraints. With this subdivision, it appears that Equation (2) can be rewritten as  
+For statically indeterminate flexible bodies, the number of constraints is larger than the number of nodal DOFs, i.e., $N_{\mathrm{c}}>6N$ implying that $\mathbf{C_{r}}$ is a rectangular matrix. Conventionally, the degree of static indeterminacy $D_{\mathrm{s}}$ is used for quantifying the number of redundant forces, which in the present context means that ${D_{\mathrm{s}}}={N_{\mathrm{c}}}-6{N_{\mathrm{}}}$ In order to calculate the Lagrange multipliers 入 for the flexible body, it is therefore necessary to include both relations in Equation (2). To this end, the geometric constraints are subdivided into a set of $6N$ determinate constraints and the remaining $D_{\mathrm{s}}$ indeterminate constraints. With this subdivision, it appears that Equation (2) can be rewritten as  
+对于静定柔性体，总约束数 $N_{\mathrm{c}}=\Sigma_{\mathrm{e}}N_{\mathrm{c}}^{\mathrm{e}}$ 等于节点自由度数 $6N$ ，即 $N_{\mathrm{c}}=6N$，这意味着 $\mathbf{C}_{\mathrm{r}}$ 是一个方阵。在这种情况下，挠曲体的拉格朗日乘子 $\lambda$ 由线性方程组 $\mathbf{C}_{\mathrm{r}}{}^{T}{\pmb{\lambda}}=\mathbf{p}_{\mathrm{r}}$ 按照约束矩阵 $\mathbf{C}_{\mathrm{r}}$ 可逆的假设计算得出。提取的单元拉格朗日乘子 $\lambda^{\mathrm{{e}}}$ 随后用于通过关系式 $\mathbf{f}_{0}^{\mathrm{e}}=\mathbf{C}_{\mathrm{r}}^{\mathrm{e}T}\lambda^{\mathrm{e}}-\mathbf{p}_{\mathrm{r}}^{\mathrm{e}}$ 计算单元端节点产生的内力，该关系式由方程 (1) 推导得出。 采用这种方法，内力由拉格朗日乘子确定，这些乘子代表单元中的内约束力。
+
+对于静不定柔性体，约束数大于节点自由度数，即 $N_{\mathrm{c}}>6N$，这意味着 $\mathbf{C_{r}}$ 是一个矩形矩阵。通常使用静不定度 $D_{\mathrm{s}}$ 来量化冗余力的数量，在本上下文中意味着 ${D_{\mathrm{s}}}={N_{\mathrm{c}}}-6{N_{\mathrm{}}}$ 为了计算挠曲体的拉格朗日乘子 入，因此需要包含方程 (2) 中的两个关系式。为此，将几何约束划分为一组 $6N$ 个确定性约束和剩余的 $D_{\mathrm{s}}$ 个不定性约束。通过这种划分，似乎可以将方程 (2) 重写为：
 
 $$
-\left[{\bf C}_{\mathrm{rr}}^{{\mathrm{\tiny~\textit{T}}}}\;\;{\bf C}_{\mathrm{er}}^{{\mathrm{\tiny~\textit{T}}}}\right]\left[\boldsymbol{\lambda}_{\mathrm{r}}\right]=\left[\boldsymbol{\mathbf{p}}_{\mathrm{r}}\;\;\right]}\\ {{\bf C}_{r\epsilon}^{{\mathrm{\tiny~\textit{T}}}}\;\;{\bf C}_{\mathrm{e}\epsilon}^{{\mathrm{\tiny~\textit{T}}}}\!\!\!\int\left[\boldsymbol{\lambda}_{\mathrm{e}}\right]=\left[{\bf p}_{\epsilon}-{\bf K}_{\epsilon\epsilon}\epsilon\right]}\end{array}
+\begin{equation}
+\left[\begin{array}{cc}
+{\mathbf{C}_{\mathrm{rr}}}^T & {\mathbf{C}_{\mathrm{er}}}^T \\
+{\mathbf{C}_{r \epsilon}}^T & {\mathbf{C}_{\mathrm{e \epsilon}}}^T
+\end{array}\right] 
+\left[\begin{array}{c}
+\boldsymbol{\lambda}_{\mathrm{r}} \\
+\boldsymbol{\lambda}_{\mathrm{e}}
+\end{array}\right]
+=
+\left[\begin{array}{c}
+\mathbf{p}_{\mathrm{r}} \\
+\mathbf{p}_{\epsilon} - \mathbf{K}_{\epsilon \epsilon} \boldsymbol{\epsilon}
+\end{array}\right]
+\end{equation}
 $$  
 
-where $\lambda_{\mathrm{r}}$ and $\lambda_{\mathrm{e}}$ include partial permutations of $\lambda$ , while $\mathbf{C}_{\mathrm{rr}}$ and $\mathbf{C}_{\mathrm{r}\epsilon}$ and respectively $\mathbf{C}_{\mathrm{er}}$ and $\mathbf{C}_{\mathrm{e}\epsilon}$ include partial permutations of $\mathbf{C}_{\mathrm{r}}$ and $\mathbf{C}_{\epsilon}$ . In principle, the subdivision of the geometric constraints into determinate and indeterminate partitions corresponds to the introduction of virtual cuts in the structure. The subdivision is done automatically, (Wehage and Haug, 1982), in such a way that the cut structure is statically determinate, which enables the calculation of internal forces by the equilibrium approach. With the application of the constraint elimination method,  
+where $\lambda_{\mathrm{r}}$ and $\lambda_{\mathrm{e}}$ include partial permutations of $\lambda$ , while $\mathbf{C}_{\mathrm{rr}}$ and $\mathbf{C}_{\mathrm{r}\epsilon}$ and respectively $\mathbf{C}_{\mathrm{er}}$ and $\mathbf{C}_{\mathrm{e}\epsilon}$ include partial permutations of $\mathbf{C}_{\mathrm{r}}$ and $\mathbf{C}_{\epsilon}$ . In principle, the subdivision of the geometric constraints into determinate and indeterminate partitions corresponds to the introduction of virtual cuts in the structure. The subdivision is done automatically, (Wehage and Haug, 1982), in such a way that the cut structure is statically determinate, which enables the calculation of internal forces by the equilibrium approach. With the application of the constraint elimination method,  (Géradin and Cardona, 2001), it appears that the Lagrange multipliers $\lambda_{\mathrm{r}}$ associated with the determinate constraints can be eliminated in Equation (3). After a minor rearrangement of terms, it appears that the resulting transformed system can be written in the symmetric form  
-
+其中 $\lambda_{\mathrm{r}}$ 和 $\lambda_{\mathrm{e}}$ 包含 $\lambda$ 的partial排列，而 $\mathbf{C}_{\mathrm{rr}}$ 和 $\mathbf{C}_{\mathrm{r}\epsilon}$ 以及 $\mathbf{C}_{\mathrm{er}}$ 和 $\mathbf{C}_{\mathrm{e}\epsilon}$ 分别包含 $\mathbf{C}_{\mathrm{r}}$ 和 $\mathbf{C}_{\epsilon}$ 的partial排列。 原理上，将几何约束划分为确定性和不确定性分区，对应于在结构中引入虚拟切割。 这种划分是自动进行的 (Wehage and Haug, 1982)，以确保切割结构是静态确定的，从而能够通过平衡法计算内力。 采用约束消除法 (Géradin and Cardona, 2001) 后，可以发现与确定性约束相关的拉格朗日乘子 $\lambda_{\mathrm{r}}$ 可以从方程 (3) 中消除。 在对项进行轻微调整后，可以发现得到的变换系统可以写成对称形式。
-(Géradin and Cardona, 2001), it appears that the Lagrange multipliers $\lambda_{\mathrm{r}}$ associated with the determinate constraints can be eliminated in Equation (3). After a minor rearrangement of terms, it appears that the resulting transformed system can be written in the symmetric form  
 
 $$
-\left[\mathbf{C}_{\epsilon\epsilon}^{*}\quad\mathbf{C}_{\epsilon\epsilon}^{*\ T}\right]\left[\boldsymbol{\epsilon}^{\prime}\right]=\left[\mathbf{p}_{\epsilon}^{*}\right]
+\begin{equation}
+\left[\begin{array}{cc}
+K_{\epsilon \epsilon}^* & C_{\mathrm{e} \epsilon}^{*T} \\
+C_{\mathrm{e} \epsilon}^* & O
+\end{array}\right]
+\left[\begin{array}{c}
+\boldsymbol{\epsilon}' \\
+\boldsymbol{\lambda}_{\mathrm{e}}
+\end{array}\right]
+=
+\left[\begin{array}{c}
+\boldsymbol{p}_{\epsilon}^* \\
+\boldsymbol{o}
+\end{array}\right]
+\end{equation}
 $$  
 
 where $\mathbf{C}_{\mathrm{e}\epsilon}^{*}=\mathbf{C}_{\mathrm{e}\epsilon}+\mathbf{C}_{\mathrm{er}}\mathbf{T}_{\mathrm{r}\epsilon}$ and $\mathbf{p}_{\epsilon}^{*}=\mathbf{p}_{\epsilon}+\mathbf{T}_{\mathrm{r}\epsilon}{}^{T}\mathbf{p}_{\mathrm{r}},$ while ${\bf K}_{\epsilon\epsilon}^{*}={\bf K}_{\epsilon\epsilon}$ is introduced for notational consistency. The transformation matrix $\mathbf{T}_{\mathbf{r}\epsilon}$ relates a variation of the generalised strains to the variation of the nodal DOFs, and it is calculated from the partial permutations $\mathbf{C}_{\mathrm{rr}}$ and $\mathbf{C}_{\mathrm{r}\epsilon}$ assuming that $\mathbf{C}_{\mathrm{rr}}$ is invertible. In principle, the indeterminate constraints $\lambda_{\mathrm{e}}$ together with generalised strains $\epsilon^{\prime}$ can be calculated from the linear system (4), but the actual calculation is done in a more numerically stable way in order to deal with the fact that the coefficient matrix on the left-hand side is ill-conditioned. The determinate Lagrange multipliers $\lambda_{\mathrm{r}}$ corresponding to the indeterminate constraints $\lambda_{\mathrm{e}}$ are then calculated from the equilibrium relation of Equation (3) resulting in the linear system $\mathbf{C}_{\mathrm{rr}}^{\phantom{\dagger}}{}^{T}\lambda_{\mathrm{r}}=\mathbf{p}_{\mathrm{r}}-\mathbf{C}_{\mathrm{er}}{}^{T}\lambda_{\mathrm{e}}$ With known Lagrange multipliers, the resulting internal forces can be calculated by the approach described above and used for statically determinate flexible bodies.  
 
 It is noted that the alternative equilibrium-based method for calculating internal forces is essentially different from the conventional displacement-based finite element method, where the internal forces at the end nodes of a beam element are calculated in terms of the internal elastic forces using the element stiffness matrix. An important difference is that the equilibrium-based method calculates the internal forces at the current modal truncated state during time integration. The number of modes used for describing the modal truncated state can vary significantly depending on the desired accuracy of the analysis. In particular, it is even possible to do analysis without any modes, for flexible bodies that are temporarily assumed rigid, which may be convenient for some initial investigations.  
 
