vault backup: 2025-07-07 10:25:13

多体+耦合求解器/参考文献/Craig-Coupling of substructures for dynamic an/auto/Craig-Coupling of substructures for dynamic an.md

@@ -39,11 +39,11 @@ a. Components and coupled system.
 
 ![](973bf8ca2cdd7d1654e066b492c9c21debc986eb3758ee1e1a9a806b1682d9fe.jpg)  
 
-b. Typical component with redundant boundary.  
+b. Typical component with redundant boundary.  $\pmb{{\cal B}} = \mathcal{R}+\mathcal{E}$
 
-As noted in Fig. 2, the coordinate sets $\tau,\mathcal{R},\mathcal{E}$ and $\pmb{{\cal B}}$ denote interior coordinates (i.e., not shared with an adjacent component), rigid-body coordinates, excess coordinates (i.e., redundant boundary coordinates), and boundary coordinates (i.e., shared with adjacent components). The numbers of coordinates in these sets are $N_{i},\,N_{r},\,N_{e},$ and $N_{b}$ , respectively, with $N_{b}=N_{r}+N_{e}$ and $N=N_{i}+N_{b}$  
+As noted in Fig. 2, the coordinate sets $I,\mathcal{R},\mathcal{E}$ and $\pmb{{\cal B}}$ denote interior coordinates (i.e., not shared with an adjacent component), rigid-body coordinates, excess coordinates (i.e., redundant boundary coordinates), and boundary coordinates (i.e., shared with adjacent components). The numbers of coordinates in these sets are $N_{i},\,N_{r},\,N_{e},$ and $N_{b}$ , respectively, with $N_{b}=N_{r}+N_{e}$ and $N=N_{i}+N_{b}$  
 
-如图2所示，坐标系 $\tau,\mathcal{R},\mathcal{E}$ 和 $\pmb{{\cal B}}$ 分别表示内坐标（即与相邻部件不共享）、刚体坐标、过余坐标（即冗余边界坐标）和边界坐标（即与相邻部件共享）。这些坐标系中坐标的数量分别为 $N_{i},\,N_{r},\,N_{e},$ 和 $N_{b}$ ，其中 $N_{b}=N_{r}+N_{e}$ 且 $N=N_{i}+N_{b}$。
+如图2所示，坐标系 $I,\mathcal{R},\mathcal{E}$ 和 $\pmb{{\cal B}}$ 分别表示内坐标（即与相邻部件不共享）、刚体坐标、过余坐标（即冗余边界坐标）和边界坐标（即与相邻部件共享）。这些坐标系中坐标的数量分别为 $N_{i},\,N_{r},\,N_{e},$ 和 $N_{b}$ ，其中 $N_{b}=N_{r}+N_{e}$ 且 $N=N_{i}+N_{b}$。
 
 The equation of motion of $\mathbf{a}$ typical undamped component, labeled $^c$ may be written as  
 典型无阻尼构件 $\mathbf{a}$，标记为$c$，其运动方程可写为：

线性化求解器/参考文献/Jonkman-Full-system linearization for floating/auto/Jonkman-Full-system linearization for floating.md

@@ -40,6 +40,7 @@ Jason M Jonkman, Alan D Wright, Greg J Hayman, Amy N Robertson National Renewabl
 # ABSTRACT  
 
 The wind engineering community relies on multiphysics engineering software to run nonlinear time-domain simulations (e.g.,  for  design-standards-based  loads  analysis). Although most  physics  involved  in  wind  energy  are  nonlinear, linearization of the underlying nonlinear system equations is often advantageous to understand the system properties and exploit well-established methods and tools for analyzing linear systems. This paper presents the development of the new linearization functionality of the open-source engineering tool OpenFAST for floating offshore wind turbines, as well as the concepts and mathematical background needed to understand and apply it.  
+风能工程领域依赖于多物理场工程软件来进行非线性时域仿真（例如，用于基于设计标准荷载分析）。 尽管风能涉及的大部分物理现象是非线性的，但对底层非线性系统方程进行线性化通常有利于理解系统特性，并利用成熟的方法和工具来分析线性系统。本文介绍了开源工程工具OpenFAST针对海上漂浮风电机组的新线性化功能开发，以及理解和应用该功能所需的概念和数学基础。
 
 # INTRODUCTION  
 
@@ -56,6 +57,19 @@ The  overall  linearization  approach  that  the  FAST modularization  framework
 The mathematical background presented in this paper is currently being implemented in the OpenFAST source code. Unfortunately, the implementation at the time of this writing has not yet been completed enough to produce results. Results will be presented in future work to highlight the functionality and verify the implementation (e.g., for a sample case whereby the natural frequencies and damping of the NREL 5-MW baseline wind turbine atop a spar buoy will be calculated as a function of rotational speed and presented in Campbell-diagram form).  
 
