diff --git a/力学书籍/Kinematically nonlinear finite element model of a horizontal axis wind turbine/auto/Kinematically nonlinear finite element model of a horizontal axis wind turbine. Part 2.md b/力学书籍/Kinematically nonlinear finite element model of a horizontal axis wind turbine/auto/Kinematically nonlinear finite element model of a horizontal axis wind turbine. Part 2.md index 656cf96..45ef218 100644 --- a/力学书籍/Kinematically nonlinear finite element model of a horizontal axis wind turbine/auto/Kinematically nonlinear finite element model of a horizontal axis wind turbine. Part 2.md +++ b/力学书籍/Kinematically nonlinear finite element model of a horizontal axis wind turbine/auto/Kinematically nonlinear finite element model of a horizontal axis wind turbine. Part 2.md @@ -28,15 +28,21 @@ Jorgen Thirstrup Petersen # Abstract. -A mathematical time domain model for simulation of the dynamic response of a horizontal axis wind turbine is presented. The model concentrates on the correct representation of the inertia loads in the equations of motion, and aims at a final model well suited for parametric studies, offering the user the possibility of chosing the appropriate level of representation of dynamical effects. +A mathematical time domain model for simulation of the dynamic response of a horizontal axis wind turbine is presented. The model concentrates on the correct representation of the inertia loads in the equations of motion, and aims at a final model well suited for parametric studies, offering the user the possibility of chosing the appropriate level of representation of dynamical effects. +本文介绍了一个用于模拟水平轴风力涡轮机动态响应的数学时域模型。该模型着重于运动方程中惯性载荷的正确表示，旨在构建一个非常适合参数化研究的模型，为用户提供选择适当的动态效应表示等级的可能性。 A general kinematic analysis, which includes the elastic rotations at the tower top and at the shaft end, is the basis for the derivation of the local inertia loads. Nonlinear kinematic terms are retained in the expressions for these loads. The wind turbine structure is subdivided into three substructures, comprising the tower, the nacelle-shaft, and the rotor blades. Each substructure is described with reference to a local coordinate system. +一项通用的运动学分析，其中包括塔顶和轴端部的弹性转动，是推导局部惯性载荷的基础。在这些载荷的表达式中，保留了非线性运动学项。风力发电机结构被划分为三个次结构，包括塔柱、舱盖-轴和叶片。每个次结构都以参考局部坐标系进行描述。 The model is discretized by use of the finite element modelling technique. A simple two node prismatic beam element is used. General element inertia matrices and vectors are derived through consistent transformation of the inertia loads to the nodes. The kinematic analysis provides for the geometric compatibility at the coupling nodes between the substructures. Final assembly of the substructure equations of motion is carried through by imposing force equilibrium at the coupling nodes. The resulting coefficient matrices are nonsymmetric. Substructuring permits easy updating of changes in geometry, corresponding to rigid body rotations of the substructures and extends the limitations on the allowable displacements. +模型采用有限元建模技术进行离散化。使用了一个简单的双节点棱柱形梁单元。通过对惯性载荷进行一致性变换到节点，推导出了通用的单元惯性矩阵和向量。运动学分析保证了子结构之间耦合节点上的几何兼容性。子结构的运动方程最终组装是通过在耦合节点上施加力平衡来实现的。得到的系数矩阵是非对称的。子结构化允许轻松更新几何变化，对应于子结构的刚体旋转，并扩展了允许位移的限制。 + The loading on the wind turbine structure includes both the gravity and aerodynamic loads on the blades. Aerodynamic loads are derived by use of a quasi-steady theory. The model is fully aeroelastic, in that the inffuence of the elastic deformations on the aerodynamic force is taken into account. The free wind vector is composed of a deterministic contribution including wind shear and tower interference, and a stochastic component, which is generated by the Sandia method for time simulation of turbulence. +风力涡轮机结构的载荷包括重力载荷和叶片上的气动载荷。气动载荷的计算采用准稳态理论。该模型是完全气弹耦合的，即考虑了弹性变形对气动力的影响。自由风矢量由一个包含风切变和塔干涉的确定性分量和一个随机分量组成，该随机分量由桑迪亚实验室的方法用于模拟湍流的时间演变。 Based on the mathematical model a computer program has been developed. The program runs on a personal computer. The equations of motion are solved in the time domain by use of an implicit Newmark integration scheme, in combination with Newton-Raphson iterations, performed to ensure equilibrium at each time step. Eigensolutions are obtained for the structure including only the dominating influence of the rotating substructures. The eigenvalues are found by use of the Sturm-sequence property for the symmetric equations of motion. These eigenvalues are then used as starting values in an inverse iteration performed on the nonsymmetric equations from the substructure formulation to give the mode shapes. Results from a simulation of the response on a typical Danish three bladed stall regulated wind turbine are presented and compared with measurements. +基于数学模型，已开发出一套计算机程序。该程序可在个人电脑上运行。通过采用隐式Newmark积分方案，并结合牛顿-拉夫逊迭代以确保每个时间步长的平衡，求解运动方程，该方案应用于时域。针对包括旋转次结构主导影响的结构，获得特征值。利用对称运动方程的Sturm序列性质，确定这些特征值。随后，这些特征值被用作次结构配方中非对称方程的反迭代的初始值，以获得模态形状。展示了对典型丹麦三叶片失速调速风力涡轮机的响应模拟结果，并与测量结果进行了比较。 Thesis submitted to the Technical University of Denmark in partial fulfilment of the require ments for the degree of Ph.D. (lic. techn.). @@ -190,7 +196,7 @@ Finally, Michael S. Courtney is thanked for his conscientious assistance in avoi # 1 Introduction. -The dominating design concept used for commercially manufactured wind turbines in the world today is the horizontal axis type connected to the electrical supply grid. The rotor configuration is usually two- or three-bladed, and power regulation is carried out either by stall- or pitchregulation. Two-bladed rotors are often equipped with a teeter hinge, but the majority of wind turbines have a rigid hub. +The dominating design concept used for commercially manufactured wind turbines in the world today is the horizontal axis type connected to the electrical supply grid. The rotor configuration is usually two- or three-bladed, and power regulation is carried out either by stall- or pitchregulation. Two-bladed rotors are often equipped with a teeter hinge, but the majority of wind turbines have a rigid hub. 目前全球商业化生产的风力涡轮机普遍采用水平轴式设计，并连接至电力供应网络。 旋转器配置通常为两叶或三叶，功率调节则采用失速调节或迎角调节。 两叶旋转器通常配备摆动铰链，但大多数风力涡轮机采用刚性轮毂。 The majority of the Danish wind turbines are of the three-bladed stall regulated type. Only three bladed turbines of the rigid hub type are commercially available in Denmark today. Very little attempt has been made to develop a more flexible and probably more optimal design, as for example, the two bladed teetered type. @@ -202,11 +208,18 @@ During the last years the industrial approach has changed more in the direction Altogether, it seems as if the industry, to a certain degree, is prepared to accept the theoretical approach and that it will be considered an important ingredient in the optimization process. When the conditions change in that direction, it is much easier to establish a fertile interaction between the research communities and the industry. Of course, the research communities have a responsibility in that process, in that the results of the research must be put in a form which makes it practicable for the recipient to use them. -During the last couple of years, work has been going on at Risg National Laboratory which aims at the development of a computer program (Design Basis [Lil) which is efficient for parametric studies on a personal computer. The first edition of the program has been published and distributed. This is a simple