+其中 $\mathbf{C}_{\mathrm{e}\epsilon}^{*}=\mathbf{C}_{\mathrm{e}\epsilon}+\mathbf{C}_{\mathrm{er}}\mathbf{T}_{\mathrm{r}\epsilon}$ 和 $\mathbf{p}_{\epsilon}^{*}=\mathbf{p}_{\epsilon}+\mathbf{T}_{\mathrm{r}\epsilon}{}^{T}\mathbf{p}_{\mathrm{r}},$ 引入 ${\bf K}_{\epsilon\epsilon}^{*}={\bf K}_{\epsilon\epsilon}$ 以保持符号一致性。变换矩阵 $\mathbf{T}_{\mathbf{r}\epsilon}$ 将广义应变的变化与节点自由度 (DOF) 的变化联系起来，它由偏置排列 $\mathbf{C}_{\mathrm{rr}}$ 和 $\mathbf{C}_{\mathrm{r}\epsilon}$ 计算得出，假设 $\mathbf{C}_{\mathrm{rr}}$ 可逆。原则上，不定约束 $\lambda_{\mathrm{e}}$ 连同广义应变 $\epsilon^{\prime}$ 可以从线性方程组 (4) 中计算得出，但实际计算采用更数值稳定的方法，以应对左侧系数矩阵病态的现象。然后，根据方程 (3) 的平衡关系计算出与不定约束 $\lambda_{\mathrm{e}}$ 对应的确定拉格朗日乘子 $\lambda_{\mathrm{r}}$，得到线性方程组 $\mathbf{C}_{\mathrm{rr}}^{\phantom{\dagger}}{}^{T}\lambda_{\mathrm{r}}=\mathbf{p}_{\mathrm{r}}-\mathbf{C}_{\mathrm{er}}{}^{T}\lambda_{\mathrm{e}}$。在已知拉格朗日乘子后，可以使用上述方法计算出结果内力，并将其用于静定柔性体。
+
+需要注意的是，基于平衡的内力计算方法与传统的基于位移的有限元方法本质上不同，后者通过使用单元刚度矩阵来计算梁单元端节点的内力。一个重要的区别是，基于平衡的方法在时间积分过程中计算当前模态截断状态下的内力。用于描述模态截断状态的模态数量可以根据所需的分析精度而有很大差异。特别地，即使可以在不使用任何模态的情况下进行分析，这适用于暂时假设为刚体的柔性体，这对于一些初步研究可能很方便。
+
 Last updated 25-10-2024  
 
-# Introduction to Flexible Components  
+# Modelling Flexible Components
+## Introduction to Flexible Components  
 
 In Bladed, the support structure is modelled as a single linear flexible component (body). The blade is also simulated as a flexible component, with the option to be subdivided into multiple linear flexible components in order to accurately model large deflections.  
 
@@ -400,6 +452,15 @@ For the blades, the finite element model degrees of freedom can be used directly
 
 The applied beam element may be considered as an extension to the standard three-dimensional engineering Timoshenko beam element (Przemieniecki, 1968) with two nodes or stations located at the two ends. This element has twelve fundamental degrees of freedom. These are the three translational and three rotational degrees of freedom at both stations. The deflection at all intermediate points is calculated via interpolation functions that are derived from a set of homogenous equilibrium equations valid for prismatic beam elements. It is important to note that this beam element includes the effect of shear deformation that may be important for some support structures, in particular complicated offshore foundations. The magnitude of the shear deformations relative to bending deformations for an element may be evaluated by the element shear parameter conveniently defined as  
 
+在Bladed中，支撑结构被建模为单个线性柔性构件（body）。叶片也被模拟为柔性构件，并可以选择将其细分为多个线性柔性构件，以便准确模拟大变形。
+
+基本的有限元模型假设柔性构件是线性空间梁或更通用的空间框架，它们由多个由Timoshenko梁单元建模的构件组成。由于模型是线性的，因此假设每个body内的变形是小的。
+
+对于叶片，**有限元模型的自由度可以直接用作广义自由度**。对于叶片和支撑结构，可以进行模态简化以减少广义自由度的数量。使用模态方法，变形由一些预先计算的模态形状的线性组合来表示。这种线性组合的标量是模态振幅（Clough and Penzien, 1993），它们代表构件的广义应变，进而代表构件的自由度。需要注意的是，**模态形状函数随时间是恒定的**。
+
+所应用的梁单元可以被认为是标准三维工程Timoshenko梁单元（Przemieniecki, 1968）的扩展，该单元在两端有两个节点或位置。该单元具有十二个基本的自由度。这些是两个位置处的三个平动自由度和三个旋转自由度。所有中间点的变形通过从一组适用于棱柱形梁单元的齐次平衡方程推导出的插值函数进行计算。需要注意的是，该梁单元包括剪切变形的影响，这对于某些支撑结构，特别是复杂的离岸地基可能很重要。相对于梁单元的弯曲变形的剪切变形的量可以通过以下方式进行评估：单元剪切参数。
+
+
 $$
 \Omega^{e}=\frac{12E I^{e}}{l^{e^{2}}G A^{e}},
 $$  
@@ -408,6 +469,10 @@ where
 $E I^{e}$ is the bending stiffness,   
 $G A^{e}$ is the corresponding shear stiffness, $l^{e}$ is the element length.  
 
+其中
+$E I^{e}$ 为弯曲刚度，
+$G A^{e}$ 为相应的剪切刚度，$l^{e}$ 为单元长度。
+
 The beam element supports an arbitrary spatial variation of the stiffness along the beam element, but in the present implementation it is assumed that the bending, torsional, axial and shear stiffness vary linearly. The orientation of the element is defined by the position of the two ends as well as the orientation of the principal axis around the neutral axis (elastic centre). The effect of possible coupling between bending and torsion is included in terms of the position of the shear centre (torsion centre), and a transformation between displacements and forces relating to the shear centre and the neutral axis is included. The resulting stiffness matrix is constant and calculated by numerical integration. For prismatic elements, where the shear centre is located at the neutral axis, the stiffness matrix is identical to the standard engineering Timoshenko beam element (Przemieniecki, 1968).  
 
 An important feature of the derived method is that some fundamental degrees of freedom may be constrained, which is particularly useful in cases where the effect of elongation and/or torsion can be neglected. In order to enable the description of rigid connection the deflection of a beam element may also be constrained completely. The constraints are modelled in terms of a constant constraint matrix together with Lagrange's method (Cook, Malkus and Plesha, 1989).  
@@ -418,25 +483,44 @@ Further second-order effects of the internal shear forces are accounted for by a
 
 Inertia forces acting on the element and the proximal node are derived as described in the multibody dynamics approach from the fundamental displacement hypothesis using the principle of virtual work. An important feature of the derived method is that the inertia forces are expressed directly in terms of the modal amplitudes, i.e. the strains and corresponding derivatives as originally suggested in (Shabana, 1998). In principle this means that the time for calculating the accelerations scales with the number of mode shapes rather than the number of beam elements of the flexible component.  
 
+梁单元支持沿梁单元任意的空间刚度变化，但在目前的实现中，假设弯曲、扭转、轴向和剪切刚度呈线性变化。单元的取向由两端的位置以及围绕中性轴（弹性中心）的主轴取向来定义。弯曲和扭转之间可能存在的耦合效应，通过剪切中心（扭转中心）的位置来体现，并且包含与剪切中心和中性轴相关的位移和力之间的变换。得到的刚度矩阵是常数，并通过数值积分计算得出。对于具有剪切中心位于中性轴的棱柱形单元，刚度矩阵与标准的工程 Timoshenko 梁单元（Przemieniecki, 1968）相同。
+
+该衍生方法的一个重要特征是，一些基本的自由度可以被约束，这在可以忽略拉伸和/或扭转效应的情况下特别有用。为了能够描述刚性连接，梁单元的变形也可以完全被约束。约束通过一个常数约束矩阵以及拉格朗日方法（Cook, Malkus and Plesha, 1989）进行建模。
+
+内部轴向力的二阶效应通过一个几何刚度矩阵（应力加刚度）来体现，该矩阵由插值函数的导数计算得出（Clough and Penzien, 1993）。对于叶片，轴向力的主要部分是由离心力引起的，因此在这种情况下传统上使用“离心刚度”一词。由于重力载荷，支撑结构中偶尔可以观察到类似效应，有时被称为“几何去刚度”。
+
+内部剪切力的进一步二阶效应通过应用基于（Krenk, 2009）的方法的额外外部载荷来考虑。该模型可以特别提高对具有扭转自由度的叶片的扭转变形的预测能力。
+
+作用在单元和邻近节点上的惯性力，根据虚拟功原理，从基本位移假设中推导出来，如同多体动力学方法所描述的那样。该衍生方法的一个重要特征是，惯性力直接用模态振幅（即应变及其对应导数）来表示，正如最初由（Shabana, 1998）建议的那样。从原理上讲，这意味着计算加速度所需的时间与柔性构件的模态数量成比例，而不是与梁单元的数量成比例。
+
 Last updated 30-08-2024  
 
-# Normal and Attachment Modes  
+## Mode Shapes and Damping
+### Normal and Attachment Modes  
 
-The selection and calculation of mode shape functions follows the idea that was originally suggested by (Craig, 2000) as a modification of the widely used Craig-Bampton method from (Craig, 1968). For both methods the stations are subdivided into boundary stations that may couple to other components and interior stations that do not couple. The boundary stations also represent the component nodes that may link to nodes of other components. In particular the station representing the proximal node is constrained completely in order to exclude rigid body displacement modes.  
+The selection and calculation of mode shape functions follows the idea that was originally suggested by (Craig, 2000) as a modification of the widely used Craig-Bampton method from (Craig, 1968). For both methods the stations are subdivided into **boundary stations** that may couple to other components and **interior stations** that do not couple. The boundary stations also represent the component nodes that may link to nodes of other components. In particular the station representing the proximal node is constrained completely in order to exclude rigid body displacement modes.  
 
-With the applied method the modes are generally selected as the union between attachment modesthat may couple to other components and normal modes that may be considered as internal vibration modes.  
+With the applied method the modes are generally **selected as the union between attachment modes** that may couple to other components and **normal modes** that may be considered as internal vibration modes.  
+
+模式形状函数的选择和计算遵循了(Craig, 2000)最初提出的思路，是对(Craig, 1968)广泛使用的 Craig-Bampton 方法的一种改进。对于这两种方法，站点被划分为可以与其他组件耦合的边界站点，以及不耦合的内部站点。边界站点也代表了可能链接到其他组件节点的组件节点。特别是，代表近端节点的站点被完全约束，以排除刚体位移模式。
+
+采用该方法时，模式通常被选择为连接模式attachment modes（可能与其他组件耦合）与正常模式**normal modes**（可被视为内部振动模式）的并集。
 
 ![](images/dfb0559f901bb76d71fca50c912bd62e34efba55b69f91f14bad8f5cd5c2161b.jpg)  
 
-Figure 1: llustration of the boundary conditions for attachment and normal modes.  
+Figure 1: illustration of the boundary conditions for attachment and normal modes.  
 
-# Attachment Modes  
+#### Attachment Modes  
 
 The attachment modes are calculated from the component stiffness matrix by a static equilibrium, where the component is fixed at the proximal node and point loads are applied in turn at the distal nodes, as seen in Figure 1. This method yields an attachment mode for every degree of freedom of the distal node of the component (fixed-free boundary condition).  
 
 For the attachment modes of the support structure, the calculation of the mode frequencies include the mass of the rotor and nacelle assembly, and would usually be of lower frequencies than the corresponding normal modes.  
 
-# Normal Modes  
+连接模式attachment modes是通过静态平衡计算得到的，其中组件固定在近端节点，并在远端节点依次施加点载point loads，如图1所示。 这种方法会为组件远端节点distal node的每个自由度产生一个连接模式（固定-自由边界条件）。
+
+对于支撑结构的连接模式，模态频率的计算包括风轮 (rotor) 和轮舱 (nacelle) 组合的质量，通常比对应的固有模态频率更低。
+
+#### Normal Modes  
 
 The normal modes are determined directly from the fully assembled finite element mass and stiffness matrices using a generalised eigenvalue problem. This calculation is performed with the component fixed at the proximal node (fixed-free) or at both the proximal and distal nodes (fixedfixed), as illustrated in Figure 1. The type of normal mode (fixed-free vs. fixed-fixed) depends solely on the presence of distal nodes.  
 
@@ -444,15 +528,27 @@ For instance, in the case of a tower, there is always a distal node at the tower
 
 This will produce one normal mode for every degree of freedom in the model, where the number of degrees of freedom is dependent on the boundary condition.  
 