 The linearization of OpenFAST involves: 1) finding an operating point (OP), 2) linearizing the underlying nonlinear equations of each module about the OP, 3) linearizing the module-to-module input-output coupling relationships in the OpenFAST glue code about the OP, and 4) combining all linearized matrices into the full-system linear state-space model and exporting those matrices and the OP to a file. Each step is highlighted in the following sections.  
+为了支持设计和分析——从而使风电机组具有创新性、优化性、可靠性和成本效益——风能行业和研究界依赖于能够预测风系统耦合动态载荷和响应的工程软件（即设计工具）。开放式FAST（以前称为FAST），由美国能源部国家可再生能源实验室（NREL）通过支持开发，是一个开源的多物理场工具，可用于设计风电机组[1]。开放式FAST模拟了重要的物理现象和系统耦合，包括环境激励（风、波浪和洋流）和全系统动态响应（风轮、传动系统、鼻锥、支撑结构和控制器），在正常（用于疲劳）和极端（用于极限）载荷条件下均可进行模拟。开放式FAST模型能够分析各种风电机组配置，包括两叶或三叶水平轴风轮、俯仰或失速调节、刚性或倾覆集卡、迎风或背风风轮，以及格子或管状塔架。风电机组可以在陆地上或海上，在底部固定或浮动结构上进行建模。
+
+开放式FAST的主要用途是运行非线性时域模拟（例如，用于基于设计标准的载荷分析）。尽管风能涉及的大多数物理现象是非线性的，但线性化底层非线性系统方程通常是有利的，以便理解系统响应并利用分析线性系统的成熟方法和工具。例如，线性状态空间模型可以转换为传递函数、脉冲响应函数或频域响应函数。生成线性化模型的能力对于特征分析（以推导结构的固有频率、阻尼比和模态）控制设计（基于线性状态空间模型）、稳定性分析、梯度用于优化问题以及支持低阶模型开发至关重要。
+
+以前的FAST线性化工作侧重于1）对FAST v8源代码进行结构化以实现线性化（FAST v8是开放式FAST的前身）；2）开发线性化模块间、输入输出耦合关系中网格映射的通用方法，包括旋转；3）线性化FAST陆基模块（InflowWind、AeroDyn、ServoDyn和ElastoDyn）及其核心（但非全部）特征及其耦合；以及4）通过将工具应用于样本案例来验证此实现[2]。最近，引入并验证了使用BeamDyn模块对具有气动弹性定制叶片的开放式FAST模型进行线性化的能力[3]。这项工作将这些努力扩展到线性化开放式FAST中浮式海上风电机组（FOWT）模块及其核心（但非全部）特征及其耦合，包括HydroDyn中的水动力学和MAP++中的系泊，并验证更新后的实现。
+
+开放式FAST的HydroDyn水动力学模块[1]允许采用多种方法来计算浮动平台上的水动力载荷，包括势流理论解、剥离线理论解或两者之间的混合组合。波浪可以是规则（周期性）或不规则（随机），可以是长波（单向）或短波（波能分布在多个方向上）。波动运动学和/或势流解的二阶项是可用的选项。开放式FAST的MAP++模块[1]准静态地模拟系泊系统，考虑了链条或拉紧线的几何非线性、弹性拉伸、线重和浮力，以及块重和浮力箱、线与线互连和海底摩擦。
+
+开放式FAST模块化框架旨在支持的整体线性化方法如[4]中所述，与本实施方案一致。有关陆基模块及其耦合的线性化细节，包括网格映射，如[2,3]中所述。在不复制大部分信息的前提下，本文采用[2,3,4]相同的策略和术语，增加了对HydroDyn和MAP++浮式海上模块及其耦合的线性化细节，包括网格映射。
+
+本文中介绍的数学背景目前正在实施到开放式FAST源代码中。不幸的是，截至本文写作时，该实施方案尚未完成到足以产生结果。未来工作将展示结果，以突出功能并验证实施（例如，对于NREL 5兆瓦基线风电机组顶部位于单筒浮体上的自然频率和阻尼，作为旋转速度的函数计算并以坎贝尔图形式呈现）。
+
+开放式FAST的线性化涉及：1）找到一个工作点（OP）；2）线性化每个模块的底层非线性方程，关于OP；3）线性化开放式FAST“胶水”代码中模块间的输入输出耦合关系，关于OP；以及4）将所有线性化矩阵组合成全系统线性状态空间模型，并将这些矩阵和OP导出到文件中。每个步骤将在以下部分中突出显示。
 
