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力学书籍/OpenFast/modeling of the uae wind turbine for refinement of fast_ad/auto/modeling of the uae wind turbine for refinement of fast_ad.md

@@ -1040,12 +1040,14 @@ Equation (3.37) is applicable to either blade.  When representing blade 1, $\bet
 # 3.2 Blade and Tower Deflections  
 
 The structural model of FAST_AD considers the blades and tower to be flexible cantilevered beams with continuously distributed mass and stiffness.  In theory, such bodies possess an infinite number of DOFs, since an infinite number of coordinates are needed to specify the position of every point on the body.  In practice, such bodies are modeled as a linear sum of known shapes of the dominant normal vibration modes.  This technique is known as the normal mode summation method and reduces the number of DOFs from infinity to $N_{\ast}$ the number of normal modes considered to be dominant.  With this method, the lateral deflection (perpendicular to the undeformed beam) anywhere on the flexible beam at any time, $u(z,t)$ , is given as the summation of the products of each normal mode shape, $\phi_{a}(z)$ , and their associated generalized coordinate, $q_{a}(t)$ :  
+FAST_AD结构的模型将叶片和塔架视为具有连续分布的质量和刚度的柔性悬臂梁。理论上，这种结构体拥有无限多的自由度（DOF），因为需要无限多个坐标来指定结构体上每个点的位移。但在实践中，这种结构体被建模为由主振动模式的已知形状的线性组合。这种技术被称为正模叠加法，它将自由度数从无限减少到$N_{\ast}$，即考虑的主振动模式数量。通过这种方法，在任何时间和在柔性梁上的任意位置，横向挠度（垂直于未变形的梁），表示为$u(z,t)$，可以表示为每个正模形状函数$\phi_{a}(z)$及其相关广义坐标$q_{a}(t)$的乘积之和：
 
 $$
 u(z,t)\!=\!\sum_{a=l}^{N}\phi_{a}\!\left(z\right)\!q_{a}\!\left(t\right)
 $$  
 
 The normal mode shape for mode $a$ , $\phi_{a}(z)$ , is purely a function of the distance $z$ along the beam $[z=0$ at the fixed end and $z=Z$ at the free end) and the generalized coordinate associated with normal mode $a$ , $q_{a}(t)$ , is purely a function of time $t$ .  Each normal mode has an associated natural frequency, $\omega_{a}$ , and phase, $\psi_{a}$ .  The generalized coordinate associated with a normal mode is customarily allowed to be the deflection of the free end of the cantilever beam; thus, each normal mode shape is dimensionless and normalized so it is equal to unity at the free end.  
+对于模式 $a$ 的正常模式形状 $\phi_{a}(z)$，它纯粹是沿梁的距离 $z$ 的函数（其中 $z=0$ 为固定端，$z=Z$ 为自由端），而与该正常模式 $a$ 相关的广义坐标 $q_{a}(t)$ 纯粹是时间的函数 $t$。每个正常模式都具有相关的固有频率 $\omega_{a}$ 和相位 $\psi_{a}$。与正常模式相关的广义坐标通常允许是悬臂梁自由端的挠度；因此，每个正常模式形状都是无量纲的，并且被归一化，使其在自由端处等于一。
 
 When each normal mode shape is known, $N$ parameters are required to specify the deflection of the flexible body at any time.  Thus, alternatively, the lateral deflection of the flexible body could be expressed using $N$ other functions, $\varphi_{b}(z)$ , not unique to each normal mode:  
 

力学书籍/力学/Dynamics of Multibody Systems (Shabana A.A.) (Z-Library)/auto/Dynamics of Multibody Systems (Shabana A.A.) (Z-Library).md

@@ -5292,16 +5292,22 @@ $$
 # 4 MECHANICS OF DEFORMABLE BODIES  
 
 Thus far, only the dynamics of multibody systems consisting of interconnected rigid bodies has been discussed. In Chapter 2, methods for the kinematic analysis of the rigid frames of reference were presented and many useful kinematic relationships and identities were developed. These kinematic equations were used in Chapter 3 to develop general formulations for the dynamic differential equations of motion of multi-rigid-body systems. In rigid body dynamics, it is assumed that the distance between two arbitrary points on the body remains constant. This implies that when a force is applied to any point on the rigid body, the resultant stresses set every other point in motion instantaneously, and as shown in the preceding chapter, the force can be considered as producing a linear acceleration for the whole body together with an angular acceleration about its center of mass. The dynamic motion of the body, in this case, can be described using Newton–Euler equations, developed in the preceding chapter.  
+到目前为止，我们讨论的是由相互连接的刚体组成的多元体系统的动力学。在第二章中，我们介绍了刚体参考系的运动学分析方法，并推导出了许多有用的运动学关系和恒等式。这些运动学方程在第三章中被用来推导多元刚体系统的运动微分方程的一般形式。在刚体动力学中，我们假设刚体上任意两点之间的距离保持不变。这意味着，当一个力施加到刚体的任意一点时，产生的应力会瞬间使其他所有点发生运动，正如前一章所展示的，这个力可以被认为是为整个刚体产生一个线性加速度，以及关于其质心的角加速度。在这种情况下，该刚体的动力学运动可以使用前一章推导出的牛顿-欧拉方程来描述。
 