-# Frequencies and Structural Damping  
+简正模式可以直接从完整组装的有限元质量和刚度矩阵，利用广义特征值问题计算得到。此计算以近端节点固定（fixed-free）或近端和远端节点同时固定（fixedfixed）的情况下进行，如图1所示。简正模式的类型（fixed-free vs. fixedfixed）完全取决于远端节点是否存在。
+
+例如，对于塔架而言，塔顶总是存在一个远端节点，而单片叶片则没有远端节点。多片叶片对于所有部件都具有近端节点和远端节点，除了最后一个部件，它仅有近端节点。因此，最后一个部件产生fixed-free简正模式，而其余部件产生fixedfixed简正模式。可能存在多个远端节点，例如在动态系泊系统中，在这种情况下，所有远端节点都被固定。
+
+这将为模型中的每个自由度产生一个简正模式，自由度的数量取决于边界条件。
+
+
+#### Frequencies and Structural Damping  
 
 The frequencies of the attachment modes are calculated by Rayleigh's method (Clough and Penzien, 1993), while the frequencies of the normal modes result from the eigenvalue problem. These frequencies are solely used for describing damping.  
 
 Structural damping is modelled as modal damping (Clough and Penzien, 1993). in terms of a set of damping coefficients (damping ratio) that relate to the mode shape functions. These coefficients are defined as input parameters for the model and may usually be measured directly, for example by exiting a mode and measure the decay of the succeeding oscillation.  
 
+采用瑞利法（Clough and Penzien, 1993）计算附着模式的频率，而简正模式的频率则源于特征值问题。这些频率仅用于描述阻尼。
+
+结构阻尼被建模为模态阻尼（Clough and Penzien, 1993），用一组阻尼系数（阻尼比）来描述模态函数。这些系数被定义为模型的输入参数，通常可以通过激发模态并测量后续振荡的衰减来直接测量。
+
+
 Last updated 10-10-2024  
 
-# Blade Modes  
+### Blade Modes  
 
 The motion of the tapered, twisted and very flexible rotor blades is among the most complex phenomena related to the structural dynamics of a wind turbine. In Bladed, a blade can be represented by one component or several rigidly connected components. Use of several components allows rigorous modelling of large deflections.  
 
@@ -470,7 +566,23 @@ where the subscript $_i$ indicates properties related to the $i^{t h}$ mode.
 
 For blade with several parts, it is still desirable to calculate and review the coupled eigenmodes for the whole blade. To facilitate this, Bladed performs a subsequent eigen analysis of the blade parts to calculate the modes corresponding to the natural modes of the blade. This is useful both for physical interpretation of the blade mode shapes and for applying damping, as explained in the next section.  
 
-# Specifying blade damping (whole blade damping)  
+锥形tapered、扭转且极具柔性的风轮叶片的运动是风力发电机结构动力学中最为复杂的现象之一。在Bladed中，**叶片可以由一个组件或若干刚性连接的组件来表示**。使用若干组件允许对大变形进行严格建模。
+
+对于单段叶片模型，仅使用简正模式，边界条件为固定-自由。这是选择风轮叶片振动模式的经典方法。在多段multi-part叶片模型中，内部分件使用both简正模式（边界条件为固定-固定）和连接模式（边界条件为固定-自由），而最外部分件仅使用简正模式（边界条件为固定-自由）。更多细节请参见简正模式和连接模式。
+
+叶片模式的命名约定详见题为“命名模态形状”的文章。
+
+每个模态由以下参数定义：
+
+模态频率，$\omega_{i}$
+模态阻尼系数，$\xi_{i}$
+模态形状，由各站点的位移矢量表示
+
+其中下标 $_i$ 表示与第 $i^{t h}$ 模态相关的属性。
+
+对于多片叶片，仍然需要计算和审查整个叶片的耦合特征模态。为了便于此，Bladed 对叶片部件执行后续特征分析，以计算与叶片的自然模态对应的模态。这对于物理解释叶片模态形状以及应用阻尼都很有用，如下一节所述。
+
+#### Specifying blade damping (whole blade damping)  
 
 The blade damping is specified for the natural modes of the whole blade. To allow this, Bladed must calculate the damping on each blade part mode (or generalised finite element freedom) based on the damping of the natural modes of the whole blade. This is done according to theory presented in (Clough and Penzien, 1993) pp240-242.  
 
@@ -478,6 +590,11 @@ Damping should be specified for the coupled modes which would be expected to con
 
 The coefficient $\displaystyle a_{1}$ is defined according to the highest mode for which damping is specified:  
 
+风轮叶片阻尼的设定是针对整个叶片的简正模式。为了实现这一点，Bladed 需要根据整个叶片的自然模式阻尼，计算每个叶片部分模式（或广义有限元自由度）上的阻尼。这遵循了 (Clough and Penzien, 1993) pp240-242 中提出的理论。
+
+阻尼应针对耦合模式进行设定，这些模式预计会显著影响叶片的动态响应（通常是前 ~10 个模式）。对于后续频率更高的耦合模式，阻尼被假定与模态刚度成正比，计算公式为 $\mathbf{C}=a_{1}\mathbf{K}$ 。这导致阻尼与模态频率成正比，从而使高阻尼比的高频模态响应被有效消除。
+
+系数 $\displaystyle a_{1}$ 的定义取决于阻尼被设定的最高模态：
 $$
 a_{1}=\frac{2\xi_{c}}{\omega_{c}}
 $$  
@@ -487,16 +604,20 @@ where
 · the subscript c refers to highest mode with damping specified, ? $\xi$ refers to the coupled mode damping ratio.  
 
 Damping on these higher frequency coupled modes is given by  
+其中
 
+· 下标 c 指的是具有阻尼规格的最高模态，? $\xi$ 指的是耦合模态阻尼比。
+
+这些较高频率耦合模态的阻尼由：
 $$
 \xi_{n}=\xi_{c}\left(\frac{\omega_{n}}{\omega_{c}}\right).
 $$  
 
 Once the coupled mode damping values are calculated, the blade part modal frequencies are calculated in Bladed according to (Clough and Penzien, 1993).  
-
+在计算出耦合模阻尼值后，Bladed软件会根据（Clough and Penzien, 1993）计算叶片（blade）的模态频率。
 Last updated 11-12-2024  
 
-# Tower Modes  
+### Tower Modes  
 
 The standard axisymmetric tower model in Bladed has one proximal node at the base and one distal node at the top. This implies that the mode shape functions are represented by a combination of attachment modes with fixed-free boundary conditions and normal modes with fixed-fixed boundary conditions. These mode shapes represent the deflection in the fore-aft and side-side directions as well as the torsional deflection and axial elongation. The tower modes are defined in terms of their modal frequency, modal damping and mode shape.  
 
@@ -506,7 +627,15 @@ The naming conventions for tower modes are detailed in the article titled Naming
 
 Mass and inertia of the nacelle and rotor: For the calculation of the tower support structure modes, the nacelle and rotor are modelled as lumped mass and inertia located at the nacelle centre of gravity and rotor hub, respectively. As the normal modes do not couple to other components it appears that only the frequency of the attachment modes are affected. This means that the mass and inertia of the nacelle and rotor only affect the resulting frequency of the attachment modes.  
 
-# Note on the Difference between Normal and Attachment modes forTowers  
+Bladed软件中的标准轴对称塔架模型在塔基处有一个近端节点，在塔顶处有一个远端节点。这意味着模态函数由带有固定-自由边界条件的附着模态和带有固定-固定边界条件的简正模式组合表示。**这些模态形状代表了前后方向和侧向方向的变形，以及扭转变形和轴向伸长**。**塔架模态由其模态频率、模态阻尼和模态形状定义**。
+
+多段塔架使用相同的方法，但此时允许由任意数量的直线相互连接的段组成任意结构。由于塔架不被假定为轴对称，模态通常是三维的，并且每个节点包含所有六个自由度，并且可能在多个节点（近端节点）处存在地基。
+
+塔架模态的命名约定详见题为“命名模态形状”的文章。
+
+机舱和风轮的质量和惯性：为了计算塔架支撑结构模态，**机舱和风轮被建模为位于机舱重心和风轮毂处的集中质量和惯性**。**由于简正模式不与其他组件耦合，似乎只有附着模态的频率会受到影响。这意味着机舱和风轮的质量和惯性仅影响附着模态的最终频率**。
+
+#### Note on the Difference between Normal and Attachment modes forTowers  
 
 For the support structure, the interpretation of the definition of norma/ modes can sometimes cause confusion.  
 
@@ -516,35 +645,78 @@ This means that if the free vibration modes for the tower structure are calculat
 
 Bladed also calculates coupled vibration modes in the Campbell diagram, shows how the normal and attachment modes combine into coupled vibration modes at a specific operating point. Typically, these coupled modes correspond well to normal modes calculated in other software, with the tower base constrained and the tower top free to move.  
 
-# Note on Comparing Coupled and Uncoupled Modes for Floating Turbines  
+对于支撑结构，对简正模式的定义解读有时会造成困惑。
+
+支撑结构的传统自由振动模式包括塔顶可以自由移动且不受外部力作用的模式。在Bladed中，简正模式仅包括使用固定-固定边界条件进行的特征分析模式，如图所示。然而，Bladed也包括更真实的附加模式，因为在现实中，塔顶会由于上方结构的外部激励而移动。
+
+这意味着，如果仅约束塔基计算塔结构的自由振动模式，其结果将与Bladed计算的模式频率不匹配。
+
+Bladed还计算坎贝尔图中的耦合振动模式，展示了简正模式和附加模式在特定工作点如何组合成耦合振动模式。通常，这些耦合模式与在其他软件中，约束塔基并允许塔顶自由移动时计算出的简正模式相符。
+
+#### Note on Comparing Coupled and Uncoupled Modes for Floating Turbines  
 
 For a floating turbine with soft moorings, the deflection shape of the tower when a load is applied to the top in the Campbell diagram calculation does not correspond to the first attachment mode, as the structure is fixed at the modal reference node during the modal analysis. So, the cantilever attachment mode shapes calculated in modal analysis are not seen individually in the Campbell diagram deflections. When a load is applied to the tower top of a floating turbine, the deflection in the tower will take a form that is a combination of various normal and attachment modes, so the first coupled tower mode will be a combination of various uncoupled modes.  
 
+对于带有柔性锚泊的浮式风机，在 Campbell 图计算中，当载荷施加到顶部时，塔架的变形形状并不对应于第一简正模式，因为模态分析期间结构固定在模态参考节点上。因此，模态分析中计算出的悬臂式附着模态形状在 Campbell 图的变形中无法单独观察到。当载荷施加到浮式风机的塔架顶部时，塔架的变形将呈现出各种简正模式和附着模态的组合形式，因此第一阶耦合塔架模态将是各种未耦合模态的组合。
+
+
 Last updated 11-12-2024  
 
-# Naming Mode Shapes  
+### Naming Mode Shapes  
 
-A structural mode shape describes an allowable pattern of translational and rotational displacements of any point of a flexible body. Using FEM, a mode shape is conveniently represented by a column vector of nodal displacements and rotations. In most cases, a mode shape represents a characteristic displacement pattern, such as the flapwise displacement of a blade or the side-to-side displacement of a tower. Naming the mode shapes with industrystandard terms can therefore facilitate easier reference.  
+A structural mode shape describes an allowable pattern of translational and rotational displacements of any point of a flexible body. Using FEM, a mode shape is conveniently represented by a column vector of nodal displacements and rotations. In most cases, a mode shape represents a characteristic displacement pattern, such as the flapwise displacement of a blade or the side-to-side displacement of a tower. Naming the mode shapes with industry-standard terms can therefore facilitate easier reference.  
 
 Lower-order mode shapes are often well-defined, with displacements primarily occurring in a single direction. These cases are typically easy to categorise. However, for prebend blades with beam-level cross couplings, the mode shapes typically contain large displacements in various directions. Similarly, for multi-member towers, such as jacket support structures, where the structural members are connected in different directions, determining the primary displacement direction is not straightforward.  
 
 The problem of identifying mode shape names cannot be solved theoretically, as the problem is not rigorously defined. Hence, the presented method is a practical approach, implementing certain conditions that result in meaningful mode shape names in most cases.  
 
