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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

@@ -620,6 +620,15 @@ The present use of the "substructure" is not in total agreement with any of the
 
 The purpose with the present division is primarily that we want to incorporate the influence on the inertia load of the elastic angular rotations at the tower top and at the shaft end. The chosen substructuring facilitates the introduction of these angular rotations in the kinematic analysis. The considered bearing controlled rotations are furthermore located at the same two points, or at least so close that they can be assumed coincident with good approximation, and no further substructuring is necessary. Other important positive implications follw from the substructuring which will be commented on later in Sec. 2.3, when the degrees of freedom havebeen defined.  
 
+目前的模型被划分为3个子结构，分别对应于3个主要组成部分。
+
+“子结构”一词被不同的研究人员以不同的方式使用，但通常与不同的理论应用相关。Bathe [B1, p. 454] 将其术语与“子结构分析”联系起来，该分析利用静态缩聚来消除内部自由度。总结构被认为是子结构的组合。每个子结构反过来被理想化为有限元的组合，并且所有内部自由度都经过静态缩聚。每个子结构被用作有限元，该方法基本上是一种减少自由度数量的工具。当存在相同的子结构时，该方法特别有效。
+
+Meirovitch [M6, pp.384-409] 使用“子结构综合”一词来描述一系列基本相同的技术，这些技术通过线性组合模态形状来分析结构，或者更一般地，通过使用满足几何边界条件的更广泛的可接受函数 [M6, pp. 242-252] 来离散子结构。该方法被描述为有限元方法的一种替代方案。运动方程是通过使用能量原理（拉格朗日方程）获得的。一篇非常具有说明性的分析 [M6, pp. 401-409] 描述了应用于旋转子结构的理论，具有一般性。这项分析对本工作具有很大的启发性。
+
+本研究中“子结构”的使用并不完全符合上述两种应用，而只能被描述为对结构的适当细分，从而便于实际分析。
+
+目前划分的目的主要是为了将塔顶和轴端弹性角转动的影响纳入惯性载荷中。所选择的子结构化便于在运动学分析中引入这些角转动。所考虑的轴承控制转动也位于这两个点，或者至少非常接近，可以很好地近似认为它们重合，因此不需要进一步的子结构化。子结构化还产生其他重要的积极影响，将在第2.3节中对自由度定义后进行评论。
 Each substructure has a Cartesian coordinate system attached to it as shown in Fig. 4. The figure shows the undeformed structure. The final model is based on the finite element method using a simple two node prismatic beam element. A typical division of the structure in elements is shown in the figure. The local deformations are assumed small all over the structure, so that linear elastic material properties can be assumed, which means that the relation between the stress and strain is linear. This implies that the linear constitutive relations given in Eq. 4.2.1 are valid.  
 
 The elastic rotations of an element within a substructure are assumed infinitesimal, so that rotations stil can be described as a vector ([M7, pp. 104-110], [G4, Pp. 164-174]). This assumption is made in order to retain a simple relation between rotational degrees of freedom and transformation matrices, which transform vectors from one substructure coordinate system to another or from a finite element to a substructure coordinate system. If finite elastic rotations were allowed the rotations could no longer be described as a vector, which means that the order of the rotations encountered in a transformation would no longer be indifferent and more complicated expressions for the transformations would be necessary.  
@@ -632,6 +641,17 @@ In [R2] and [R3] the Euler angles have been used for a geometric nonlinear model
 
 If finite rotations should be allowed within a substructure a reasonable description should at the same time allow for an updating of the equilibrium equations in accordance with the change in geometry. This possibility is inherent in many methods, especially those developed for the finite element models.  
 
+如图4所示，每个子结构都附带一个笛卡尔坐标系。该图显示了未变形的结构。最终模型基于有限元方法，使用简单的双节点棱柱形梁单元。图中显示了结构元素的典型划分。假设整个结构中的局部变形很小，因此可以假设线性弹性材料属性，这意味着应力和应变之间的关系是线性的。这暗示了在公式4.2.1中给出的线性本构关系是有效的。
+
+子结构中一个梁单元的弹性转动被认为是无穷小，因此转动仍然可以描述为向量 ([M7, pp. 104-110], [G4, Pp. 164-174])。 做出此假设是为了保持转动自由度和变换矩阵之间的简单关系，该变换矩阵将向量从一个子结构坐标系变换到另一个子结构坐标系，或从有限元到子结构坐标系。如果允许有限弹性转动，则无法再将转动描述为向量，这意味着变换中遇到的转动不再不敏感，并且需要更复杂的变换表达式。
+
+一种可能性是使用欧拉角来描述变换 ([M7, pp. 140- 143], [G4, pp. 143-148])，这些欧拉角是指相对于3个特定轴定义的3个转动角。这些轴不是垂直的，它们的相对位置随着角度而变化。此外，规定了转动的特定序列，这使得变换是唯一的。
+
+由于上述原因，使用欧拉角来描述部分由弹性变形组成的有限转动会导致相当复杂的表达式，并且必须引入额外的自由度才能建立运动方程。使用简单的双节点梁单元被排除在外，并且在当前情况下，该方法并不具有吸引力。
+
+在[R2]和[R3]中，欧拉角已被用于描述移动和旋转杆的几何非线性模型，从而导致相当复杂的运动方程。
+
+如果在子结构中允许有限转动，那么合理的描述应该同时允许根据几何变化更新平衡方程。这种可能性蕴含在许多方法中，特别是为有限元模型开发的方法。
 ![](1d437510a4127dc4f31f91359731b83a5209472f79166e233f87d0e863d7b570.jpg)  
 Figure 4: Substructures and coordinate systems. Undeformed state.  
 