 # OPERATING-POINT DETERMINATION  
 
@@ -65,6 +79,12 @@ An OP is defined by given values for the continuous-time states, $x\big|_{o p}$
 
 The number of states, inputs, and outputs (i.e., the size of the vectors $x\;,\;x^{d}\;,\;z\;,\;u$ , and $y$ ) depend on the features enabled in OpenFAST.  
 
+OP (或固定点) 确定是线性化过程中的重要第一步，因为非线性系统的线性表示仅对 OP 周围的小偏差（扰动）有效。在 OpenFAST 当前版本中，OP 可以由给定的初始条件（零时刻）或非线性时间步进过程中的给定时间（或多个时间）来定义。通常，OP 应该是一个静态平衡条件（对于停机/空转风轮）或稳态条件（对于运行中的风轮）；否则，它可能会对线性系统矩阵产生不良影响。
+
+OP 由每个模块的连续时间状态值 $x\big|_{o p}$ 、离散时间状态值 $\left.x^{d}\right|_{o p}$ 、输入值 $u\Big|_{o p}$ 和时间值 $t\Big|_{o p}$ 定义。然后可以使用 [4] 中的方程 (1a)、(1c) 和 (1d) 来计算每个模块的连续时间状态的一阶时间导数值 $\dot{x}\big|_{o p}$ 、约束（代数）状态值 $z{\Big|}_{o p}$ 和输出值 ${y\Big|}_{o p}$ 。这些变量中的每一个都可以围绕其各自的 OP 值进行扰动（由 $\Delta$ 表示），如 [4] 中的方程 (11) 所示（例如，对于模块输入 $\boldsymbol{u}=\boldsymbol{u}\Big|_{o p}+\Delta\boldsymbol{u}\,\Big)$ 。[2,3] 阐明了如何将此操作扩展到三维 (3D) 中的旋转（姿态），这些旋转不位于线性空间中。
+
+状态、输入和输出的数量（即向量 $x\;,\;x^{d}\;,\;z\;,\;u$ 和 $y$ 的大小）取决于在 OpenFAST 中启用的功能。
+
 # MODULE LINEARIZATION  
 
 As explained in [4], the FAST modularization framework supports  a  very  general  (need  not  be  linear)  state-space formulation, with any combination of continuous-time-state, discrete-time-state, constraint- (algebraic-) state, other- (e.g., logical) state, and output equations. However, for a module to support linearization, the formulation is limited to a hybrid semiexplicit differential-algebraic equation (DAE) of index 1,2 which has the following limitations: 1) the continuous-timestate derivatives and discrete-time-state updates must be written as an explicit function of the states, inputs, and parameters; 2) the constraints must be of index 1; and 3) other states are used only for time-integration or when acting as parameters in the linearization process.  
@@ -73,6 +93,13 @@ To support linearization, a module must also be able to export Jacobian matrices
 
 The linearized form of a general module is given by Eqs. (12) and (13) from [4]; the simplified forms for the land-based modules InflowWind, AeroDyn, ServoDyn, ElastoDyn, and BeamDyn are given in [2,3]. The simplified forms for the newly linearized offshore modules HydroDyn and $\mathrm{MAP++}$ are given next, along with a description of how each module is linearized.  
 