-# The Naming Procedure  
+结构模态形状描述了柔性体任意一点允许的平动和转动位移模式。利用有限元方法，模态形状可以方便地用节点位移和转动的列向量来表示。在大多数情况下，模态形状代表一种特征位移模式，例如叶片的挥舞位移或塔架的侧向位移。因此，使用行业标准术语命名模态形状可以方便参考。
+
+低阶模态形状通常定义明确，位移主要发生在单一方向。这些情况通常容易分类。然而，对于具有梁级beam-leve横向耦合的预弯叶片，模态形状通常包含各个方向上的大位移。同样，对于多构件塔架，例如裙式支撑结构，由于结构构件连接方向不同，确定主要位移方向并不直接。
+
+模态形状命名的确定问题无法通过理论解决，因为该问题没有严格的定义。因此，所提出的方法是一种实用的方法，它实施了某些条件，在大多数情况下可以产生有意义的模态形状名称。
+
+#### The Naming Procedure  
 
 This method uses a set of predefined conditions to identify and label mode shapes based on their primary characteristics. The mode shape vector is defined as a column of the mode shape matrix, shown here, where each entry corresponds to a degree of freedom of a node in the proximal frame of the component.  
 
 Example of a normalised mode shape vector containing 2 nodes with 6 degrees of freedom each:  
 
+本方法使用一组预定义的条件，根据其主要特征识别和标记模态形状。模态形状向量被定义为模态矩阵的一列，如图所示，其中每个条目对应于部件临近框架中一个节点的自由度。
+
+例如，一个包含 2 个节点，每个节点具有 6 个自由度的归一化模态形状向量：
+
 $$
-\boldsymbol{\psi}_{\mathrm{prox}}=\left[\underbrace{\!\!\left[u_{x_{1}}\quad u_{y_{1}}\quad u_{z_{1}}\quad\theta_{x_{1}}\quad\theta_{y_{1}}\quad\theta_{z_{1}}\quad u_{x_{2}}\quad u_{y_{2}}\quad u_{z_{2}}\quad\theta_{x_{2}}\quad\theta_{y_{2}}\quad\theta_{z_{2}}\right]}_{\mathrm{Node}\;1}\right]^{T}
+\begin{equation} 
+{\psi_{prox}} = \left[\begin{matrix}
+\underbrace{\begin{aligned}
+u_{x_1} \quad
+u_{y_1} \quad  
+u_{z_1} \quad
+\theta_{x_1} \quad
+\theta_{y_1} \quad
+\theta_{z_1}
+\end{aligned}}_{\text{Node 1}} \quad
+\underbrace{\begin{aligned}
+u_{x_2} \quad
+u_{y_2} \quad  
+u_{z_2} \quad
+\theta_{x_2} \quad
+\theta_{y_2} \quad
+\theta_{z_2}
+\end{aligned}}_{\text{Node 2}}
+\end{matrix}\right]^T
+\end{equation}
 $$  
 
 where the first 6 entries correspond to the translational and rotational degrees of freedom for the first node, while the next 6 entries correspond to those of the second node. By examining the values in the vector, one can observe that the first entry is the largest. This would indicate a fore-aft or flapwise mode, depending on whether it pertains to a tower or a blade. However, this approach is not suitable, as the values of the translational and rotational degrees of freedom use differing units. Therefore, a more sophisticated method is necessary.  
 
 The procedure for naming mode shapes in Bladed involves a series of checks to determine the primary characteristic of the mode shape.  
 
-# 1. Determine if the Mode is Longitudinal $(u_{z})$  
+其中前6个条目对应于第一个节点的平动和转动自由度，而接下来的6个条目对应于第二个节点的自由度。通过检查向量中的值，可以观察到第一个条目最大。这可能表明是前后方向或挥舞模式，具体取决于它是否与塔架或叶片相关。然而，这种方法并不适用，因为平动和转动自由度的值使用不同的单位。因此，需要一种更复杂的方法。
+
+Bladed中命名模态形状的过程涉及一系列检查，以确定模态形状的主要特征。
+##### 1. Determine if the Mode is Longitudinal $(u_{z})$  确定模式是否为纵向 $(u_{z})$
 
 Evaluate the following condition:  
 
@@ -558,11 +730,14 @@ $$
 \begin{array}{r l}&{u_{z\operatorname*{max}}=\operatorname*{max}\big(u_{z_{1}}\quad u_{z_{2}}\quad\cdot\cdot\cdot\quad u_{z_{n}}\big),}\\ &{u_{x y_{m a x}}=\operatorname*{max}\left(\sqrt{u_{x_{1}}^{2}+u_{y_{1}}^{2}}\quad\sqrt{u_{x_{2}}^{2}+u_{y_{2}}^{2}}\quad\cdot\cdot\cdot\quad\sqrt{u_{x_{n}}^{2}+u_{y_{n}}^{2}}\right),}\end{array}
 $$  
 
-$\kappa$ is a constant calibration factor.  
+$K$ is a constant calibration factor.  
 
 The left-hand side of the condition is the maximum longitudinal displacement in the mode shape vector, and the right-hand side is the maximum displacement magnitude in the $\times$ and y directions compared across all nodes.  
 
-# 2. Determine if the Mode is Torsional $(\theta_{z})$  
+$K$ 是一个常数标定因子。
+
+条件左侧为模态向量中的最大纵向位移，右侧为在 $x$ 和 $y$ 方向上，对所有节点进行比较的最大位移幅值。
+##### 2. Determine if the Mode is Torsional $(\theta_{z})$  
 
 If the mode is not axial, evaluate if it is a torsional mode:  
 
@@ -579,52 +754,41 @@ $$
 The left-hand side of the condition is the maximum torsional displacement in the mode shape vector, and the right-hand side is the maximum rotation (bending) magnitude in the x and y directions compared across all nodes.  
 
 Additionally, the following condition must also be true:  
+条件左侧为模态向量中的最大扭转位移，右侧为在x和y方向上对所有节点进行比较的最大旋转（弯曲）幅值。
 
+此外，以下条件也必须满足：
 $$
 \frac{1}{2}\frac{L}{G_{r a t i o}}\theta_{z m a x}>K\cdot u_{x y_{m a x}}
 $$  
 
-The left-hand side of the condition is the approximate tangential displacement resulting from the torsional rotation. This assumes a small angle approximation and uses the constant $G_{r a t i o}=15,$ which is the typical ratio between the length and the cross-section of a wind turbine structure (both tower and blades). $L$ is a characteristic length: for whole-blade modes, it is the length of the blade; for blade part modes, it is the part length; and for towers, it is the tower height.  
+The left-hand side of the condition is the approximate tangential displacement resulting from the torsional rotation. This assumes a small angle approximation and uses the constant $G_{r a t i o}=15,$ which is the typical ratio between the length and the cross-section of a wind turbine structure (both tower and blades). $L$ is a characteristic length: for whole-blade modes, it is the length of the blade; for blade part modes, it is the part length; and for towers, it is the tower height. 
 
-# 3. Determine if the Mode is Transverse $(u_{x},u_{y})$  
+条件左侧是由于扭转旋转引起的近似切向位移。这基于小角度近似，并使用常数 $G_{r a t i o}=15$，这是风轮结构（包括风轮叶片和风轮）的典型长度与横截面之比。$L$ 是一个特征长度：对于整个叶片的简正模式，它是叶片长度；对于叶片部分的简正模式，它是部分长度；对于风轮塔架，它是塔架高度。
+
+##### 3. Determine if the Mode is Transverse $(u_{x},u_{y})$  确定模式是否为横向 $(u_{x},u_{y})$
 
 If the mode is neither axial nor torsional, it must be a mode in either the $u_{x}$ Or $u_{y}$ direction (transverse). This is determined by evaluating the largest $\times$ or y displacement at the maximum displacement magnitude, Urymax'  
+如果该模式既非轴向也非扭转，那么它必须是$u_{x}$或$u_{y}$方向上的模式（横向）。这可以通过评估最大位移幅值处最大的x或y位移来确定，即${{u_{xy}}_{max}}$。
 
-# List of Mode Shape Names  
+##### List of Mode Shape Names  
 
 Table 1: Translation of mode types to blade or tower mode names. Note that the axes refer to the proximal frame of the component.  
+表1：模态类型到叶片或塔架模态名称的翻译。请注意，坐标轴指的是部件的近端参考系。
 
-Mode Type  
+| Mode Type                | Blade              | Tower               |
-
+| ------------------------ | ------------------ | ------------------- |
-Blade  
+| Longitudinal ($u_z$)     | Axial              | Vertical            |
-
+| Torsional ($\theta_{z}$) | Torsional          | Torsional           |
-Tower  
+| Transverse ($u_x, u_y$)  | Flapwise, Edgewise | Fore-aft, Side-side |
-
-Longitudinal $\left(u_{z}\right)$  
-
-Vertical  
-
-Tower  
-
-Torsional $(\theta_{z})$  
-
-Torsional  
-
-Torsional  
-
-Transverse $(u_{x},u_{y})$  
-
-Flapwise, Edgewise  
-
-Fore-aft, Side-side  
 
 Last updated 11-12-2024  
 
-# Modal Analysis Output  
+## Modal Analysis Output  
 
-# Generalised mass and stiffness properties  
+### Generalised mass and stiffness properties  
 
 During modal analysis, Bladed calculates the "generalised mass matrix" and "generalised stiffness matrix" by transforming the flexible component finite element mass/stiffness matrices using the normalised mode shape matrix $\Psi$ as shown below. The purpose of this is to transform the finite element mass and stiffness matrices into modal space (often referred to as the generalised coordinates).  
+在模态分析过程中，Bladed 通过使用归一化模态形状矩阵 $\Psi$ 转换柔性部件有限元质量/刚度矩阵，从而计算出“广义质量矩阵”和“广义刚度矩阵”，具体如下所示。 这样做是为了将有限元质量和刚度矩阵转换到模态空间（通常被称为广义坐标）。
 
 $$
 \begin{array}{r}{\mathbf{M}_{g e}=\boldsymbol{\Psi}^{T}\mathbf{M}_{f e}\boldsymbol{\Psi}}\\ {\mathbf{K}_{g e}=\boldsymbol{\Psi}^{T}\mathbf{K}_{f e}\boldsymbol{\Psi}}\end{array}
@@ -636,23 +800,55 @@ where
 · $\mathbf{M}_{g e}$ and ${\bf K}_{g e}$ are the modal mass and stiffness matrices. They are described in terms of $N_{m}$ -by- $.N_{m}$ square matrices, where $N_{m}$ is the total number of modes specified for the flexible component $\Psi$ is the mode shape matrix, which is used to transform between the finite element and modal domain. Consequently, the mode shape matrix has the dimensions below, with one mode shape defined on each column.  
 