one bladed model (see Sec. 1.1), which is based on earlier research [M4] covering the complete rotor. +During the last couple of years, work has been going on at Risg National Laboratory which aims at the development of a computer program (Design Basis [L1]) which is efficient for parametric studies on a personal computer. The first edition of the program has been published and distributed. This is a simple one bladed model (see Sec. 1.1), which is based on earlier research [M4] covering the complete rotor. The main purpose of the present work is to support the continued development of the Design Basis code, especially concerning the proper inclusion of the dynamical effects when a relatively flexible structure is considered. Therefore the present model focuses on the modelling of the inertia loads. The model is developed in a form that makes it possible to identify the origin of the inertia loads and eliminate the associated contributions if they are found to be negligible for a given application. In order to furnish the reader with some insight in the subject, it is found valuable to go through a presentation of the capability of existing, comparable models to include the inertia loads and place the present model in this context. This is done in the following Sec. 1.1. The introduction concludes with an overview of the elements of the present model and the scope of the thesis. +总的来说，似乎该行业在一定程度上已经准备好接受理论方法，并且会将它视为优化过程中的重要组成部分。当条件朝着那个方向变化时，更容易在研究社区和行业之间建立富有成效的互动。当然，研究社区在这个过程中负有责任，即必须将研究成果转化为便于接收方使用的形式。 + +在过去的几年里，挪威国家实验室一直在进行工作，旨在开发一个高效的计算机程序（设计基础[Lil]，Design Basis [Lil]），可在个人电脑上进行参数化研究。该程序的第一个版本已经发布并分发。这是一个简单的单叶模型（见第1.1节），基于之前涵盖整个转子的研究[M4]。 + +目前工作的目的是支持设计基础代码的持续开发，尤其是在考虑相对柔性结构时，要恰当地包含动力学效应。因此，目前的模型侧重于惯性载荷的建模。该模型以一种可以识别惯性载荷的来源并消除与其相关的贡献（如果发现它们对于特定应用而言可以忽略不计）的形式开发。 + +为了向读者提供一些关于该主题的见解，我们认为回顾一下现有、可比模型包含惯性载荷的能力，并将当前模型置于该背景下的介绍是很有价值的。这将在第1.1节中进行说明。引言最后对当前模型的要素和本论文的范围进行了概述。 # 1.1 Placing the model in relation to existing models. @@ -215,6 +228,11 @@ The following survey is not intended to be a general evaluation of the quality o In the present model a kinematic analysis results in nonlinear expressions for the inertia loads, which to a great extent are retained in the equations of motion. Anticipating the results from later sections, the inertia loads for the blade will be the basis for the following survey. The present model makes use of the finite element model (FEM) for discretization and the inertia loads are represented as node loads, resulting from consistent transformation of the distributed inertia loads to the nodes. When the origin of the coefficient matrices and vectors are made clear, it is possible to compare the present model with others that make use of different techniques for discretization. +以下调查并非旨在对模型的整体质量进行评估，而仅仅是对其纳入重要动力学效应的能力，即惯性载荷的能力进行评估。本研究主要关注惯性载荷的影响，而其他模型则是在截然不同的主要目标下开发的。 + +在本模型中，运动学分析导出了惯性载荷的非线性表达式，这些表达式在很大程度上被保留在运动方程中。 预料到后续章节的结果，叶片的惯性载荷将成为以下调查的基础。 + +本模型采用有限元模型（FEM）进行离散化，惯性载荷被表示为节点载荷，这是通过将分布的惯性载荷一致性地转换到节点而得到的。 当系数矩阵和向量的来源明确后，就可以将本模型与其他采用不同离散化技术进行比较。 # 1.1.1 Simplified expression for the blade inertia load. @@ -223,14 +241,36 @@ When dealing with rotating substructures, important inertia forces may arise. If However, the work going on with optimization of the wind turbines results inevitably in more flexible structures, and additional inertia forces are likely to be important. An example is given below of the origin of such loads. At the same time the example covers the inertia loads which are very likely to be important on some existing designs, but often neglected because the existing design tools are often incapable of including them. The basis for the example is the schematic drawing in Fig.1 and a reduced expression for the inertia loads which are part of the mathematical model. The kinematic analysis in the model includes deformation at the shaft end, teeter and yaw but only the tower top deformation will be included in the following survey. The angular rotor velocity is assumed constant, in order to keep the expression for the blade inertia force as simple as possible. +在处理旋转次结构时，可能会产生重要的惯性力。如果结构构件相对刚 stiff， 仅需假设次结构表现为刚体，即可确定惯性力。 此外，如果角速度可以认为是均匀的，则惯性力可以简单地被视为离心力。 对于风力发电机而言，这相当于将离心力包含在叶片中。当转子对称时，离心力仅影响转子的载荷，而其他结构部分不受影响。 + +然而，风力发电机优化工作不可避免地导致更灵活的结构，额外的惯性力很可能变得重要。 下面给出一个此类载荷来源的例子。 同时，该例子涵盖了某些现有设计中很可能重要的惯性载荷，但由于现有设计工具通常无法包含它们而被忽略。 + +该例子的基础是图1中的示意图以及惯性载荷的简化表达式，这些载荷是数学模型的一部分。 该模型中的运动学分析包括轴端变形、倾斜和偏航，但以下调查仅包含塔顶变形。 为了保持叶片惯性力表达式尽可能简单，假设转子角速度恒定。 ![](a28a0e154d2585e6e10e55a3c856167d4f7ddf20ddd744bca82878589ea298ac.jpg) Figure 1: Tower top elastic rotations. -Applying these simplifications the following formal expression for the inertia load at a blade point is obtained +Applying these simplifications the following formal expression for the inertia load at a blade point is obtained 应用这些简化后，可以得到以下关于叶片某一点惯性载荷的正式表达式。 + $$ -\begin{array}{r l}&{-\left\{F_{I B}\right\}=}\\ &{\qquad\left[M\right]\left\{\bar{q}_{B}\right\}\qquad.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... +\begin{aligned} +- \{F_{IB}\} =\ +& [M] \{\ddot{q}_B\} + && \quad \text{(mass)} \\ ++ & [C(\omega, \dot{\theta}_T)] \{\dot{q}_B\} + && \quad \text{(Coriolis)} \\ ++ & [K(\omega^2, \omega \dot{\theta}_T, \dot{\theta}_T \dot{\theta}_T, \ddot{\theta}_T)] \{q_B\} + && \quad \text{(softening)} \\ ++ & \{F(\omega^2, \omega \dot{\theta}_T, \dot{\theta}_T \dot{\theta}_T, r)\} + && \quad \text{(centrifugal + gyroscopic)} \\ ++ & [M_{\ddot{\theta}}(\ell_{\text{shaft}}, r)] \{\ddot{\theta}_T\} + && \quad \text{(rigid body rotation)} \\ ++ & [C_{\dot{\theta}}(\omega, \dot{\theta}_T, \ell_{\text{shaft}}, r)] \{\dot{\theta}_T\} + && \quad \text{(gyroscopic)} \\ ++ & [M_{\ddot{u}}] \{\ddot{u}_T\} + && \quad \text{(rigid body translation)} +\end{aligned} $$ The expression has been derived in App. B, and it is possible through the description there to trace the origin of the terms in the model equations. Eq. 1.1.1 is a simplified version of Eq. B.0.11 in App. B. The relation between the two equations is kept unique by retaining the order of the terms. @@ -246,6 +286,19 @@ The influence of the inertia force on the terms related to local deformation $\p In addition to the centrifugal forces gyroscopic forces are present in the equation, due to the resulting time varying rotation vector for the rotor. The gyroscopic terms are represented both in the vector of line 4 and the matrix of line 6. The gyroscopic forces appear in both terms due to the chosen discretization technique and to the level of reduction of the equations. For many designs it is probably very important to incorporate the gyroscopic forces. The last terms to mention are the terms denoted rigid body terms in the equation. They are usually incorporated in the most models, which integrate the complete turbine. Their incorporation do not depend on a kinematic analysis. +表达式已在附录B中推导得出，通过附录B中的描述，可以追溯模型方程中各术语的来源。方程1.1.1是附录B中方程B.0.11的简化版本。通过保留各项的顺序，保持这两个方程之间的关系保持唯一。 + +除了列出的自由度和参数外，所有矩阵和离心陀螺向量都取决于材料参数和几何形状。除了包含方位角位置和塔顶角弹性变形的质量矩阵。 + +所讨论的叶片径向位置由r表示。假设轴是刚性的，其长度用$\ell_{s h a f t}$表示。转子相对于塔顶以恒定角速度$(\omega)$旋转。$\pmb{\omega}$角速度向量的方向受轴承约束。假设塔是柔性的，使得塔顶的弹性旋转不容忽视。这些旋转被假定为较小，以便它们可以参考坐标轴并描述为向量。它们通常用$\theta_{i}$表示，索引为$\pmb{x},\pmb{y},$和$\pmb{z}$，分别对应于各个轴，如图所示。在正式方程中，不对轴进行区分，$T$索引表示旋转可能是三个中的任何一个。 + +在此上下文中，我们的主要兴趣是展示柔性塔如何影响惯性载荷。必须将发生的弹性角速度加到旋转速度$(\omega)$中，才能获得转子的总速度。