+正如[4]中所述，FAST模块化框架支持非常通用的（不必是线性的）状态空间公式，它可以结合连续时间状态、离散时间状态、约束（代数）状态、其他（例如逻辑）状态以及输出方程。然而，为了支持模块线性化，公式必须限定为指数为1,2的混合半显式微分代数方程（DAE），这具有以下限制：1）连续时间状态的导数和离散时间状态的更新必须显式地写成状态、输入和参数的函数；2）约束必须具有指数1；3）其他状态仅用于时间积分或在线性化过程中充当参数。
+
+为了支持线性化，模块还必须能够导出状态方程和输出方程关于状态和输入的雅可比矩阵。表1总结了OpenFAST模块在迄今为止线性化的特征的状态、输入和输出。迄今为止线性化的OpenFAST模块特征仅包括连续时间状态和约束状态（尚未线性化任何具有离散时间状态的特征）。
+
+一般模块的线性化形式由[4]中的公式(12)和(13)给出；陆基模块InflowWind、AeroDyn、ServoDyn、ElastoDyn和BeamDyn的简化形式由[2,3]给出。接下来给出新线性化的海上模块HydroDyn和$\mathrm{MAP++}$的简化形式，并描述了如何线性化每个模块。
+
+
 TABLE 1. MODULE STATES, INPUTS, AND OUTPUTS KEPT IN THE OPENFAST LINEARIZATION PROCESS TO DATE.   
 
 
@@ -81,20 +108,27 @@ TABLE 1. MODULE STATES, INPUTS, AND OUTPUTS KEPT IN THE OPENFAST LINEARIZATION P
 The linearization of the HydroDyn (HD) module applies to both the strip-theory solution, potential-flow solution, or a hybrid combination of the two. Linear state-space-based waveexcitation and wave-radiation models have been added to HydroDyn  to  enable  linearization  of  the  potential-flow solution, including wave-excitation loads (with diffraction) from disturbances of wave elevation and wave-radiation loads with free-surface memory effects.  
 
 The linear state-space-based wave-radiation model was previously available in HydroDyn within FAST v8. As shown in [5], the model approximates the convolution term in the Cummins equation as shown in Eq. (1), with a linear statespace model as shown in Eq. (2), where $F_{R d t n}$ is the waveradiation  loads; $K_{R d t n}$ is  the  wave-radiation-retardation kernels; $\dot{q}_{P t f m}$ is the floating platform translation and rotational velocities; $x_{R d t n}$ are the wave-radiation states; and $A_{R d t n}$ ,  
-
 $B_{R d t n}$ , and $C_{R d t n}$ are the matrices of the linear state-space wave-radiation model. The more wave-radiation states there are, the better the state-space model can approximate the convolution. Within HydroDyn, the user can select either the convolution-based wave-radiation  model,  which  is implemented via numerical convolution using discrete-time states  (which,  if  linearized,  would  overly  complicate application of the linear state-space model), or the linear statespace-based wave-radiation model, which is implemented using matrices  derived  from  the  SS_Fitting  preprocessor  [6]  or equivalent.  
 
+机组的HydroDyn (HD)模块线性化适用于带叶片理论解、势流解或两者的混合组合。为了能够对势流解进行线性化，包括来自波高扰动引起的波激载以及具有自由表面记忆效应的波辐射载，线性状态空间波激和波辐射模型已被添加到HydroDyn中。
+
+这种线性状态空间波辐射模型以前在FAST v8版本的HydroDyn中可用。如[5]所示，该模型将Cummins方程中的卷积项近似为状态空间模型，如公式(2)所示，其中$F_{R d t n}$是波辐射载；$K_{R d t n}$是波辐射延迟核；$\dot{q}_{P t f m}$是浮式平台的平动和转动速度；$x_{R d t n}$是波辐射状态；以及$A_{R d t n}$，
+$B_{R d t n}$，和$C_{R d t n}$是线性状态空间波辐射模型的矩阵。波辐射状态越多，状态空间模型就越能更好地近似卷积。在HydroDyn中，用户可以选择基于卷积的波辐射模型，该模型通过使用离散时间状态进行数值卷积实现（如果线性化，会过度复杂化线性状态空间模型的应用），或者选择线性状态空间波辐射模型，该模型使用从SS_Fitting预处理器[6]或其等效物派生的矩阵实现。
+
 $$
-\begin{array}{c}{{F_{R d t n}\left(t\right)=-\displaystyle\int_{}^{t}{{K_{R d t n}}\left(t-\tau\right){{\dot{q}}_{P t f m}}\left(\tau\right)d\tau}}}\\ {{\dot{x}_{R d t n}={A_{R d t n}}x_{R d t n}+{B_{R d t n}}{{\dot{q}}_{P t f m}}}}\\ {{F_{R d t n}\cong{C_{R d t n}}x_{R d t n}}}\end{array}
+\begin{array}{c}{{F_{R d t n}\left(t\right)=-\displaystyle\int_{0}^{t}{{K_{R d t n}}\left(t-\tau\right){{\dot{q}}_{P t f m}}\left(\tau\right)d\tau}}}\\ {{\dot{x}_{R d t n}={A_{R d t n}}x_{R d t n}+{B_{R d t n}}{{\dot{q}}_{P t f m}}}}\\ {{F_{R d t n}\cong{C_{R d t n}}x_{R d t n}}}\end{array}
 $$  
 