 The diagonal elements of this generalised mass and stiffness matrices are then reported by Bladed for each mode.  
+其中
 
-![](images/3a667c36c91f37aaa8c16230d06a2749a9e91ee4a0b09280cbd05651745286fe.jpg)  
+· $\mathbf{M}_{f e}$ 和 $\mathbf{K}_{f e}$ 分别是完整柔性构件的有限元质量矩阵和刚度矩阵。它们以 $6N_{n}$ -by- $6N_{n}$阶方阵表示，其中 $N_{n}$ 是柔性构件的总节点数。
+· $\mathbf{M}_{g e}$ 和 ${\bf K}_{g e}$ 分别是模态质量矩阵和刚度矩阵。它们以 $N_{m}$ 阶方阵表示，其中 $N_{m}$ 是为柔性构件 $\Psi$ 指定的总模态数。$\Psi$ 是模态形状矩阵，**用于在有限元域和模态域之间进行转换**。因此，模态形状矩阵具有以下尺寸，每个列定义一个模态形状。
 
-# Natural frequency calculation  
+Bladed 会为每个模态报告这些广义质量矩阵和刚度矩阵的对角线元素。
+
+$$
+\begin{equation}
+{\Psi} = \ \begin{bmatrix}
+. & . & . & . & . & . &   &   &   & \\
+. & . & . &   &   &   &   &   &   & \\
+. & . & . &   &   &   &   &   &   & \\
+. &   &   & . &   &   &   &   &   & \\
+. &   &   &   & . &   &   &   &   & \\
+. &   &   &   &   &   &   &   &   & \\
+  &   &   &   &   &   &   &   &   & \\
+  &   &   &   &   &   &   &   &   & \\
+< & - & - & - & {N_{n}} & - & - & - & - & >
+\end{bmatrix}\begin{matrix}
+/\backslash \\
+| \\
+| \\
+| \\
+\ \ 6{N_{m}} \\
+| \\
+| \\
+| \\
+\backslash\text{/}
+\end{matrix}
+\end{equation}
+$$
+
+### Natural frequency calculation  
 
 Using the Rayleigh principle (Clough and Penzien, 1993), the mode natural frequencies can be calculated as  
+运用瑞利原理（Clough and Penzien, 1993），简正模式的固有频率可计算如下：
 
 $$
 \omega_{i}^{2}=\;\frac{K_{g e,i}}{M_{g e,i}},
 $$  
 
 where $K_{g e,i}$ and $M_{g e,i}$ are the terms on the leading diagonal of the generalised stiffness and mass matrices.  
-
+其中，$K_{g e,i}$ 和 $M_{g e,i}$ 分别是广义刚度和广义质量矩阵的主对角线上的项。
-# Note on the "effective modal mass"  
+### Note on the "effective modal mass"  
 
 There is also a quantity called the "effective modal mass" associated with each mode shape, which is not reported by Bladed. This quantity can represent (for example) the part of the total mass in each mode shape that responds to a specified unit displacement (for example a unit translation of the whole component in a certain direction). The "effective modal mass" for a particular mode, i, can be expressed as:  
 
+每个mode shape都存在一个相关的“有效模态质量”，而Bladed软件未报告此项数据。该量可以表示（例如）在每个mode shape中，对特定单位位移（例如，某个方向上整个部件的单位平动）所响应的总质量部分。对于特定模式i的“有效模态质量”，可以表示为：
 $$
 M_{\mathrm{eff},i}=\frac{l_{i}^{2}}{M_{g e,i}},
 $$  
@@ -663,65 +859,97 @@ $$
 \mathbf{l}=\mathbf{\rho}\left[\begin{array}{c}{l_{1}}\\ {l_{2}}\\ {\cdot}\\ {\cdot}\\ {l_{\mathbf{N}_{m}}}\end{array}\right]=\,\Psi^{T}\mathbf{M}_{f e}\mathbf{r}
 $$  
 
-and r is the displacement vector of each degree of freedom when the component is subject to a unit displacement.  
+and $r$ is the displacement vector of each degree of freedom when the component is subject to a unit displacement.  
 
 It is possible to add up the "effective modal mass" for each mode and compare that to the total component mass to evaluate the contribution of each mode to a certain displacement field. This summation of "effective modal mass" is sometimes used to help choose the number of modes required for a simulation.  
 
 However, it is not clear in general which set of displacements should be used to calculate the "effective modal mass". For example, a unit displacement of the whole component is appropriate for (say) simulating an earthquake, but does not give information in general as to the significance of a mode when subject to arbitrary forcing and displacement. For this reason, the "effective modal mass" is not reported by Bladed.  
 
+并且 $r$ 是每个自由度在部件承受单位位移时产生的位移矢量。
+
+可以对每个模式累加“有效模态质量”，并将其与总部件质量进行比较，以评估每个模式对特定位移场的影响。这种“有效模态质量”的累加有时用于帮助选择模拟所需的模式数量。
+
+然而，一般来说，尚不清楚应使用哪一组位移来计算“有效模态质量”。例如，整个部件的单位位移对于模拟地震是合适的，但通常无法提供关于在任意强制和位移作用下模式重要性的信息。因此，Bladed 不报告“有效模态质量”。
+
 Last updated 11-12-2024  
 
-# Geometric Stiffness Models  
+## Geometric Stiffness Models  
 
 Geometric stiffening models account for changes in structural response due to structural deflection from the reference (not deflected) state. Bladed provides models that include contributions from element axial and shear internal forces.  
+几何刚度模型考虑了结构变形（相对于未变形的参考状态）引起的结构响应变化。Bladed 提供包含单元轴向和剪切内力贡献的模型。
 
-# Geometric stiffness due to element axial forces  
+### Geometric stiffness due to element axial forces  
 
 Traditional geometric stiffness models account for the effect of element internal axial forces on structural stiffness. This is illustrated schematically in Figure 1. Centrifugal loads in the structural elements lead to a restoring load that tends to increase the stiffness of the blade. A linear finite element model for an initially straight blade is illustrated on the left side of Figure 1. A centrifugal force applied to the blade in its deflected position does not cause a bending moment along the blade. This is normal for linear finite element (FE) models as deflections are assumed to be small. On the right side of the diagram, the effect of geometric stiffness is illustrated. As the centrifugal load is applied in the deflected blade position, a bending moment is generated in the blade. This extra bending moment can change the blade flapwise and edgewise frequencies.  
 
+传统的几何刚度模型考虑了单元内轴向力对结构刚度的影响。如图1所示的示意图所示。**结构单元中的离心载荷导致一个恢复力，该力倾向于增加叶片的刚度**。图1左侧展示了一个初始为直线的叶片的线性有限元模型。当叶片处于变形位置时，施加的离心力不会在叶片上产生弯矩。对于线性有限元 (FE) 模型，这是正常的，因为假设变形是微小的。图的右侧说明了几何刚度的影响。当离心载荷施加在变形的叶片位置时，叶片上会产生弯矩。这个额外的弯矩可以改变叶片的挥舞和摆振频率。
+
 ![](images/c48399e0e79b7a2b47ed45609a405baddc76883869eb435b58fbf5c77367a5bd.jpg)  
 Figure 1: Element axial forces causing bending moments in a blade.  
 
 Geometric stiffness forces in the axial direction are responsible for the well-known "centrifugal stiffening" effect, where blade vibrational frequencies increase with rotor speed.  
 
-# Geometric stiffness due to element shear forces  
+轴向几何刚度力导致了著名的“离心加固”效应，即风轮转速越高，叶片振动频率越高。
+
+### Geometric stiffness due to element shear forces  
 
 There are also geometric stiffening forces associated with element internal shear forces. Figure 2 illustrates how torsion moments can be generated in the blade by application of shear forces to the blade in its displaced position. On the right side of the diagram, as the drag or lift load is applied in the deflected blade position, a torsional moment is generated in the blade. This extra torsional moment can affect the blade torsional dynamics.  
+此外，元件内部剪切力还会产生几何刚度力。如图2所示，**当剪切力作用于位移后的叶片时，会产生扭转力矩**。在图的右侧，当阻力或升力作用于变形后的叶片时，会在叶片上产生扭转力矩。这个额外的扭转力矩会影响叶片的扭转动力学。
 
 ![](images/ebe52afdc3940fe0f0ddc81bcb38324d4f988c9469b26583a9d63c32b1082029.jpg)  
 Figure 2: Element shear forces causing torsion moments in a blade  
 
-When evaluating the geometric stiffness effect of shear forces, it is important to account for the change in orientation of the torsion axis of the blade elements due to deflection. This is illustrated in Figure 3, where the difference in internal torsional load between the "reference" and deflected coordinate system is shown. Whether this affect is included depends on whether IgnoreAxesorientationDifferencesForshear is set true or false . For more details see the theory section about Translation and orientation offset between neutral and shear axes.  
+When evaluating the geometric stiffness effect of shear forces, it is important to account for the change in orientation of the torsion axis of the blade elements due to deflection. This is illustrated in Figure 3, where the difference in internal torsional load between the "reference" and deflected coordinate system is shown. Whether this affect is included depends on whether IgnoreAxesOrientationDifferencesForShear is set true or false . For more details see the theory section about Translation and orientation offset between neutral and shear axes.  
+
+在评估剪切力引起的几何刚度效应时，需要考虑叶片（Blade）元素由于变形（Deflection）导致的扭转轴方向变化。如图3所示，该图展示了“参考”坐标系和变形坐标系之间的内部扭转载荷差异。是否需要考虑这种影响取决于IgnoreAxesOrientationDifferencesForShear是否设置为true or false。 更多详情请参见关于中性轴和剪切轴之间的平移和方向偏移的理论部分。
 
 ![](images/9c4d734bb0d937679f80dd7bd1d6e2e0f39ba3f179fcb63197dc4653444ffe10.jpg)  
 Figure 3: Element shear forces causing torsion moments in a blade  
 
 Last updated 13-12-2024  
 
-# Bend-Twist Coupling Relationships in Beam Elements  
+## Bend-Twist Coupling Relationships in Beam Elements  
 
 This article describes how bend-twist coupling effects can be accounted for in Bladed beam elements. The built-in Bladed model of bend-twist coupling due to shear centre offset from the neutral axis are described. Additionally, the specification of user-defined bend-twist coupling terms is discussed.  
-
+本文描述了如何在Bladed梁单元中如何考虑弯扭耦合效应。内置的Bladed弯扭耦合模型是由于剪切中心偏离中性轴的情况。此外，还讨论了用户自定义弯扭耦合项的设置。
-# Co-incident shear and neutral axes  
+### Co-incident shear and neutral axes 重合剪切和中性轴 
 
 If the elastic centre and shear centre coincide, the constitutive relationship between strain and internal load for a beam element can be expressed as a diagonal matrix as shown below. Note that this equation is formulated in the local element coordinate system (i.e. it is rotated according to blade structural twist, prebend and sweep).  
+如果弹性中心和剪切中心重合，梁单元应力和内力之间的本构关系可以表示为如下对角矩阵。需要注意的是，该方程是根据叶片扭角、预弯和展向结构旋转而建立的，即采用叶片局部坐标系。
 
-![](images/6a3c44ac0ff44832599cf6e68b2c38da0375f6ead341f2470f6cec1bb9ba4305.jpg)  
+$$
+\begin{equation}
+
+\left[\begin{matrix}{F_x}\\{F_y}\\{F_z}\\{M_x}\\{M_y}\\{M_z}\\\end{matrix}\right]=\left[\begin{matrix}{EA}&\ &\ &|&\ &\ &\ \\0&{{GA}}_y&\ &|&\ &{sym}&\ \\0&0&{{GA}}_z&|&\ &\ &\ \\-&-&-&-&-&-&-\\0&0&0&|&{{GI}}_x^\ast&\ &\ \\0&0&0&|&0&{EI_y}&\ \\0&0&0&|&0&0&{{EI}_z}\\\end{matrix}\right]\left[\begin{matrix}{\gamma_x}\\{\gamma_y}\\{\gamma_z}\\{\kappa_x}\\{\kappa_y}\\{\kappa_z}\\\end{matrix}\right]=\ {\bar{\bar{C}}}\left[\begin{matrix}{\gamma_x}\\{\gamma_y}\\{\gamma_z}\\{\kappa_x}\\{\kappa_y}\\{\kappa_z}\\\end{matrix}\right]
+
+\end{equation}
+$$
 
 The $6\!\times\!6$ constitutive matrix is referred to in this document as $\bar{\bar{\mathbf{C}}},$ where the double over-bar denotes the local element coordinate system.  
 
-It is also noted that this equation may easily be transformed to the local principal coordinate system if the principal axis direction is constant for successive beam elements in the blade. In this case, the principal x-axis equals the element z-axis and the principal y-axis equals the element yaxis in the opposite direction, which implies that the principal z-axis equals the element x-axis (neutral axis). Further details of the relation between the local principal coordinate system and the element coordinate system may be found in the article about Blade Local Element Axes System.  
+It is also noted that this equation may easily be transformed to the local principal coordinate system if the principal axis direction is constant for successive beam elements in the blade. In this case, the principal x-axis equals the element z-axis and the principal y-axis equals the element yaxis in the opposite direction, which implies that the principal z-axis equals the element x-axis (neutral axis). Further details of the relation between the local principal coordinate system and the element coordinate system may be found in the article about Blade Local Element Axes System.
 