由于弹性旋转及其时间导数（用点表示）随时间变化，方向和大小都会变化，因此很明显，所有通常由旋转坐标系中著名的四项加速度表达式描述的加速度都存在。这反映在方程中。 + +惯性力对与叶片局部变形$\pmb{\left(\left\{q_{B}\right\}\right)}$相关的项的影响，反映在方程的第2行和第3行，代表科里奥利效应和软化效应。这些的重要性在很大程度上取决于叶片的实际特性。 + +除了离心力外，由于转子的时间变化旋转向量，方程中还存在陀螺力。陀螺力既在第4行的向量中，也在第6行的矩阵中表示。由于所选择的离散化技术和方程的还原程度，陀螺力出现在这两个项中。对于许多设计，可能非常重要地将陀螺力纳入考虑。 + +最后需要提及的项是方程中称为刚体项的项。它们通常被整合到大多数完整的涡轮机模型中。它们的整合不依赖于运动学分析。 # 1.1.2 Comparison of models. @@ -254,31 +307,52 @@ Based on the expression for the inertia loads from Eq. 1.1.1 existing representa The result of the model description will be a series of models, which provides a survey of the ability of the models to incorporate the inertia loads. At the same time the overview will serve to illustrate where the present model is placed in relation to existing models. Other important characteristics of the models, e.g. special degrees of freedom and aerodynamic model, will be mentioned briefly, in order to emphasize the aim of the model in question. The models will be referred to three main categories. Within each category degrees of varying complexity exist. The first category covers models which only consider the dynamics of a single blade. Coupling to the other blades, the nacelle, and to the tower are not included, which means that the hub and the tower are assumed to be stiff. The second category covers models which consider dynamics of the whole rotor and perhaps a very simplified coupling to the tower. The models in this category assume that the tower is relatively stiff. The third category deals with the models which incorporate the complete structure, with a varying number of DOFs and with a varying degree of linearization. +基于公式 1.1.1 中惯性载荷的表达式，以下将描述现有的水平轴风力涡轮机代表性模型，从最简单的模型开始，逐步介绍复杂性递增的模型。仅考虑专门为风力涡轮机开发的模型，并且仅考虑公开领域的模型。这意味着，例如，经过修改的直升机模型和私有制造商的模型不在此范围内，前者通常非常复杂且难以提取信息，这使得它们难以与专门设计的风力涡轮机模型相提并论；后者则仅仅是因为这些模型的细节信息未知。在上述两种类别中都存在重要的模型。 + +模型描述的结果将是一系列模型，旨在提供一个关于模型整合惯性载荷能力的概览。同时，该概览也将用于说明当前模型与现有模型之间的关系。为了强调每个模型的目标，将简要提及模型的一些其他重要特征，例如特殊的自由度以及气动模型。 + +这些模型将被归类到三个主要类别中。在每个类别中都存在不同复杂程度的模型。第一类涵盖仅考虑单叶片动态的模型。不包括与其他叶片、机舱和塔架的耦合，这意味着假设轮毂和塔架是刚性的。第二类涵盖考虑整个转子动态以及与塔架简化耦合的模型。这类模型假设塔架相对刚性。第三类处理整合完整结构的模型，具有变化的自由度数和变化的线性化程度。 # 1.1.2.1 Single blade models. -This category covers models which consider the dynamics of one single blade. They are usually very fast and effective, as appropriate when the main characteristics of the wind turbine are to be established. When the tower, nacelle, shaft and hub with good approximation can be considered stiff, the models will often give answers which are sufficient for the final design, perhaps modified according to results from prototype testing. +This category covers models which consider the dynamics of one single blade. They are usually very fast and effective, as appropriate when the main characteristics of the wind turbine are to be established. When the tower, nacelle, shaft and hub with good approximation can be considered stiff, the models will often give answers which are sufficient for the final design, perhaps modified according to results from prototype testing. 本类别涵盖那些考虑单个叶片动态的模型。这些模型通常速度很快且效果显著，尤其适用于确定风力发电机的主要特性。当塔架、风机舱、轴和轮毂可以被合理地近似为刚性时，这些模型通常能给出足够用于最终设计的答案，或许会根据原型测试结果进行调整。 + +The ability of the models to incorporate the inertia loads can be roughly illustrated by use of Eq. 1.1.1, which after cancelling of the terms related to the tower DOFs is written 模型整合惯性载荷的能力，大致可以通过公式 1.1.1 进行说明，在消去与塔架自由度相关的项后，可写成： -The ability of the models to incorporate the inertia loads can be roughly ilustrated by use of Eq. 1.1.1, which after cancelling of the terms related to the tower DOFs is written $$ -\begin{array}{r l r}{-\left\{F_{I B}\right\}}&{=}&\\ &{}&{\left[M\right]\left\{\ddot{q}_{B}\right\}\quad...\cdot....\left(\sf m a s s\right)}\\ &{}&{+\left[C\left(\omega\right)\right]\left\{\dot{q}_{B}\right\}\quad...\cdot..\left(\sf C\mathrm{oriol}\backslash\dot{s}\right)}\\ &{}&{+\left[K\left(\omega^{2}\right)\right]\left\{q_{B}\right\}\quad...\left(\sf s o f t e n i n g\right)}\\ &{}&{+\left\{F\left(\omega^{2},r\right)\right\}\quad...\cdot.\left(\sf c e n t r i f u g a l\right)}\end{array} +\begin{array}{r l r}{-\left\{F_{I B}\right\}}&{=}&\\ &{}&{\left[M\right]\left\{\ddot{q}_{B}\right\}\quad...\cdot....\left(\sf m a s s\right)}\\ &{}&{+\left[C\left(\omega\right)\right]\left\{\dot{q}_{B}\right\}\quad...\cdot..\left(\mathrm{Coriolis}\right)}\\ &{}&{+\left[K\left(\omega^{2}\right)\right]\left\{q_{B}\right\}\quad...\left(\sf s o f t e n i n g\right)}\\ &{}&{+\left\{F\left(\omega^{2},r\right)\right\}\quad...\cdot.\left(\sf c e n t r i f u g a l\right)}\end{array} $$ Here the deformation vector, $\left\{{q}_{B}\right\}$ , covers only one single blade. The equation of motion is usually established in the rotating frame of reference following the blade. The equations are usually linearized and solved by a modal analysis technique, which reduces the number of degrees of freedom. The solution is often carried out in the frequency domain. It is mentioned below where the particular models deviate from the equation above. One such model is described by Larsen [L1]. Blade displacement in flap- and chord-wise directions are considered, while torsion is excluded. The two directions are not structurally coupled, but coupled through the aerodynamics. Centrifugal stiffening is included. The equations of motion are solved both in the time and in the frequency domain by use of a modal technique, based on the mode shapes. This work focuses on the calculation of fatigue and extreme response. -A similar model is described by Oye in [01] and [02]. Here a more complete structural model is used. Structural coupling between the flap- and chord-wise directions and torsion are included. The influence of the centrifugal force is included in the stiffness, both in directions perpendicular to the blade axis, but also in the torsional direction. The latter may be important for blades designed with a considerable twist and low torsional stiffness. This effect is usually neglected, as in the present model. The model operates in the time domain and uses a modal technique based on the mode shapes for reduction of the degrees of freedom. +A similar model is described by Oye in [O1] and [O2]. Here a more complete structural model is used. Structural coupling between the flap- and chord-wise directions and torsion are included. The influence of the centrifugal force is included in the stiffness, both in directions perpendicular to the blade axis, but also in the torsional direction. The latter may be important for blades designed with a considerable twist and low torsional stiffness. This effect is usually neglected, as in the present model. The model operates in the time domain and uses a modal technique based on the mode shapes for reduction of the degrees of freedom. A third example in this category is the model described by Thresher [T1]. The model is restricted to consider only the flap-wise direction. The stiffening effect of the inertia loads is fully incorporated in this direction. The model allows for a prescribed yawing motion, which may be time varying. Its representation in the model is only restricted by the introduced linearization. This gives rise to additional gyroscopic forces through the product $\omega{\dot{\theta}}_{T}$ and further, when the yaw motion is time varying, additional inertia forces through the angular acceleration $\ddot{\pmb{\theta}}_{T}$ . The solution is based on discretization and reduction of the DOFs by use of the Galerkin method. The equations of motion are integrated in the time domain. +这里变形矢量，$\left\{{q}_{B}\right\}$，仅覆盖单个叶片。