 The linear state-space-based wave-excitation model was newly introduced in HydroDyn within OpenFAST with the addition of the linearization functionality for FOWTs. Similar to the linear state-space-based wave-radiation model, the model approximates the infinite integral shown in Eq. (3) with a linear state-space model as shown in Eq. (4), where $F_{E x c t n}$ is the wave-excitation loads under unidirectional waves; $K_{E x c t n}$ is the incident wave-excitation kernels; $\zeta$ is the wave elevation; $\zeta_{c}$ is a time-shifted wave elevation; $x_{E x c t n}$ is the waveexcitation  states;  and $A_{E x c t n}$ , $B_{E x c t n}$ ,  and $C_{E x c t n}$ are  the matrices  of  the  linear  state-space  wave-excitation  model. Further details on this new linear state-space-based waveexcitation model are provided in Annex A. Within HydroDyn, the user can now select either a Fourier-transform-based waveexcitation model, which is implemented with discrete Fourier transforms (which is not conducive to linearization), or the new linear state-space-based wave-excitation model.  
 
+在OpenFAST中，HydroDyn新增了基于线性状态空间波激振模型，并增加了对浮式风电场（FOWT）的线性化功能。类似于基于线性状态空间的波辐射模型，该模型使用线性状态空间模型近似Eq. (3)中所示的无限积分，如Eq. (4)所示，其中$F_{E x c t n}$为单向波下的波激振载荷；$K_{E x c t n}$为入射波激振核；$\zeta$为波高；$\zeta_{c}$为时移波高；$x_{E x c t n}$为波激振状态；而$A_{E x c t n}$ , $B_{E x c t n}$ ,  和 $C_{E x c t n}$是线性状态空间波激振模型的矩阵。关于该新型基于线性状态空间的波激振模型的更多细节见附录A。在HydroDyn中，用户现在可以选择基于傅里叶变换的波激振模型（该模型使用离散傅里叶变换实现，不利于线性化），或该新型基于线性状态空间的波激振模型。
+
+
 $$
 \begin{array}{c}{{\displaystyle F_{E x c t n}\left(t\right)=\int_{\phantom{\bigg|}E_{E x c t n}}\left(t-\tau\right)\zeta\left(\tau\right)d\tau}}\\ {{\displaystyle\dot{x}_{E x c t n}=A_{E x c t n}x_{E x c t n}+B_{E x c t n}\zeta_{c}\vphantom{\bigg|}}}\\ {{\displaystyle F_{E x c t n}\cong C_{E x c t n}x_{E x c t n}}}\end{array}
 $$  
 
-When linear state-space-based wave excitation or wave radiation are enabled in the potential-flow solution, continuoustime states associated with the excitation and radiation damping are included in the linearized HydroDyn model. Linearization is permitted only in still water $\left(\left.\zeta_{c}\right|_{o p}=x_{E x c t n}\right|_{o p}=F_{E x c t n}\right|_{o p}=0\:)^{3}$ and is not permitted when  
+When linear state-space-based wave excitation or wave radiation are enabled in the potential-flow solution, continuoustime states associated with the excitation and radiation damping are included in the linearized HydroDyn model. Linearization is permitted only in still water $\(\left.\zeta_{c}\right|_{o p}=x_{E x c t n}\right|_{o p}=F_{E x c t n}\right|_{o p}=0\:)^{3}$ and is not permitted when  
 
 Fourier-transform-based  wave-excitation,  convolution-based wave-radiation, wave directional spreading, second-order wave kinematics, or second-order potential-flow loads are enabled. The  linearized  form  of  the  HydroDyn  state  and  output equations is given by Eq. (5), where  ∆x(HD)= $\varDelta x^{(H D)}=\left\{{\varDelta x_{E x c t n}}\atop{\varDelta x_{R d t n}}\right\},$ $\varDelta u^{(H D)}$ includes contributions from $\varDelta\zeta$ and $\varDelta\dot{q}_{P t f m}$ , and ∆y(HD) includes contributions from ∆FExctn and ∆FRdtn.  
 

软件组工作讨论/2025.7.3 控制模块讨论.md

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