-# Translational offset between neutral and shear axes  
+本文件中，$6\!\times\!6$ 组分矩阵被称为 $\bar{\bar{\mathbf{C}}}$, 其中双重横线表示局部单元坐标系。
+
+需要注意的是，如果叶片的连续梁单元的主轴方向恒定，该方程可以很容易地转换为局部主坐标系。在这种情况下，主 x 轴等于单元 z 轴，主 y 轴等于单元 y 轴（方向相反），这意味着主 z 轴等于单元 x 轴（中性轴）。关于局部主坐标系与单元坐标系之间的关系详情，请参阅关于叶片（叶片）局部单元轴系的文章。
+### Translational offset between neutral and shear axes中性轴与剪切轴的平移偏移  
 
 It is possible to define a translational offset between the neutral axis and the shear centre within the blade section, as illustrated in Figure 1.  
 
+如图1所示，叶片截面中可以定义中性轴与剪切中心之间的平移偏移量。
+
 ![](images/2232c8006b1981075f53ab84f9ac8e8216e9b12e69cab2faf4743d7390036300.jpg)  
 Figure 1: Shear centre offset from neutral axis  
 
 The translational offset between shear and neutral axes is taken into account using the following calculation, which transforms the shear properties onto the neutral axis position.  
+考虑到剪切轴和中性轴之间的平移偏移，采用以下计算将其将剪切特性转换到中性轴位置。
+$$
+\begin{equation}
 
-![](images/9fc72b2fea82719d5e0d97c537f08630b11f1aa83453593ef611fb2df3a700ec.jpg)  
+\left[\begin{matrix}{F_x}\\{F_y}\\{F_z}\\{M_x}\\{M_y}\\{M_z}\\\end{matrix}\right]=\left[\begin{matrix}{I}&{0}\\{Y_S}^T&{I}\\\end{matrix}\right]\left[\begin{matrix}{EA}&\ &\ &|&\ &\ &\ \\0&{{GA}}_y&\ &|&\ &{sym}&\ \\0&0&{{GA}}_z&|&\ &\ &\ \\-&-&-&-&-&-&-\\0&0&0&|&{{GI}}_x^\ast&\ &\ \\0&0&0&|&0&{EI_y}&\ \\0&0&0&|&0&0&{{EI}_z}\\\end{matrix}\right]\left[\begin{matrix}{I}&{Y_S}\\{0}&{I}\\\end{matrix}\right]\left[\begin{matrix}{\gamma_x}\\{\gamma_y}\\{\gamma_z}\\{\kappa_x}\\{\kappa_y}\\{\kappa_z}\\\end{matrix}\right],
+
+\end{equation}
+$$
 
 where  
 
@@ -732,38 +960,71 @@ $$
 Entries $y_{c s}$ and $z_{c s}$ define the position of the shear centre relative to the neutral axis.  
 
 Expanding the above expression gives the following constitutive relationship around the neutral axis. The effect of shear centre offset is to introduce additional coupling between shear strain and torsional moment, and between bending strain and shear force.  
+$y_{c s}$ 和 $z_{c s}$ 表达式定义了剪切中心相对于中性轴的位置。
 
+展开上述表达式得到以下本构关系，即关于中性轴的本构关系。 剪切中心偏移的影响是引入了剪切应变和扭转力矩之间的额外耦合，以及弯曲应变和剪切力之间的额外耦合。
 $$
-\left[\!\!{\begin{array}{c}{F_{x}}\\ {F_{y}}\\ {F_{z}}\\ {M_{x}}\\ {M_{y}}\\ {M_{z}}\end{array}}\right]=\left[\!\!{\begin{array}{c c c c c c c c}{\mathbf{\big[}E A}&&&&{\big|}&&&\\ {0}&{G A_{y}}&&{\big|}&&{\mathbf{sym}}&\\ {0}&{0}&{G A_{z}}&{\big|}&&&\\ {-}&{-}&{-}&{-}&{-}&{-}&{-}\\ {0}&{-z_{c s}G A_{y}}&{y_{c s}G A_{z}}&{\big|}&{G I_{x}}&&\\ {0}&{0}&{0}&{\big|}&{0}&{E I_{y}}&\\ {0}&{0}&{0}&{\big|}&{0}&{0}&{E I_{z}}\end{array}}\!\!\right]\left[\!{\begin{array}{c}{\gamma_{x}}\\ {\gamma_{y}}\\ {\gamma_{z}}\\ {\kappa_{x}}\\ {\kappa_{y}}\\ {\kappa_{z}}\end{array}}\right]
+\begin{equation}
+
+\left[\begin{matrix}{F_x}\\{F_y}\\{F_z}\\{M_x}\\{M_y}\\{M_z}\\\end{matrix}\right]=\left[\begin{matrix}{EA}&\ &\ &|&\ &\ &\ \\0&{{GA}}_y&\ &|&\ &{sym}&\ \\0&0&{{GA}}_z&|&\ &\ &\ \\-&-&-&-&-&-&-\\0&{{-z}_{cs}}{{GA}}_y&{y_{cs}}{{GA}}_z&|&{{GI}}_x&\ &\ \\0&0&0&|&0&{EI_y}&\ \\0&0&0&|&0&0&{{EI}_z}\\\end{matrix}\right]\left[\begin{matrix}{\gamma_x}\\{\gamma_y}\\{\gamma_z}\\{\kappa_x}\\{\kappa_y}\\{\kappa_z}\\\end{matrix}\right]
+
+\end{equation}
 $$  
 
 where, $G I_{x}=G I_{x}^{\mathrm{~*~}}+\;G A_{y}z_{c s}^{2}+\;G A_{z}y_{c s}^{2}$ and $G{I_{x}}^{*}$ is the torsional stiffness defined around the shear (torsional) axis.  
-
+其中，$G I_{x}=G I_{x}^{\mathrm{~*~}}+\;G A_{y}z_{c s}^{2}+\;G A_{z}y_{c s}^{2}$，且 $G{I_{x}}^{*}$ 为围绕剪切（扭转）轴定义的扭转刚度。
-# Translation and orientation offset between neutral and shear axes  
+### Translation and orientation offset between neutral and shear axes  中性轴与剪切轴的平移和方向偏移
 
 Optional by selecting "ignore blade shear centre axis orientation transformation" in Additional Items.  
 
 In general, the elastic and shear axes are not parallel, so it can be important to take account of the orientation difference between them. The orientation difference between the shear axis and the elastic axis is illustrated by the $\theta$ terms in Figure 2.  
 
+可在“其他选项”中选择“忽略叶片剪切中心轴向变换”以进行可选设置。
+
+一般来说，弹性轴和剪切轴通常不平行，因此考虑它们之间的方向差异可能很重要。图2中的$\theta$项展示了剪切轴和弹性轴之间的方向差异。
+
 ![](images/5dadf76cb98bcff1f0577218862c9577ce405f9aac0dc87622dcbfda2bcfded4.jpg)  
 Figure 2: Orientation difference between shear and elastic axes  
 
 The combined translational and orientation offset between shear and neutral axes is taken into account using the following calculation, which transforms the shear properties onto the neutral axis position.  
+结合剪切面和中性轴之间的平移和姿态偏移，采用以下计算进行考虑，该计算将剪切特性转换到中性轴位置。
 
-![](images/680f8193c8469e09860cf395332fffdb1655a1d252d0b84377df481c7a5d1a09.jpg)  
+$$
+\begin{equation}
+
+\left[\begin{matrix}{F_x}\\{F_y}\\{F_z}\\{M_x}\\{M_y}\\{M_z}\\\end{matrix}\right]=\left[\begin{matrix}{I}&{0}\\{Y_S}^T&{R_s}^T\\\end{matrix}\right]\left[\begin{matrix}{EA}&\ &\ &|&\ &\ &\ \\0&{{GA}}_y&\ &|&\ &{sym}&\ \\0&0&{{GA}}_z&|&\ &\ &\ \\-&-&-&-&-&-&-\\0&0&0&|&{{GI}}_x^\ast&\ &\ \\0&0&0&|&0&{EI_y}&\ \\0&0&0&|&0&0&{{EI}_z}\\\end{matrix}\right]\left[\begin{matrix}{I}&{Y_S}\\{0}&{R_s}\\\end{matrix}\right]\left[\begin{matrix}{\gamma_x}\\{\gamma_y}\\{\gamma_z}\\{\kappa_x}\\{\kappa_y}\\{\kappa_z}\\\end{matrix}\right],
+
+\end{equation}
+$$
 
 where  
 
 $$
-\mathbf{Y_{S}}=\begin{array}{c c c}{\left[\begin{array}{c c c}{0}&{0}&{0}\\ {-z_{c s}}&{0}&{0}\\ {y_{c s}}&{0}&{0}\end{array}\right]}&{\qquad\quad\mathbf{R_{s}}=\frac{1}{L^{e}}\begin{array}{c c c}{\left[\begin{array}{c c c}{L_{s}^{e}}&{0}&{0}\\ {-\Delta y_{c s}}&{L^{e}}&{0}\\ {-\Delta z_{c s}}&{0}&{L^{e}}\end{array}\right]}\end{array}
+\begin{equation*}
+
+{Y_S}=\ \left[\begin{matrix}0&0&0\\{{-z}_{cs}}&0&0\\{y_{cs}}&0&0\\\end{matrix}\right]  \qquad \qquad
+
+{R_s}=\frac{1}{{L^e}}\ \left[\begin{matrix}{L_s^e}&0\ &0\\{-{\Delta y_{cs}}}&{L^e}&0\\{-{\Delta z_{cs}}}&0&{L^e}\\\end{matrix}\right]
+
+\end{equation*}
 $$  
 
 Entries $\Delta y_{c s}$ and $\Delta z_{c s}$ describe the change in position of the shear centre within the beam element, in order to describe the shear axis orientation.  
 
 The effect of shear centre translation and orientation offset is to introduce additional coupling between bending and torsional moments, resulting in the following constitutive relationship around the neutral axis.  
 
+条目 $\Delta y_{c s}$ 和 $\Delta z_{c s}$ 描述了梁单元剪切重心位置的变化，以描述剪切轴的取向。
+
+剪切重心平移和取向偏移的影响是引入了弯矩和扭矩之间的附加耦合，从而导致围绕中性轴的以下本构关系。
+
 $$
-\left[\begin{array}{c}{F_{x}}\\ {F_{y}}\\ {F_{z}}\\ {M_{x}}\\ {M_{y}}\\ {M_{z}}\end{array}\right]=\begin{array}{c c c c c c c}{\displaystyle\left[\begin{array}{c c c c c c c}{\scriptstyle E A}&&&&&{\mid}&&&\\ {0}&{\scriptstyle G A_{y}}&&&{\mid}&&&{\mathbf{sym}}&\\ {0}&{0}&{\scriptstyle G A_{z}}&&{\mid}&&&\\ {-}&{-}&{-}&{-}&{-}&{-}&{-}&{-}\\ {0}&{-z_{c s}G A_{y}}&{y_{c s}G A_{z}}&{\mid}&{\scriptstyle G I_{x}}&&\\ {0}&{\scriptstyle0}&&{0}&{\mid}&{-\frac{\Delta y_{c s}E I_{y}}{L e}}&{\scriptstyle E I_{y}}&\\ {0}&{0}&{0}&{\mid}&{-\frac{\Delta z_{c s}E I_{z}}{L e}}&{0}&{\scriptstyle E I_{z}}\end{array}\right]\left[\begin{array}{c}{\gamma_{x}}\\ {\gamma_{y}}\\ {\gamma_{z}}\\ {\kappa_{x}}\\ {\kappa_{y}}\\ {\kappa_{z}}\end{array}\right],
+\begin{equation}
+
+\left[\begin{matrix}{F_x}\\{F_y}\\{F_z}\\{M_x}\\{M_y}\\{M_z}\\\end{matrix}\right]=\ \left[\begin{matrix}{EA}&\ &\ &|&\ &\ &\ \\0&{{GA}}_y&\ &|&\ &{sym}&\ \\0&0&{{GA}}_z&|&\ &\ &\ \\-&-&-&-&-&-&-\\0&{{-z}_{cs}}{{GA}}_y&{y_{cs}}{{GA}}_z&|&{{GI}}_x&\ &\ \\0&0&0&|&-\frac{{\Delta y_{cs}}{EI_y}}{{L^e}}&{EI_y}&\ \\0&0&0&|&-\frac{{\Delta z_{cs}}EI_z}{{L^e}}&0&{{EI}_z}\\\end{matrix}\right]
+
+\left[\begin{matrix}{\gamma_x}\\{\gamma_y}\\{\gamma_z}\\{\kappa_x}\\{\kappa_y}\\{\kappa_z}\\\end{matrix}\right],
+
+\end{equation}
 $$  
 
 where  
@@ -773,36 +1034,50 @@ G I_{x}=\bigg(\frac{L^{e}}{L_{s}^{e}}\bigg)^{2}G I_{x}^{*}+G A_{y}z_{c s}^{2}+G
 $$  
 
 and $G{I_{x}}^{*}$ is the torsional stiffness defined around the shear (torsional) axis. This transformation results in extra bend-twist off-diagonal coupling terms, as well as a change to the torsional stiffness around the neutral axis.  
-
+and $G{I_{x}}^{*}$ 是围绕剪切（扭转）轴定义的扭转刚度。这种变换会导致额外的弯扭耦合非对角项，以及围绕中性轴的扭转刚度发生变化。
-# User-defined bend-twist coupling  
+### User-defined bend-twist coupling  
 