运动方程通常是在跟随叶片的旋转参考系中建立的。这些方程通常线性化，并使用模态分析技术求解，从而减少自由度的数量。求解通常在频域进行。下面将说明特定模型如何偏离上述方程。 +Larsen [L1] 描述了一种这样的模型。考虑叶片的flap-wise和chord-wise位移，扭转被排除。这两个方向结构上不耦合，但通过气动耦合。包括离心刚化。运动方程使用基于模态形状的模态技术，在时域和频域中求解。这项工作侧重于疲劳和极端响应的计算。 + +Oye 在 [O1] 和 [O2] 中描述了一种类似的模式。这里使用了一个更完整的结构模型。包括flap-wise和chord-wise方向和扭转之间的结构耦合。离心力的影响包含在刚度中，不仅在垂直于叶片轴的方向上，还在扭转方向上。后者对于设计具有相当大的扭转和低扭转刚度的叶片可能很重要。这种影响通常被忽略，就像目前的模型一样。该模型在时域运行，并使用基于模态形状的模态技术来减少自由度。 + +在这个类别中的第三个例子是 Thresher [T1] 描述的模型。该模型仅限于考虑flap-wise。惯性载荷的加固效应完全包含在该方向上。该模型允许规定偏航运动，该运动可能是时变的。它在模型中的表示仅受引入的线性化所限制。这会产生额外的陀螺力，通过乘积 $\omega{\dot{\theta}}_{T}$，并且进一步，当偏航运动随时间变化时，通过角加速度 $\ddot{\pmb{\theta}}_{T}$ 产生额外的惯性力。解决方案基于离散化和使用伽辽金方法减少自由度。运动方程在时域中积分。 # 1.1.2.2 Coupled rotor models. Models in this category are characterized primarily by their ability to describe the dynamics of the coupled rotor, i.e. the rotor hub is no longer considered stiff. Apart from this, the basic approach may be very different. Also, the included dynamic effects are treated at different levels of refinement, primarily governed by the actual modelling employed and the solution technique. In spite of that, the inertia forces accounted for in the models can roughly be described by a common equation, which results from reduction of Eq. 1.1.1 +这类模型的主要特征在于其能够描述rotor的动力学，即不再将轮毂视为刚体。除此之外，基本方法可能差异很大。所包含的动态效应的处理精度也不同，主要取决于所采用的建模方法和求解技术。尽管如此，模型中考虑的惯性力大致可以用一个共同的方程来描述，该方程源于对公式1.1.1的简化。 $$ -\begin{array}{r l}&{-\left\{F_{I B}\right\}=}\\ &{\quad\quad\left[M\right\}\left\{\bar{q}_{B}\right\}\quad\ldots\ldots\ldots\ldots\ldots\ldots\left(\mathrm{mass}\right)}\\ &{\quad\quad+\left[C\left(\omega\right)\right]\left\{\bar{q}_{B}\right\}\quad\ldots\ldots\ldots\ldots\ldots\left(\mathrm{Coriolis}\right)}\\ &{\quad\quad+\left[K\left(\omega^{2}\right)\right]\left\{q_{B}\right\}\quad\ldots\ldots\ldots\ldots\left(\mathrm{softening}\right)}\\ &{\quad\quad+\left\{F\left(\omega^{2},\omega\bar{\theta}_{T},r\right)\right\}\quad\ldots\ldots\ldots\left(\mathrm{centriugal\}+\mathrm{gyroscopic}\right)}\\ &{\quad\quad+\left[M_{\delta}\left(\ell_{\delta u\,o\,f\,},r\right)\right]\left\{\bar{\theta}_{T}\right\}\quad\ldots\quad\cdot\left(\mathrm{rigid\body\rotation}\right)}\\ &{\quad\quad+\left[M_{\sharp}\right]\left\{\bar{u}_{T}\right\}\quad\ldots\ldots\ldots\ldots\ldots\ldots\left(\mathrm{rigid\body\transiation}\right)}\end{array} +\begin{align} +-\left\{ F_{IB} \right\} = & \\ +& \quad \left[ M \right] \left\{ \ddot{q}_B \right\} \quad \ldots \quad \text{(mass)} \\ +& \quad + \left[ C(\omega) \right] \left\{ \dot{q}_B \right\} \quad \ldots \quad \text{(Coriolis)} \\ +& \quad + \left[ K(\omega^2) \right] \left\{ q_B \right\} \quad \ldots \quad \text{(softening)} \\ +& \quad + \left\{ F(\omega^2, \omega \dot{\theta}_T, r) \right\} \quad \ldots \quad \text{(centrifugal + gyroscopic)} \\ +& \quad + \left[ M_{\ddot{\theta}}(\ell_{shaft}, r) \right] \left\{ \ddot{\theta}_T \right\} \quad \ldots \quad \text{(rigid body rotation)} \\ +& \quad + \left[ M_{\ddot{u}} \right] \left\{ \ddot{u}_T \right\} \quad \ldots \quad \text{(rigid body translation)} +\end{align} $$ It is mentioned below, where the actual model deviates from this equation. @@ -286,15 +360,24 @@ It is mentioned below, where the actual model deviates from this equation. One model in this category is described by Madsen [M4]. The model is developed for solution in the frequency domain, and this is naturally reflected in the choice of representation of the dynamic terms. The modelling is carried through in the rotating frame of reference. A constant angular velocity of the rotor, $\omega$ , is asumed, but a periodic yaw motion , $\dot{\theta}_{T}$ ,isallowed for, resulting in gyroscopic forces, represented in the vector $\{F\}$ . The yaw acceleration is neglected, assuming slowly varying yaw velocity. The centrifugal stiffening is taken into account as geometric stiffness and softening in the plane of rotation is included through $[K]$ Coriolis forces are accounted for through $[C]$ , yet, neglecting the influence of the yaw motion on the Coriolis force. The tower top DOFs are not included in the model, i.e. $[M_{\tilde{u}}]$ and $[M_{\tilde{\theta}}]$ are zero. As was the case with the model by Larsen [L1], the main purpose with this model is to calculate fatigue and extreme response. A model using a finite element approach is described by Garrad [G1]. The model is almost identical to the one by Lobitz [L2]. Garrad integrates the tower, although without taking into account the additional gyroscopic and Coriolis forces on the rotor resulting from tower top angular motion. The tower is modelled in a fixed frame of reference and the rotor in a rotating frame of reference. The two sets of equations are coupled before solution by use of a time varying transformation matrix reflecting the hub configuration and taking care of the relative movement of the structures. The tower top deformations are treated as real DOFs, but because the rotor axis is assumed to be fixed in space in the kinematic analysis, the inertia termsdependingon $\left\{{\dot{\theta}}_{T}\right\}$ are not taken into account. The solution of the equations is carried out in the time domain. +以下所述，实际模型偏离该方程的地方是： + +该类别中的一种模型由Madsen [M4] 描述。该模型是为在频域中求解而开发的，这自然体现在动态项表示形式的选择上。建模是在旋转参考系中进行的。假设转子的恒定角速度 $\omega$ ，但允许周期性的偏航运动 $\dot{\theta}_{T}$，从而产生陀螺力，表示在向量 $\{F\}$ 中。忽略偏航加速度，假设偏航速度变化缓慢。离心加固被视为几何刚度，并且通过 $[K]$ 包含旋转平面的软化。通过 $[C]$ 考虑了科里奥利力，但忽略了偏航运动对科里奥利力的影响。塔顶自由度 (DOFs) 未包含在模型中，即 $[M_{\tilde{u}}]$ 和 $[M_{\tilde{\theta}}]$ 为零。与Larsen [L1] 的模型一样，该模型的目的是计算疲劳和极端响应。 + +Garrad [G1] 描述了一种使用有限元方法的模型。该模型几乎与Lobitz [L2] 的模型相同。Garrad 整合了塔架，但没有考虑塔顶角运动引起的转子附加陀螺力和科里奥利力。塔架采用固定参考系建模，转子采用旋转参考系建模。在求解之前，通过使用时变变换矩阵耦合这两个方程组，该矩阵反映了轴架配置并处理了结构的相对运动。塔顶变形被视为真实的自由度 (DOFs)，但由于在运动学分析中假设转子轴固定在空间中，因此不考虑取决于 $\left\{{\dot{\theta}}_{T}\right\}$ 的惯性项。在时域中进行方程求解。 # 1.1.2.3 Integrated models. The main characteristic of the models described in this category is their ability to incorporate the DOFs at the tower top in such a way that the influence of time varying angular rotations includes the angular velocities at the tower top. Still, the level of refinement is very broad, from those where the coefficient matrices do not depend on the DOFs to models that can be characterized as fully nonlinear. The discussion is carried through with basis in the original Eq. 1.1.1. -The first model to mention in this category is one specially developed for stability analysis of two bladed, teetered wind turbines by Janetzke [J1]. The model has only five DOFs including the teeter. The remaining DOFs are two generalized coordinates corresponding to the fundamental flapping mode for each blade, and two rotations at the tower top, corresponding to yaw and tilt directions, respectively. The equations of motion are linearized so that the coefficient matrices do not depend on the DOFs. The Coriolis term is neglected so that $\left[C\right]=\left[0\right]$ The softening is not inciuded in the form of Eq. 1.1.1, but the stiffening due to the centrifugal force is taken into account, both on the blade and on the teeter DOF. The influence of the angular accelerations at the tower top on