@@ -647,6 +667,17 @@ However, the experience with existing wind turbines shows that a model which is
 
 In the following two subsections the coordinate systems and the transformation matrices involved in the model are described with reference to Fig. 4.  
 
+另一种处理有限转动的方案，尤其适用于计算机数值解，可以在目前已有的、处理有限元方法中非线性结构问题的解的文献中找到。
+
+例如，在[B1, pp. 301-406]中有详细的描述，该理论与特殊有限元联系在一起。这里简要提及该方法的主要原理：在有限转动状态下的单元的姿态由与单元节点连接的一组向量来描述。通过增量过程实现对给定载荷历史的结构响应的求解，载荷以步进增加，从而产生微小转动。在每个步进中，转动可以被视为向量，并且与模型自由度（DOFs）相关的转动可以被直接处理。通常，几何形状会遵循某些指导方针进行更新，但更新频率通常低于载荷步进。
+
+结果是获得与真实结构中施加载荷时可观察到的自由度随时间变化的历史记录。为了实现参与力的可接受平衡状态，通常需要在每个载荷增量后将该方法与迭代相结合。
+
+该方法同样适用于静态和动态问题，以及各种类型的非线性问题，并且必须被认为是求解结构问题的最通用的方法，但它构成了一个相当复杂的模型，通常需要在大型计算机上运行。 [F1] 中使用的该方法与之类似。
+
+然而，现有风力涡轮机的经验表明，能够处理中等程度几何非线性的模型将足以模拟下一代涡轮机，因为考虑到面向全球市场的行业，下一代涡轮机可能只会比现有一代略微更灵活，不太可能出现在该方向上的巨大飞跃。因此，当前的该模型避免使用更复杂的有限元，并接受只能允许中等程度的转动。
+
+在接下来的两个子节中，将参考图 4 描述模型中使用的坐标系和变换矩阵。
 # 2.2 Definition of coordinate systems.  
 
 In Fig. 4 the wind turbine structure is shown in the undeformed state. The figure serves to define the substructures, the attached coordinate systems and the degrees of freedom.  
@@ -659,9 +690,23 @@ The boundaries between substructures are assumed to be single point connections
 
 The node numbering is well suited for referencing the actual points on the structure. It has been chosen as follows  
 
-Tower: Nodes are numbered from 1 to $\ell$ Tower node $T1$ is at the tower support. Tower node $_{T\ell}$ is at the tower top, and common with shaft node A1.   
-Shaft: Nodes are numbered from 1 to $\mathbf{\nabla}m$ Shaft node $_{A1}$ is at the joint to the tower, and common with tower node Te. Shaft node Am is at the joint to the blade hub, and common with blade node $B1$   
-Blade: Nodes are numbered from 1 to $\pmb{n}$ Blade node $\boldsymbol{B1}$ is at the joint to the shaft, and common with shaft node Am.  
+图 4 所示为风力发电机结构在未变形状态下的示意图。该图用于定义子结构、附着的坐标系和自由度。
+
+图中仅显示一个叶片，因为整个转子通过一个公共的叶片子结构坐标系进行描述。因此，对一个叶片点的运动学分析可以代表整个转子。转子的几何形状完全体现在有限元模型中，这意味着例如锥角、迎角和叶片扭转不会出现在运动学分析中。
+
+此外，还展示了有限元划分示例以及节点编号的顺序。叶片的节点编号未详细显示，仅显示了连接到轴的耦合节点 (B1) 和最后一个节点 (Bn) 的编号。在组装单元方程时，叶片上的实际节点编号至关重要，并在第 6 节图 18中与子结构运动方程的推导相关联。
+
+子结构之间的边界被假定为单点连接，可以用一个节点来表示。
+
+节点编号非常适合于引用结构上的实际点。其选择方式如下：
+
+Tower: Nodes are numbered from 1 to $\ell$.
+	Tower node $T1$ is at the tower support. 
+	Tower node ${T\ell}$ is at the tower top, and common with shaft node A1.   
+Shaft: Nodes are numbered from 1 to $m$.
+	Shaft node ${A1}$ is at the joint to the tower, and common with tower node Te. Shaft node ${Am}$ is at the joint to the blade hub, and common with blade node $B1$   
+Blade: Nodes are numbered from 1 to ${n}$.
+	Blade node ${B1}$ is at the joint to the shaft, and common with shaft node ${Am}$.  
 
 The reference systems are right handed, rectangular coordinate systems (Cartesian). The axis indices are designated by the symbols $\pmb{x},\pmb{y}$ and $_z$ , or synonymously by 1, 2, and 3, respectively, throughout the thesis.