 The user can directly add extra off-diagonal terms to the constitutive matrix as shown  
-
+用户可以直接按照所示方式，在刚度矩阵中添加额外的非对角线项。
 $$
 \begin{array}{r}{\bar{\bar{\mathbf{C}}}=\left[\begin{array}{c c c c c c c}{E A}&&&{|}&&&\\ {0}&{G A_{y}}&&{|}&&{\mathrm{sym}}&\\ {0}&{0}&{G A_{z}}&{|}&&&\\ {-}&{-}&{-}&{-}&{-}&{-}&{-}\\ {0}&{0}&{0}&{|}&{G I_{x}^{*}}&&\\ {0}&{0}&{0}&{|}&{C_{x y}}&{E I_{y}}&\\ {0}&{0}&{0}&{|}&{C_{x z}}&{C_{y z}}&{E I_{z}}\end{array}\right]}\end{array}
 $$  
 
 The transformations described in previous sections based on shear axis position relative to neutral axis are also applied, resulting in the following relationship by selecting the option "ignore blade shear centre axis orientation transformation" in Additional Items.  
 
+前几节中基于剪切轴相对于中性轴的位置所描述的变换同样适用，通过在附加选项中选择“忽略叶片剪切重心轴向变换”选项，可得到以下关系。
+
 $$
-{\left[\begin{array}{l}{F_{x}}\\ {F_{y}}\\ {F_{z}}\\ {M_{x}}\\ {M_{y}}\\ {M_{z}}\end{array}\right]}\,=\,{\left[\begin{array}{l l l l l l l l}{\mathbf{I}}&{\mathbf{0}}&{}&{}&{}&{}&{}&{}&{}\\ {\mathbf{I}}&{\mathbf{0}}&{}&{}&{}&{}&{}&{}\\ {\mathbf{I}}&{\mathbf{0}}&{}&{}&{}&{}&{}&{}\\ {\mathbf{T}}&{-}&{-}&{-}&{-}&{-}&{-}&{}&{-}\\ {\mathbf{0}}&{}&{\mathbf{0}}&{\mathbf{0}}&{}&{|}&{G\mathbf{I}_{x}^{*}}&{}&{}\\ {\mathbf{0}}&{\mathbf{0}}&{\mathbf{0}}&{|}&{C_{x y}}&{E I_{y}}&{}&{}\\ {\mathbf{0}}&{\mathbf{0}}&{\mathbf{0}}&{|}&{C_{x z}}&{C_{y z}}&{E I_{z}}\end{array}\right]}{\left[\begin{array}{l l}{\mathbf{I}}&{\mathbf{Y}}\\ {\mathbf{Y}}\\ {\mathbf{0}}&{\mathbf{I}}\end{array}\right]}\,\Biggr[{\boldsymbol{Y}}\cdot\mathbf{Y}\left[{\begin{array}{l}{\mathbf{X}}\\ {\mathbf{Y}}\\ {\mathbf{X}}\\ {\mathbf{H}}\\ {\mathbf{X}}\end{array}}\right]}\end{array}\right]}
+\begin{equation}
+
+\left[\begin{matrix}{F_x}\\{F_y}\\{F_z}\\{M_x}\\{M_y}\\{M_z}\\\end{matrix}\right]=\left[\begin{matrix}{I}&{0}\\{Y_S}^T&{I}\\\end{matrix}\right]\ \left[\begin{matrix}{EA}&\ &\ &|&\ &\ &\ \\0&{{GA}}_y&\ &|&\ &{sym}&\ \\0&0&{{GA}}_z&|&\ &\ &\ \\-&-&-&-&-&-&-\\0&0&0&|&{{GI}}_x^\ast&\ &\ \\0&0&0&|&{C_{xy}}&{{EI_y}}&\ \\0&0&0&|&{C_{xz}}&{C_{yz}}&{{EI}_z}\\\end{matrix}\right]\left[\begin{matrix}{I}&{Y_S}\\{0}&{I}\\\end{matrix}\right]\left[\begin{matrix}{\gamma_x}\\{\gamma_y}\\{\gamma_z}\\{\kappa_x}\\{\kappa_y}\\{\kappa_z}\\\end{matrix}\right]
+
+\end{equation}
 $$  
 
 The following transformation is used as default without enabling the option.  
 
-<html><body><table><tr><td rowspan="7">Fc Fy Fz I Mx [Ys T My Mz</td><td rowspan="7"></td><td colspan="3">EA 0 0</td><td rowspan="2"></td><td colspan="3"></td><td rowspan="2">Yy</td></tr><tr><td>GAy 0</td><td>GA</td><td></td><td>sym</td><td>[I</td></tr><tr><td>0 一 RsT 0</td><td>一</td><td>一</td><td>一 一</td><td></td><td>Ys 0 Rs</td><td>Y2</td><td>(9)</td></tr><tr><td></td><td>0</td><td>0</td><td>GI* c</td><td></td><td></td><td>Kc</td><td></td></tr><tr><td>0</td><td>0</td><td>0</td><td>Cry</td><td>EIy</td><td></td><td>Ky</td><td></td></tr><tr><td>0</td><td>0</td><td>0</td><td>Cxz</td><td>EIz C yz</td><td></td><td>Kz.</td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table></body></html>  
+$$
+\begin{equation}
+
+\left[\begin{matrix}{F_x}\\{F_y}\\{F_z}\\{M_x}\\{M_y}\\{M_z}\\\end{matrix}\right]=\left[\begin{matrix}{I}&{0}\\{Y_S}^T&{R_s}^T\\\end{matrix}\right]\ \left[\begin{matrix}{EA}&\ &\ &|&\ &\ &\ \\0&{{GA}}_y&\ &|&\ &{sym}&\ \\0&0&{{GA}}_z&|&\ &\ &\ \\-&-&-&-&-&-&-\\0&0&0&|&{{GI}}_x^\ast&\ &\ \\0&0&0&|&{C_{xy}}&{EI_y}&\ \\0&0&0&|&{C_{xz}}&{C_{yz}}&{{EI}_z}\\\end{matrix}\right]\left[\begin{matrix}{I}&Y_S\\{0}&{R_s}\\\end{matrix}\right]\left[\begin{matrix}{\gamma_x}\\{\gamma_y}\\{\gamma_z}\\{\kappa_x}\\{\kappa_y}\\{\kappa_z}\\\end{matrix}\right]
+
+\end{equation}
+$$
 
 It is noted that the transformation of these equations to the local principal coordinate system as described in Co-incident shear and neutral axes will introduce a change of the sign of the offdiagonal terms that relate to bending about the element y-axis.  
 
+需要注意的是，按照“共轭剪切面和中性轴”中所述，将这些方程变换到局部主坐标系，会改变与关于单元 y 轴弯曲相关的非对角项的符号。
+
 Last updated 13-12-2024  
 
-# Support Structure Superelement  
+## Support Structure Superelement  
 
 For offshore turbines, the jacket support structure is sometimes modelled as a superelement. The superelement can be included as a component in the multibody framework in a similar way to the other flexible components.  
+对于海上风机，套筒支撑结构有时会被建模为超单元。超单元可以像其他柔性部件一样，被包含在多体动力学框架中。
+### Model Creation模型创建  
 
-# Model Creation  
+The finite element model of the jacket must be converted into reduced form and exported from a offshore modelling code for example Sesam, SACS or ROSA. The Bladed superelement feature has been designed with (Craig-Bampton, 2000), reduction for the superelement in mind. Craig-Bampton is the preferred method as an accurate dynamic response of the jacket can be retained in the superelement. Other reduction methods could in principle be used but are not considered in this document.  
-
-The finite element model of the jacket must be converted into reduced form and exported from a offshore modelling code for example Sesam, SACS or ROSA. The Bladed superelement feature has been designed with (Craig-Bampton, 2000), reduction for the superelement in mind. CraigBampton is the preferred method as an accurate dynamic response of the jacket can be retained in the superelement. Other reduction methods could in principle be used but are not considered in this document.  
 
 Craig-Bampton modes for the superelement consist of constraint and norma/ mode shapes. An important concept is division of the jacket nodes into the boundary node and interior nodes. The boundary node is the node where the superelement connects to the tower base. The interior nodes are all of the other nodes in the jacket.  
 
@@ -810,10 +1085,28 @@ Constraint modes describe the displacement of the interior jacket nodes when six
 
 For the normal modes, the jacket structure is constrained at the base and the interface node and the eigenmodes are calculated. As the jacket is constrained at both ends, these are also referred to as interior modes (they include motion of the interior nodes only). The normal modes enhance the dynamic response of the superelement. The union of these two sets of modes can provide an accurate dynamic model of the jacket and motion of the interface node.  
 