the blade inertia force as well as gyroscopic forces from the tower top angular velocities are taken into account, so that both $[M_{\sun}]$ and $[C_{\dot{\theta}}]$ are included. The solution is carried out in the time domain. +The first model to mention in this category is one specially developed for stability analysis of two bladed, teetered wind turbines by Janetzke [J1]. The model has only five DOFs including the teeter. The remaining DOFs are two generalized coordinates corresponding to the fundamental flapping mode for each blade, and two rotations at the tower top, corresponding to yaw and tilt directions, respectively. The equations of motion are linearized so that the coefficient matrices do not depend on the DOFs. The Coriolis term is neglected so that $\left[C\right]=\left[0\right]$ The softening is not included in the form of Eq. 1.1.1, but the stiffening due to the centrifugal force is taken into account, both on the blade and on the teeter DOF. The influence of the angular accelerations at the tower top on the blade inertia force as well as gyroscopic forces from the tower top angular velocities are taken into account, so that both $[M_{\ddot{\theta}}]$ and $[C_{\dot{\theta}}]$ are included. The solution is carried out in the time domain. Although the model is rather simple, it demonstrates the important steps in the derivation of the equations of motion and how it is possible with models of this type to include only those DOFs that are important in an actual analysis . +这类模型的主要特征是它们能够将塔顶的自由度（DOFs）纳入考虑，从而使随时间变化的角旋转的影响包括塔顶的角速度。然而，精细程度差异很大，从系数矩阵不依赖于自由度的模型，到可以被认为是完全非线性的模型。讨论基于原始公式 1.1.1 进行。 +在本类别中，首先需要提及的是 Janetzke [J1] 为两叶片摆动式风力涡轮机稳定性分析特别开发的模型。该模型仅包含五个自由度，包括摆动自由度。其余的自由度是两个对应于每个叶片的基频摆动模式的广义坐标，以及两个塔顶的旋转，分别对应于偏航和俯仰方向。运动方程进行了线性化处理，使得系数矩阵不依赖于自由度。科里奥利力项被忽略，因此$\left[C\right]=\left[0\right]$。软化效应未包含在公式 1.1.1 的形式中，但考虑了因离心力引起的加 stiffening，既考虑了叶片，也考虑了摆动自由度。考虑了塔顶的角加速度对叶片惯性力和塔顶角速度产生的陀螺力矩的影响，因此同时包含了 $[M_{\ddot{\theta}}]$ and $[C_{\dot{\theta}}]$。求解过程在时域内进行。 + +尽管该模型相对简单，但它展示了运动方程推导中的重要步骤，以及如何使用这类模型仅包含实际分析中重要的自由度。 Almost the same can be stated about the model by Madsen [M3]. It has been specially developed for investigation of the influence of rotationally sampled turbulence on the rotor loads. The model has only three DOFs which are the tower top displacement in the wind direction, the yaw and the tit angle. The local blade dynamics is thus ignored and the rotor is modelled as a rigid body, and the structural mass and stiffness coupling has been disregarded. The tilt and yaw rotation couple through the gyroscopic forces. Considering the terms of Eq. 1.1.1 it is equivalent to exclude the three first lines of the equation. The model operates in the frequency domain and allows for experimentally determined turbulence properties. A model similar to the one by Janetzke [J1] has been developed by Garrad [G2]. As regards the representation of the inertia forces, it is basically almost identical to the model in the present work and differs in that respect only in the extent to which linearization has been introduced. The Garrad model has been linearized, so that the coefficient matrices are independent of the DOFs. However, by omitting the linearization, equations identical to Eq. 1.1.1 wouid be arrived at. The presentation of the model by Garrad in [G2] focuses on the application of the symbolic manipulation program Reduce and stability analysis. @@ -302,7 +385,13 @@ A model similar to the one by Janetzke [J1] has been developed by Garrad [G2]. A The model presented by Schottl [S1] is rather comprehensive, allowing for transient analysis in that the angular velocity of the rotor may be time varying. This means that Eq. 1.1.1 should be extended to include $\dot{\omega}$ as well, in order to describe the inertia forces in this model correctly, or more precisely, the azimuthal rotation should be treated as a real DOF. Further this model implements a vortex model in the aerodynamic analysis. The equations are solved in the time domain. The last example to be mentioned in this survey is the nonlinear finite beam element model by Fabian [F1]. The equations of motion are established in the fixed frame of reference. The kinematic analysis is not carried through explicitly. Instead, the local accelerations are calculated numerically during solution in the time domain and treated as distributed forces. The basic tools in this calculation are position and orientation vectors for each finite element node. The time derivatives of these vectors are calculated during the solution process and used to generate the inertia forces. The model allows for arbitrarily large rotations of the elements and must be characterized as fully nonlinear. As regards the inertia loads, the model is capable of simulating all load situations for a wind turbine at the expense of computer storage and time. +几乎可以对 Madsen [M3] 的模型做出类似的陈述。该模型是专门为研究旋转采样湍流对转子载荷的影响而开发的。该模型仅有三个自由度 (DOF)，分别是风向塔顶位移、偏航角和倾角。因此，忽略了局部叶片动力学，将转子建模为刚体，并且忽略了结构质量和刚度耦合。倾斜和偏航旋转通过陀螺力耦合。考虑到等式 1.1.1 中的项，这相当于排除该等式的首三行。该模型在频域中运行，并允许使用实验确定的湍流特性。 +Garrad [G2] 开发了一个类似于 Janetzke [J1] 的模型。在惯性力的表示方面，它基本上与本研究中的模型几乎完全相同，仅在引入线性化的程度方面有所不同。Garrad 模型已经线性化，使得系数矩阵与自由度无关。但是，如果省略线性化，将会得到与等式 1.1.1 相同的方程。Garrad 在 [G2] 中对该模型的介绍重点在于符号操纵程序 Reduce 的应用和稳定性分析。 + +Schottl [S1] 提出的模型相当全面，允许进行瞬态分析，因为转子的角速度可以是随时间变化的。这意味着为了正确描述该模型中的惯性力，需要将等式 1.1.1 扩展以包含 $\dot{\omega}$，或者更准确地说，方位角旋转应被视为一个真实的自由度。此外，该模型在气动分析中实现了涡流模型。方程在时域中求解。 + +在本综述中最后提到的例子是 Fabian [F1] 的非线性有限梁单元模型。运动方程是在固定参考系中建立的。运动学分析没有显式进行。相反，在时域求解过程中，数值计算局部加速度，并将其视为分布力。该计算的基本工具是每个有限元节点的位移和姿态矢量。这些矢量的时导数在求解过程中计算出来，并用于生成惯性力。该模型允许单元进行任意大的旋转，必须将其归类为完全非线性。在惯性载荷方面，该模型可以在计算机存储和时间成本的代价下，模拟风力发电机的所有载荷情况。 # 1.1.3 The present model, its near relatives and why is it developed? From the previous survey of representative existing models it can be seen that the present model is placed in the third category covering models of the complete integrated structure, somewhere in the vicinity of the Garrad [G2] and Schottl [S1] models, at least when the ability to incorporate the influence of inertia loads is considered. So, in that sense these two models must be characterized as near relatives to the present model. But still the modelling approach differs at essential points, which will become clear when a comparison is made of other characteristics. @@ -310,27 +399,44 @@ From the previous survey of representative existing models it can be seen that t Both Garrad and Schottl use modal expansion of the response as a discretization technique and for reducing the number of DOFs, while the present work makes use of the finite element discretization. The difference between the models can primarily be attributed to these choices. This difference in approach is reflected in the choice of technique for derivation of the equations of motion (EOMs). Both Garrad and Schottl make use of energy principles when deriving the EOMs, Lagrange's equations and Hamilton's principle, respectively. This is a natural choice to make, when the mode shapes are used as the basis for a series expansion, because the choice of mode shapes defines the generalized coordinates or equivalently the DOFs, which are used as the basic independent variables in the energy expressions. At the same time the mode shapes act as interpolation functions and enter the coefficient matrices as integrands in a straightforward manner. The method thus requires that the mode shapes, or good approximations (functions satisfying the geometric boundary contitions are admissible functions, [M6, pp. 242-252]) are known in advance, and further that integrais depending on the actual mode shapes must be evaluated. Also, a means for truncation of the series expansion must be available. +从之前对现有代表性模型的调查可以看出，目前的模型属于第三类，涵盖具有完整集成结构的型号，至少在考虑惯性载荷的影响时，其位置大致在 Garrad [G2] 和 Schottl [S1] 模型附近。因此，从这个意义上来说，这两个模型可以被认为是目前模型的近亲。但建模方法在本质上存在差异，当比较其他特性时，这一点将变得清晰。 +**Garrad 和 Schottl 都使用响应模态展开作为离散化技术，并用于减少自由度 (DOFs) 的数量，而目前的工作则采用有限元离散化**。这两个模型之间的差异主要归因于这些选择。 + +**这种方法上的差异反映在运动方程 (EOMs) 推导技术上的选择**。Garrad 和 Schottl 都使用能量原理来推导运动方程，分别使用**拉格朗日方程和哈密顿原理**。当模态形状被用作级数展开的基础时，这是一个自然的选择，因为模态形状的选择定义了广义坐标，或者等价地说是自由度，这些自由度被用作能量表达式中的基本独立变量。同时，模态形状也充当插值函数，并以一种直接的方式进入系数矩阵作为积分元。因此，该方法要求预先知道模态形状，或者良好的近似值（满足几何边界条件的函数是允许的函数，[M6, pp. 242-252]），并且还需要评估取决于实际模态形状的积分。此外，还需要一种截断级数展开的方法。 As long as the structure is simple with respect to geometry and distribution of mass and stiffness and further relatively stiff, these requirements do not constitute a problem. Often a good understanding of the dynamic behaviour of the structure will be sufficient to make a good choice as regards the mode shapes and the number of terms in the expansion. Experience with similar structures may contribute valuable information. The evaluation of the integrals will in this case often be simple and is usually carried out numerically. When the structural design gets more complicated or much more flexible, it will often be necessary to make use of a complementary model, which can be used to calculate the mode shapes. Further, this model can help in providing knowledge about the dynamic behaviour and thus provide a basis for truncation of the series expansion. A more complex structure may further give rise to more complicated integrals over the structural elements. However, the structure might be so complex that a satisfactory solution by use of the modal expansion is precluded [M6, pp. 328-329]. One of the main objectives with the present model is to develop a tool, which can be used to decide to what degree a model should be refined in order to include the important dynamic effects on an actual wind turbine structure, especially including the more complex and flexible structures, which are likely to emerge during the process of optimization. A model will be much more efficient for that study if the decision on model resolution is more integrated than is the case with the modal technique described above. Further, it has been found valuable to use a model which is better suited to deal with complex geometry and distribution of mass and stiffness than those based on modal expansion. An example of such a complex structure is the aerodynamic tip brake of a stall regulated horizontal axis wind turbine. These are the main reasons for chosing the finite element method in the present work. +只要结构在几何形状和质量、刚度分布方面保持简单，并且相对刚性，这些要求就不会构成问题。通常，对结构的动力学行为有良好的理解就足以做出关于模态形状和展开式项数的良好选择。与类似结构的经验可能会提供有价值的信息。在这种情况下，积分的评估通常会很简单，并且通常采用数值方法进行。 +当结构设计变得更加复杂或更加柔顺时，通常需要使用互补模型来计算模态形状。此外，该模型可以提供关于动力学行为的知识，从而为截断级数展开式提供基础。更复杂的结构可能会导致结构元素上的更复杂的积分。然而，结构可能非常复杂，以至于使用模态展开法无法获得令人满意的解决方案 [M6, pp. 328-329]。 + +目前模型的首要目标之一是开发一种工具，用于确定模型应在多大程度上进行细化，以包含实际风力涡轮机结构的重要的动态效应，尤其包括在优化过程中可能出现的更复杂、更柔顺的结构。如果模型分辨率的决策更加集成，那么该模型对于此类研究将更加高效，这与上述模态技术的情况不同。此外，发现使用一种更适合处理复杂的几何形状和质量、刚度分布的模型是有价值的，而这些模型基于模态展开法。例如，一种此类复杂结构的横轴失速调节风力涡轮机的气动尖端制动器。这些是选择有限元方法作为本工作的首要原因。 When the equations of motion are derived for the FEM formulation, the use of the energy methods are less advantageous than they are in connection with the modal technique. Therefore, it has been chosen to use Newton's second law directly for derivation of the inertia loads. Through a general kinematic analysis (Sec. 3) the acceleration is derived for each material point on the structure, and the dynamic problem is converted to the equivalent static problem by use of d'Alembert's principle, which, according to Newton's second law, expresses that the inertia force on a mass particle is equal to the product of mass and acceleration and directed opposite to the acceleration. The interpolation functions for the actual element are simple polynomials. The inertia force is treated as a distributed force and transformed to the nodes, consistent with the principle of virtual displacement. The integration over the element can be performed analytically resulting in closed form expressions for the terms of the coefficient matrices, thus eliminating errors that may result from numerical integration. Use is made of the substructuring technique. The wind turbine structure is divided into substructures, which are coupled by imposing force equilibrium at the coupling nodes. The geometric compatibility at the coupling nodes is automatically satisfied by the chosen kinematic procedure. This is very much equivalent to what is done in the modal formulations, allthough very little attention has been payed to the subject by the authors of the texts, which describe the models mentioned above. Schottl states that the FEM method prohibits the treatment of the angular rotation as time varying. This is certainly not the case. Basically the methods are identical, they only differ in the choice of discretization technique. -All three models discussed in this section make wide use of the algebraic programming system + +在有限元方法（FEM）公式推导运动方程时，能量法的使用不如与模态技术结合时那样有利。因此，我们选择直接使用牛顿第二定律来推导惯性载荷。通过一般的运动学分析（第3节），可以推导出结构中每个质点的加速度，并通过使用达朗贝尔原理，将动力学问题转化为等效的静态问题。根据牛顿第二定律，达朗贝尔原理表达了惯性力等于质量与加速度的乘积，并且方向与加速度相反。 -Reduce [H2] for derivation of the EOMs. This reflects the common need for a convenient and safe method for handling very large and complicated algebraic expressions. Both Garrad and Schottl accomplish the complete derivation of the energy expressions by use of Reduce, after having defined the position vectors. The present kinematic analysis is carried through by hand, resulting in expressions for the acceleration composed of matrix and vector products, and then only the final multiplication and reduction is done by Reduce. +实际单元的插值函数是简单的多项式。惯性力被视为分布力，并转换为节点，这与虚拟位移原理一致。对单元进行解析积分，可以得到系数矩阵各项的闭合形式表达式，从而避免了数值积分可能导致的误差。 + +我们使用了次结构技术。风力发电机结构被划分为次结构，这些次结构通过施加耦合节点处的力平衡条件进行耦合。耦合节点处的几何兼容性由所选的运动学程序自动满足。这与模态公式中采用的方法非常相似，尽管在描述上述模型的文本中，作者对该主题关注较少。Schottl 认为，有限元方法禁止将角旋转视为随时间变化的量。这显然不是事实。基本上，这些方法是相同的，它们的不同之处仅在于离散化技术的选择。 + +All three models discussed in this section make wide use of the algebraic programming system Reduce [H2] for derivation of the EOMs. This reflects the common need for a convenient and safe method for handling very large and complicated algebraic expressions. Both Garrad and Schottl accomplish the complete derivation of the energy expressions by use of Reduce, after having defined the position vectors. The present kinematic analysis is carried through by hand, resulting in expressions for the acceleration composed of matrix and vector products, and then only the final multiplication and reduction is done by Reduce. This procedure is followed primarily because the matrix-vector form of the acceleration expression is found to be very informative, when the origin of the acceleration terms is considered. This may prove to be very useful, when a reduction is carried out during the process of optimization of the equations for special wind turbine configurations. But further, the application of the more advanced and complex procedures in Reduce, requires, in the present authors opinion, a good deal of confidence in the system, which can only be achieved through experience with tasks of which the results can be checked. +本节讨论的所有三个模型都广泛使用了代数编程系统Reduce。 [H2] 以推导 EOMs。这反映了对一种方便且安全处理非常大且复杂的代数表达式的方法的普遍需求。**Garrad 和 Schottl 在定义位置向量后，通过使用 Reduce 完成了能量表达式的完整推导。本研究的运动学分析是手工完成的，得到由矩阵和向量乘积组成的加速度表达式，然后仅使用 Reduce 进行最终的乘法和简化。 +采用此过程的主要原因是，当考虑加速度项的来源时，发现加速度的矩阵-向量形式非常具有信息量。这在优化针对特殊风力涡轮机配置的方程的过程中进行简化时，可能会非常有用。 + +但进一步来说，在作者看来，使用 Reduce 中更高级和复杂的程序需要对系统有相当的信心，这只能通过对结果可以验证的任务积累经验来实现。 # 1.2 Survey of the main elements of the model, and scope of the thesis. In order to give a more coherent representation of the elements in the model than indirectly given through the previous comparison, the main elements of the model are described next. Further the scope of the thesis is described parallel with this exposition. @@ -345,6 +451,18 @@ This is primarily done in order to incorporate the elastic rotations at the towe Both yaw rotation and teeter rotation are also present in the model, and like the rotor azimutha! rotation they are treated as bearing rotations. +为了更连贯地呈现模型中的各个要素，而不是通过之前的比较间接呈现，下面将描述模型的主要组成部分。同时，本论文的范围也将与此阐述并行地进行描述。 + +风力发电机结构被划分为三个子结构，如第2节所述： + +1. 塔架 (Tower)。 +2. 轴-机舱 (Shaft-nacelle)。 +3. 叶片（转子）(Blades (rotor))。 + +这样做主要是为了在惯性载荷推导中包含塔顶和轴端的弹性转动。但与此同时，这种划分还有两个其他重要的影响。一是现在子结构的变形是相对于局部坐标系进行描述的，这放松了如果子结构在公共坐标系中描述，否则必须施加的关于允许变形的限制，以便能够将转动表示为向量。结果是，叶片相对于塔架支撑的总体变形可以大于如果子结构在公共坐标系中描述的情况下允许的变形。另一个重要的影响是，从划分得出的方程结构，使得在每个时间步长上，可以简单地根据塔顶和轴端的弹性转动引起的几何变化来更新方程。这使得模型更加准确，允许存在适度的几何非线性。仅是适度的，因为局部角转动仍然必须较小，才能使向量表示有效，包括塔顶和轴端的弹性转动。轴承约束的转动不会引起此类问题，因为转动轴是唯一定义的，轴承转动以标量形式进入方程。 + +偏航转动和teeter转动也存在于模型中，与转子方位角转动一样，它们被视为轴承转动。 + To summarize, the division into substructures has as a result that the rigid body motions of the substructures are fully taken into account, both as regards inertia loads and change in geometry, and moderate geometric nonlinearities, as seen from a common coordinate system, are allowed for. The finite element model is used for discretization of the structure. The reasons for that choice was given in Sec. 1.1.3. @@ -357,11 +475,27 @@ Initially, the kinematic analysis is carried through, (Sec. 3). The aim is to de Input to the kinematic analysis is basically the position vector to the point in question, $\{r_{S0}\}$ The velocity and the acceleration are derived as the first and the second time derivative to the position vector, respectively. +总而言之，子结构划分的结果是，子结构的刚体运动被完全考虑在内，既涵盖惯性载荷，也涵盖几何变化，并且允许适度的几何非线性，从共同坐标系观察。 + +有限元模型用于结构的离散化。选择该方法的原因见1.1.3节。 + +运动方程（EOM）的推导以及与子结构EOM的组装，参照图2进行描述，该图说明了叶片子结构的流程。 + +牛顿直接法用于推导运动方程。选择该方法的原因见上一节，与1.1.3节中对其他模型的比较相关。 + +首先，进行运动学分析（见第3节）。目的是推导材料点上的速度和加速度。速度用于气动计算[第2部分，F节]。 + +运动学分析的输入基本上是待分析点的位移向量 $\{r_{S0}\}$。速度和加速度分别被推导为位移向量的一阶和二阶时间导数。 + Information about geometry and DOFs is used in the kinematic analysis. important variables are 1. Rotor azimuthal position: $\theta,\,\dot{\theta}=\omega,\,\ddot{\theta},$ (bearing controlled). -3. Coupling DOFs between substructures: Tower top: $\left\{q_{T\ell}^{T}\right\},\,\left\{\dot{q}_{T\ell}^{T}\right\},\,\left\{\ddot{q}_{T\ell}^{T}\right\}$ (elastic deformation). Yaw: $\theta_{3N}^{N},\,\dot{\theta}_{3N}^{N},\,\ddot{\theta}_{3N}^{N}$ (bearing controlled). Shaft end: $\left\{q_{A m}^{A}\right\},\,\left\{\dot{q}_{A m}^{A}\right\},\,\left\{\ddot{q}_{A m}^{A}\right\}$ (elastic deformation). Teeter: $\theta_{1H}^{H},\,\dot{\theta}_{1H}^{H},\,\ddot{\theta}_{1H}^{H}$ (bearing controlled). -4. Shaft length: lshaft. +2. Coupling DOFs between substructures: +3. Tower top: $\left\{q_{T\ell}^{T}\right\},\,\left\{\dot{q}_{T\ell}^{T}\right\},\,\left\{\ddot{q}_{T\ell}^{T}\right\}$ (elastic deformation). +4. Yaw: $\theta_{3N}^{N},\,\dot{\theta}_{3N}^{N},\,\ddot{\theta}_{3N}^{N}$ (bearing controlled). +5. Shaft end: $\left\{q_{A m}^{A}\right\},\,\left\{\dot{q}_{A m}^{A}\right\},\,\left\{\ddot{q}_{A m}^{A}\right\}$ (elastic deformation). +6. Teeter: $\theta_{1H}^{H},\,\dot{\theta}_{1H}^{H},\,\ddot{\theta}_{1H}^{H}$ (bearing controlled). +7. Shaft length: $\ell_{shaft}$. The transformation matrices needed in the description of the position vector $\{r_{S0}\}$ ,arederived from the elastic- and bearing-rotations, listed as number 3, and further the azimuthal position $\pmb{\theta}$ and the tit angle $\theta_{1R}^{R}$ Also the shaft length, $\ell_{s h a f t}$ , appears in the position vector. @@ -369,6 +503,21 @@ The finite element used in the model is a simple two node prismatic beam element The elastic stiffness matrix is derived by use of the constitutive relations and the principle of virtual displacements. A general expression for consistent transformation of the distributed loads to the nodes is derived. This expression is used for transformation of the inertia load to the nodes, resulting in element mass-, Coriolis-, and softening matrices as described in Sec. 4.11. +在运动学分析中，几何信息和自由度（DOFs）被用于以下变量： + +1. rotor方位角位置：$\theta,\,\dot{\theta}=\omega,\,\ddot{\theta}$，（轴承控制）。 +2. 子结构之间的耦合自由度： + 1. 塔顶：$\left\{q_{T\ell}^{T}\right\},\,\left\{\dot{q}_{T\ell}^{T}\right\},\,\left\{\ddot{q}_{T\ell}^{T}\right\}$ (弹性变形)。 + 2. 偏航：$\theta_{3N}^{N},\,\dot{\theta}_{3N}^{N},\,\ddot{\theta}_{3N}^{N}$ (轴承控制)。 + 3. 轴端：$\left\{q_{A m}^{A}\right\},\,\left\{\dot{q}_{A m}^{A}\right\},\,\left\{\ddot{q}_{A m}^{A}\right\}$ (弹性变形)。 + 4. teeter：$\theta_{1H}^{H},\,\dot{\theta}_{1H}^{H},\,\ddot{\theta}_{1H}^{H}$ (轴承控制)。 +3. 轴长：$\ell_{shaft}$。 + +描述位置向量 $\{r_{S0}\}$ 所需的变换矩阵，由弹性旋转和轴承旋转（如第3项所述）推导得出，此外还包括方位角位置 $\pmb{\theta}$ 和tilt角 $\theta_{1R}^{R}$。轴长 $\ell_{s h a f t}$ 也出现在位置向量中。 + +模型中使用的有限元是一个简单的双节点棱柱形梁单元，如第4节所述，其中还描述了完整的单元分析。插值函数是简单的多项式，它们是单元静态平衡方程的解。描述中包括剪切旋转，这通常对细长结构不重要，但为了完整性，并考虑到使用该模型分析结构细节的可能性，将其包含在内。当剪切中心位于弹性轴（中性弯曲轴）之外时产生的弯曲和扭转耦合也被包含在内。 + +弹性刚度矩阵是通过使用本构关系和虚拟位移原理推导而得出的。导出了分布式载荷到节点的一致变换的通用表达式。该表达式用于惯性载荷到节点的变换，从而得到质量、科里奥利和软化矩阵，如第4.11节所述。 ![](63c46d57a880c6e590ad0f1f4f6388b54874a224d382dfa34b5be34368c39e20.jpg) Figure 2: Derivation of blade substructure equations of motion. @@ -390,9 +539,26 @@ Similar procedures are followed for the tower and the shaft substructures. As shown schematically in Fig. 3, the substructure equations are assembled (Sec. 6) by imposing force equilibrium at the coupling nodes. Further, the boundary conditions at the tower foundation are introduced, thus removing the rigid body motion of the total structure. These conditions are assumed to be purely geometric, equivalent to zero displacement. The displacement compatibility between substructures is ensured through the kinematic analysis. +额外的惯性载荷，以向量形式表示，源于此变换。这些向量由项组成，这些项是叶片次结构之外的自由度 (DOFs) 的函数，以及转子的角速度 $\omega$。可以从向量中提取包含自由度的项，如[第2部分，E节]中的示例计算所示，从而产生额外的质量矩阵和科里奥利斯矩阵。惯性向量中的剩余项主要受转子角速度的平方 $\omega^{2}$ 相关的离心力支配。 + +离心力被用于几何刚度矩阵的推导，如第4.9节所述。 + +结构阻尼以瑞利阻尼的形式被考虑在内，元素阻尼矩阵被推导为质量矩阵和弹性刚度矩阵的线性组合（第4.13节）。 + +气动力载荷的计算采用准稳态理论，如[第2部分，F节]所述。风场包括受剪切和塔干涉影响的平均风速。湍流采用桑迪亚方法进行模拟，也如[第2部分，F节]所述。 + +重力影响被包含在模型中，目前仅作用于叶片。扩展到整个结构当然是直接的，但由于其对动力学的影响可以忽略不计，因此被省略。 + +气动力载荷和重力载荷被一致地变换到节点上。 + +通过将上述项汇集到一个方程中，可以获得叶片单元的运动方程 (EOMs)。 进一步，通过组装单元方程，可以推导出次结构运动方程，关键在于节点处的位移相容性条件。 + +塔架和轴子结构也遵循类似的程序。 + +如图3所示，次结构方程通过施加耦合节点处的力平衡条件进行组装（第6节）。 此外，引入塔架地基处的边界条件，从而消除整个结构的刚体运动。 这些条件被假定为纯几何条件，相当于零位移。 通过运动学分析确保了次结构之间的位移相容性。 ![](39a778d2c49c409cff6b98530fa999ee92e39a30626250710c759388624fe358.jpg) -# Figure 3: Assembly of substructure equations of motion. +Figure 3: Assembly of substructure equations of motion. As the resulting structure EOMs are rather complicated and strongly coupled and nonsymmetric, some attention has been payed to assess the validity of the assembly procedure. In @@ -402,14 +568,25 @@ Indirectly, the validity of the substructure formulation has further been partia In Sec. 7 the procedures used for the solution of the equations are described. A schematic representation of the solution procedure is found in Fig. 19. The Newmark implicit integration scheme is applied. The method is unconditionally stable, when the coefficient matrices are not time dependent. However, the present implementation has not revealed any instability problems for the example calculations which have been carried through so far, even if the coefficient matrices are time dependent. Due to the kinematic nonlinearity of the equations it is in general necessary to combine the integration with iterations in order to achieve equilibrium at each time step. Omitting this will usually give erronous results, because an error is accumulated. +由于得到的结构运动方程 (EOMs) 结构复杂、耦合性强且非对称，因此对组装过程的有效性进行了评估。在第 5 节，我们将当前的 EOMs 形式与使用通用坐标系获得的有限元方程的常用对称形式进行比较。对于一个简单情况，我们证明了可以从一种形式转换为另一种形式。显而易见的是，两种方程组之间的真正差异在于对刚体位移的表示方式。 + +间接而言，子结构公式的有效性已通过实现特征值求解程序（第 7 节）部分地进行了研究，尽管其目的有所不同。在处理风力涡轮机动力学时，能够计算特征值非常具有指导意义，无论是在建立仿真模型还是在需要了解动态行为时都很有用。因此，我们编写了一个特征值求解程序。该程序基于计算在通用坐标系中描述的结构的特征值，仅包括几何刚度和离心刚度，即 EOMs 对称。我们使用对称矩阵的斯图尔兹数列性质来找到特征值。接下来，这些特征值被用作子结构公式 EOMs 的逆迭代的起始值，以获得特征函数（模态）。在数值精度范围内，我们发现特征值是相同的。 + +在第 7 节，我们描述了用于求解方程的程序。解决方案程序的示意图见图 19。应用了 Newmark 隐式积分方案。当系数矩阵不随时间变化时，该方法是无条件稳定的。然而，迄今为止，即使系数矩阵随时间变化，我们当前的实现也没有发现任何不稳定性问题。由于方程的运动非线性，通常需要将积分与迭代相结合，以在每个时间步长实现平衡。省略此步骤通常会导致错误的计算结果，因为误差会累积。 The mathematical model has resulted in a computer program written in Fortran 77. This closely follows the mathematical model and therefore a detailed description is unnecessary. The thesis concludes with some calculation results. Example calculations have been carried through on a typical Danish three bladed wind turbine, which has been tested at the Test Station for Windmills at Riso, and the simulated results are compared with measurements. Reasonable agreement is demonstrated (Sec. 8). -In the example calculations the yaw DOF $(\theta_{3N}^{N})$ and the tit $(\theta_{1R}^{R})$ have been omitted, and further the angular velocity $\mathbf{\Pi}(\omega)$ has been assumed constant. This is done, simply, in order to reduce the expressions for velocity and acceleration. The tower top deformations and the shaft end deformations have been given the highest priority in the calculations. Omission of the tilt approximately halves the number of terms. The yaw DOF can still be represented in the calculations by assigning an appropriate torsional stiffness to the beam element at the tower top. +In the example calculations the yaw DOF $(\theta_{3N}^{N})$ and the tit $(\theta_{1R}^{R})$ have been omitted, and further the angular velocity $\mathbf{(\omega)}$ has been assumed constant. This is done, simply, in order to reduce the expressions for velocity and acceleration. The tower top deformations and the shaft end deformations have been given the highest priority in the calculations. Omission of the tilt approximately halves the number of terms. The yaw DOF can still be represented in the calculations by assigning an appropriate torsional stiffness to the beam element at the tower top. The thesis consists of two parts. Part 1 (the present part) covers the mathematical model and the results and constitutes the chief part of the thesis, while Part 2 is a supplement, where the most space consuming inertia matrices are listed, and some of the most important matrix manipulations are shown. Further, the complete aerodynamic model is described in Part 2. References between the two parts are given as references in general. +数学模型已经导出了一个Fortran 77程序。该程序紧密遵循数学模型，因此详细描述是不必要的。 +本论文以一些计算结果作为结论。示例计算在一个典型的丹麦三叶风力涡轮机上进行，该涡轮机在瑞索风力涡轮机测试站进行了测试，并将模拟结果与测量结果进行了比较。实验结果表明具有合理的吻合度（第8章）。 + +在示例计算中，偏航自由度 (DOF) $(\theta_{3N}^{N})$ 和tilt角 $(\theta_{1R}^{R})$ 已被省略，并且进一步假设角速度 $\mathbf{(\omega)}$ 为常数。这是为了简单地减少速度和加速度的表达式。塔顶变形和轴端变形在计算中被赋予了最高优先级。省略倾斜角大约将项数减半。偏航自由度仍然可以通过为塔顶梁单元分配适当的扭转刚度来在计算中表示。 + +本论文由两部分组成。第一部分（本部分）涵盖数学模型和结果，构成论文的主要部分，而第二部分是补充，其中列出了占用最多空间的惯性矩阵，并展示了一些最重要的矩阵操作。此外，完整的空气动力学模型在第二部分中进行了描述。两部分之间的引用以常规引用的形式给出。 # 2 Model geometry. The horizontal axis wind turbines commercially available today follow a similar structural design, regarding the main components, which are important when the overall dynamic response is considered. lf the rotor shaft and its supporting connections to the tower is regarded as one main component, the wind turbine is naturally divided into 3 main components @@ -422,6 +599,15 @@ At the same time the connections between these main components are usually chara If we can accept, that the model does not provide information about local response of details in the nacelle, the simplification relating to the nacelle and the shaft is reasonable, because it is possible to assign mass, stiffness and damping to the shaft elements, so that the overall coupling between the rotor shaft and the tower reflects the real construction in such a way that acceptable agreement is found for all lower order modes of practical interest. The results obtained with this simplified model will therefore provide the correct blade and rotor loads, which then can be used as input to a model with finer resolution, where the local details of special interest are analysed. Experience with existing response models and wind turbines confirms that the simplification is acceptable, e.g. Patel [P1] and Fabian [F1, pp. 25-34]. When we want to investigate relative changes in response, due to variation of structural parameters, the influence of the simplification is even less important. +目前商业上使用的水平轴风力发电机，在主要部件的设计上遵循相似的结构，这在考虑整体动态响应时至关重要。如果将转子轴及其与塔架的支撑连接视为一个主要部件，那么风力发电机自然可以划分为3个主要部件： + +1. 支撑塔架 +2. 转子轴 +3. 旋转叶片 + +与此同时，这些主要部件之间的连接通常具有重要的内建自由度（DOFs）。塔架和机舱之间的连接处发生偏航旋转，通常由一个具有一个绕垂直轴旋转自由度的轴承控制。进一步地，如果严格按照上述简化方法，偏航旋转和转子轴的旋转都可以被认为是发生在塔架和轴之间的连接处。在轴和旋转叶片之间，放置teete轴承。 + +如果我们可以接受该模型不提供机舱内部细节的局部响应信息，那么关于机舱和轴的简化是合理的，因为我们可以将质量、刚度和阻尼分配给轴的各个部分，使得转子轴和塔架之间的整体耦合能够反映实际的构造，从而在所有实际感兴趣的低阶模态中获得可接受的一致性。因此，使用此简化模型获得的结果将提供正确的叶片和转子载荷，然后这些载荷可以作为输入到具有更高分辨率的模型中，用于分析特别感兴趣的局部细节。现有响应模型和风力发电机经验证实，这种简化是可以接受的，例如Patel [P1] 和 Fabian [F1, pp. 25-34]。当我们想要研究由于结构参数变化引起的响应相对变化时，这种简化的影响就显得更加不重要了。 # 2.1 Definition of substructures. The present model is divided into 3 substructures corresponding to the 3 main components. @@ -483,7 +669,7 @@ All the angles are defined relative to a coordinate axis, and are considered pos Not all the coordinate systems used in the transformations are shown in Fig. 4 to avoid overcrowding the figure, and others have been displaced parallel relative to the original location, in order to show how other systems move relative to them. All applied coordinate systems will be defined uniquely, when the tranformations are described in Sec. 2.4. -# Tower support. +Tower support. The tower substructure coordinate system (index $\pmb{T}$ ) has its origin at the tower support at node T1. The $\pmb{x_{T}}-$ and ${\pmb y}_{\pmb T}$ -axis are in a horizontal plane and the ${\pmb z}{\pmb T}$ -axis is vertical downward. The tower is assumed to be rigidly supported, and the displacements to be zero at the support.