-The outputs of the superelement creation process are a mass matrix and stiffness matrix for the reduced model. A wave load time history for each mode is also output. A reduction basis $[R]$ based on the constraint and normal mode shapes, is used to reduce the jacket finite element matricesasfollows  
+The outputs of the superelement creation process are a mass matrix and stiffness matrix for the reduced model. A wave load time history for each mode is also output. A reduction basis $[R]$ based on the constraint and normal mode shapes, is used to reduce the jacket finite element matrices as follows  
+导管架的有限元模型必须转换为降阶形式，并从海上建模代码（例如 Sesam、SACS 或 ROSA）中导出。Bladed 叶片超单元功能的设计考虑了（Craig-Bampton, 2000）的降阶方法。Craig-Bampton 方法是首选方法，因为它可以保留超单元中导管架的准确动态响应。原则上可以使用其他降阶方法，但本文不作考虑。
 
+Craig-Bampton 超单元模式由约束模式和简正模式组成。一个重要的概念是将导管架节点划分为边界节点和内部节点。边界节点是超单元连接到塔架基座的节点。内部节点是导管架的其他所有节点。
+
+约束模式描述了当在超单元界面节点（边界节点）处施加六个单位位移（3个平动和3个转动）时，内部导管架节点的位移。这些模式形状提供了超单元的静态响应，并允许超单元界面节点移动。
+
+对于简正模式，导管架结构在基座和界面节点处进行约束，并计算特征模式。由于导管架两端都受到约束，这些也称为内部模式（仅包括内部节点运动）。简正模式增强了超单元的动态响应。这两个模式集合的联合可以提供导管架的准确动态模型和界面节点运动。
+
+超单元创建过程的输出是降阶模型的质量矩阵和刚度矩阵。每个模式的波浪载荷时程也作为输出。基于约束模式和简正模式的基 $[R]$ 用于降阶导管架有限元矩阵，如下所示：
 $$
-\begin{array}{r l}&{\overline{{\mathbf{M}}}=\mathbf{R}^{T}\mathbf{M}_{F E}\mathbf{R}}\\ &{\overline{{\mathbf{K}}}=\mathbf{R}^{T}\mathbf{K}_{F E}\mathbf{R}}\\ &{\overline{{\mathbf{f}}}=\mathbf{R}^{T}\left[\mathbf{f}_{i}\right]}\end{array}
+\begin{flalign}
+
+\overline{{M}}  & = {R}^T {M}_{FE}{R} \\[1ex]
+
+\overline{{K}}  &=  {R}^T {K}_{FE}{R} \\[1ex]
+
+\overline{{f}} & =  {R}^T  \left[\begin{array}{c}
+
+{f}_b \\ {f}_i  \end{array} \right]
+
+\end{flalign}
 $$  
 
 where:  
@@ -821,13 +1114,24 @@ where:
 $\mathbf{R}$ is the reduction basis that transforms the jacket FE model into a superelement model   
 $\mathbf{M}_{F E}$ is the finite element mass matrix   
 $\bf{K}_{F E}$ is the finite element stiffness matrix  
+$\overline{{\bf{M}}}$ is the superelement mass matrix 
+$\overline{{\mathbf{K}}}$ is the superelement stiffness matrix 
+$\mathbf{f}_{b}$ is the external applied wave load at the boundary node 
+$\mathbf{f}_{i}$ is the external applied wave load at the interior nodes 
+$\bar{\mathbf{f}}$ is the superelement external applied wave load  
 
-$\overline{{\bf{M}}}$ is the superelement mass matrix $\overline{{\mathbf{K}}}$ is the superelement stiffness matrix $\mathbf{f}_{b}$ is the external applied wave load at the boundary node $\mathbf{f}_{i}$ is the external applied wave load at the interior nodes · f is the superelement external applied wave load  
+It is important to note that the first six degrees of freedom in the superelement model correspond to displacement of the interface node (boundary node). The remaining degrees of freedom correspond to the interior normal modes. Sufficient normal modes should be included to obtain an accurate dynamic response. The reduced equations of motion for the superelement are expressed as follows  
-
-It is important to note that the first six degrees of freedom in the superelement model correspond to displacement of the interface node (boundary node). The remaining degrees of freedom correspond to the interior normal modes. Sufficient normal modes should be included to obtain an accurate dynamic response. The reduced equations of motion for the superelement are expressed asfollows  
 
 $$
-\overline{{\mathbf{M}}}\left[\stackrel{\cdot\cdot}{\mathbf{\ddot{x}}_{b}}\right]+\overline{{\mathbf{K}}}\left[\stackrel{\mathbf{x}_{b}}{\mathbf{\Delta}}\right]=\overline{{\mathbf{f}}}
+\begin{equation}
+
+\overline{{M}} \left[\begin{array}{c} \ddot{{x}}_b \\ \ddot{{\eta}}_i  \end{array} \right] +
+
+\overline{{K}} \left[\begin{array}{c} {x}_b \\ {\eta}_i  \end{array} \right]
+
+= \overline{{f}}
+
+\end{equation}
 $$  
 
 where  
@@ -838,7 +1142,7 @@ $\pmb{\eta}_{i}$ is the amplitude of each normal mode (istands for interior)
 
 The method applied to complete modal reduction for the superelement method in Bladed is referred to as the Craig-Bampton method.  
 
-# Superelement Mode Shape Matrix  
+### Superelement Mode Shape Matrix  
 
 The reduced matrices $\overline{{\bf M}}$ and $\overline{{\mathbf{K}}}$ include allthe necessary properties to describe the dynamics of the superelement in Bladed. However, it is also necessary for Bladed to know how the interface node in the Bladed structural definition couples to the superelement modal displacements. In other words, how this interface node moves when each superelement degree of freedom is activated.  
 
@@ -848,7 +1152,7 @@ By default, a mode shape matrix for the superelement is assumed by Bladed as sho
 
 In the case that the superelement has been generated in a coordinate system different from that used in Bladed, an additional transformation is required.  
 
-# Superelement Damping Definition  
+### Superelement Damping Definition  
 
 The integrated and superelement approaches use a different structural mode basis, as illustrated in Figure 1. In the integrated approach, modes cover the entire support structure, whereas in the superelement approach, separate mode shapes are defined for the superelement and tower modes.  
 
@@ -859,7 +1163,7 @@ In Bladed, damping is defined on the modal degrees of freedom. This leads to the
 
 In this section, the methods and challenges for damping definition are discussed for both integrated and superelement approaches. A method is then described that overcomes these dificulties and allows equivalent damping to be defined on integrated and superelement approaches.  
 
-# Structural Damping for Integrated Models  
+### Structural Damping for Integrated Models  
 
 For an integrated model, typically modal damping ratios are defined on the uncoupled Craig Bampton vibration support structure modes.  
 
@@ -903,7 +1207,7 @@ $\zeta_{i}$ are the modal damping ratios,
 $\omega_{i}$ are modal angular frequencies (rad/s),   
 $\mathbf{K}_{\mathrm{SSii}}$ are the diagonal terms of the modal stiffness matrix.  
 
-# Structural Damping for Superelement Models  
+### Structural Damping for Superelement Models  
 
 Bladed allows damping to be defined separately on the tower and superelement, which each have their own mode shapes and damping definitions.  
 
@@ -933,7 +1237,7 @@ Firstly, the "cross terms" between the tower and superelement components are zer
 
 Secondly, the supplied superelement damping matrix does not typically account for the effect of the inertia of the tower and RNA.  
 
-# Unified Damping for Superelement and Integrated Methods  
+### Unified Damping for Superelement and Integrated Methods  
 
 It has been explained that integrated model damping and superelement model damping are not equivalent, for the following reasons:  
 
@@ -944,7 +1248,7 @@ It has been explained that integrated model damping and superelement model dampi
 
 To unify the damping for the superelement and integrated approaches, it is proposed to define the damping on a set of modes that is common to both approaches such as the coupled vibrational modes for the support structure. If the structural properties of the superelement are defined in a valid way, then the support structure coupled modes will be very similar to the integrated case.  
 
-# Theory Basis  
+### Theory Basis  
 
 The aim is to specify damping on the support structure coupled modes, and then transform this damping onto the actual degrees of freedom for the tower and superelement which are used in the simulation. This method is based on theory presented in (Clough, 1993).  
 
@@ -1002,7 +1306,7 @@ $$
 
 $C_{\mathrm{SS}}$ is a full matrix that has coupling terms between the tower and superelement components (for a superelement model), or coupling between the support structure modes (for an integrated model). The matrix $C_{S S}$ also includes the influence of the RNA inertia.  
 
-# Results Comparison  
+### Results Comparison  
 
 The effect of specifying damping on the support structure coupled modes is demonstrated in Figure 2.  
 
@@ -1018,7 +1322,7 @@ damping ratio for the first few coupled modes.
 
 Figure 2: Support structure coupled mode damping ratios for integrated and superelement models, including effect of specifying damping on the support structure coupled modes  
 
-# Specifying Superelement Damping  
+### Specifying Superelement Damping  
 
 In Bladed, it is recommended to specify damping on the support structure coupled modes using a unified damping.approach. However, this functionality is not available in Bladed 4.8, so for this version a different method must be used to achieve the target damping for the support structure modes when using a superelement method.  
 
@@ -1037,7 +1341,7 @@ Example results of this tuning exercise are illustrated in Figure 3, using the s
 ![](images/6a5dd588e420c40358f0c7050dc80f863482889b95a2d70bb9ad46e8a2cfc989.jpg)  
 Figure 3: Support structure coupled mode damping ratios for integrated and superelement models, including effect of tuning the damping for the superelement model.  
 
-# Craig-Bampton Reduction Basis  
+### Craig-Bampton Reduction Basis  
 
 The aim of the reduction is to reduce the number of degrees of freedom in the jacket structure and export the resulting matrices in a compact form ready for import to Bladed for dynamic analysis.  
 
@@ -1123,22 +1427,27 @@ $$
 
 Last updated 11-10-2024  
 
-# Modelling Foundation Supports  
+# Modelling Foundation Supports 建模基础支撑 
 
 This section describes how linear and non-linear foundation supports are taken into account in the model. A dynamic model for non-linear foundation loading is described in detail.  
-
+本节描述了如何在模型中考虑线性与非线性基础支撑。 详细介绍了非线性基础载荷的动力学模型。
-# Linear and non-linear foundations  
+## Linear and non-linear foundations  
 
 Linear foundations can be defined at structural nodes and have constant stiffness. The stiffness and mass is included directly in the support structure matrices in order to be included in the dynamic solution.  
 
 Non-linear foundations can be defined as a lookup table between the foundation node displacement and the reaction load from the foundation. These lookup tables are often referred to as "p-y curves" which refers to the lateral stiffness definition of a foundation, although lookup tables in all 6 nodal degrees of freedom can be defined. Non-linear curves result in a variable foundation stiffness (the p-y curve gradient), as illustrated in Figure 1.  
 
 Only a constant stiffness value can be included in the support structure stiffness matrix. Therefore, the initial slope of the curve (labelled $K_{01}$ in the figure) is included in the support structure stiffness matrix, and the non-linear part is included as an applied load.  
+线性基础可以定义在结构节点上，且具有恒定刚度。刚度和质量直接包含在支撑结构矩阵中，以便纳入动态解。
+
+非线性基础可以定义为基础节点位移与基础反作用力之间的查找表。这些查找表通常被称为“p-y 曲线”，指的是基础的横向刚度定义，尽管可以在所有 6 个节点自由度上定义查找表。非线性曲线导致基础刚度发生变化（p-y 曲线的梯度），如图 1 所示。
+
+只有恒定刚度值可以包含在支撑结构刚度矩阵中。因此，曲线的初始斜率（如图中标记为 $K_{01}$）包含在支撑结构刚度矩阵中，而非线性部分则作为施加载荷包含在内。
 
 ![](images/0fe3cc9ef4f678fc3e6e81d7a7c24f45f33b741fdb41c1851702e584d6b36263.jpg)  
 Figure 1: Example of a non-linear p-y curve  
 
-# Dynamic model for non-linear foundations  
+## Dynamic model for non-linear foundations  
 
 For non-linear foundation p-y curves, support structure modal deflections are often not accurate enough to be used for looking up the foundation load in the p-y curve. This is because the nonlinear nature of the p-y curve means that small differences in displacement can result in large  
 
@@ -1294,5 +1603,5 @@ Last updated 06-09-2024
 # Structural loads  
 
 The structural loads at the multibody nodes, e.g. the blade root, the power train and the tower base, are calculated as the joint reactions by the component equilibrium equation described in the multibody dynamics approach. The inertial loads are calculated by integration of the mass properties and the total acceleration vector at each station according to the principle of virtual work. The total acceleration vector includes modal, centrifugal, Coriolis and gravitational components. Second-order effects of axial forces due to structural deflections are taken into account, i.e. the contribution to bending moments caused by the applied axial forces acting on the deflected structure. For example, the contribution to tower base bending moment caused by the weight of the nacelle takes into account the true position of the nacelle centre of gravity including the deflection of the tower top.  
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+多体节点处的结构载荷，例如叶片根部 (blade)、传动链 (power train) 和塔架底座，通过多体动力学方法描述的部件平衡方程计算得出，作为连接反作用力。惯性载荷则根据虚拟功原理，对每个站点的质量属性和总加速度矢量进行积分计算。总加速度矢量包括简正模式 (normal mode)、离心力、科里奥利力和重力分量。考虑了由于结构变形引起的轴向力的一阶效应，即考虑施加的轴向力作用于变形结构所引起的弯矩贡献。例如，考虑了舱盖重量引起的塔架底座弯矩，其中考虑了舱盖重心在塔顶变形后的真实位置，包括变形。
 Last updated 30-08-2024