 In recent years, greater emphasis has been placed on the design of high-speed, lightweight, precision mechanical systems. These systems, in general, incorporate various types of driving, sensing, and controlling devices working together to achieve specified performance requirements under different loading conditions. In many of these industrial and technological applications, systems cannot be treated as collections of rigid bodies and the rigid body assumption is no longer valid. In such cases, a mechanical system can be modeled as a multibody system that consists of two collections of bodies. One collection consists of bulky compact solids that can be modeled as rigid bodies, while the second collection consists of elastic bodies, such as rods, beams, plates, and shells, that may deform. Many of these structural components are used constantly in industrial and technological applications, such as high-speed robotic manipulators, vehicle systems, airplanes, and space structures.  
+近年来，人们对高速、轻量化、高精度的机械系统设计越来越重视。这些系统通常整合了各种驱动、传感和控制设备，协同工作以在不同载荷条件下满足特定的性能要求。在许多工业和技术应用中，系统不能被视为刚体集合，刚体假设不再有效。在这种情况下，机械系统可以被建模为多体系统，它由两类体组成：一类是体积大、紧凑的实体，可以被建模为刚体；另一类是有弹性的体，例如杆、梁、板和壳体，它们可能会发生变形。这些结构部件中的许多在工业和技术应用中被持续使用，例如高速机器人操作器、车辆系统、飞机和空间结构。
 
 Continuum mechanics is concerned with the mechanical behavior of solids on the macroscopic scale and treats material as uniformly distributed throughout regions of space. It is then possible to define quantities such as density, displacement, and velocity as continuous (or at least piecewise continuous) functions of position. The study of continuum mechanics is focused on the motion of deformable bodies, which can change their shape. For such bodies the relative motion of the particles is important, and this introduces as significant kinematic variables the spatial derivatives of displacement and velocity. For deformable bodies, the relative motion between particles that form the body is important and has a significant effect on the body dynamics. When a force is applied to a point on a body, other points are not set in motion instantaneously. The effect of the force must be considered in terms of the propagation of waves.  
+连续介质力学关注的是固体在宏观尺度上的力学行为，并将材料视为均匀地分布于空间区域中。 这样，密度、位移和速度等量就可以被定义为位置的连续函数（或至少是分段连续函数）。连续介质力学研究的重点是可变形体的运动，这些体可以改变其形状。对于此类体，粒子之间的相对运动至关重要，这引入了空间位移和速度的导数作为重要的运动学变量。对于可变形体，构成该体的粒子之间的相对运动至关重要，并且对体的动力学产生显著影响。当对物体上的一个点施加力时，其他点不会瞬间开始运动。必须从波的传播角度来考虑力的作用。
 
 In this chapter, we briefly discuss the subject of continuum mechanics and introduce many concepts and definitions that are important in the development of computational methods for the dynamic analysis of multi-deformable body systems presented in subsequent chapters. First the kinematics of deformable bodies is discussed and important definitions such as the Jacobian matrix, the gradient of the displacement vector, the strain tensor, and the rotation tensor are introduced. These definitions are then used to express the strain vector in terms of the derivatives of the displacements. In Section 3, a brief discussion of the physical meaning of the strain components is provided, and in Section 4, other deformation measures are introduced. In Section 5, the stress components are defined and the important Cauchy stress formula is developed. The general form of the partial differential equations of equilibrium is derived and used to prove the symmetry of the stress tensor in Section 6. The kinematic and force relationships developed in the first six sections do not depend on the material of the body and, accordingly, apply equally to all materials. In Section 7, the constitutive relationships that serve to distinguish one material from another are discussed. Finally, an expression for the virtual work of the elastic forces in terms of the stress and strain components is developed in Section 8. Since in this chapter we will be concerned with deformation analysis of one body in the system, the superscript $i$ which denotes the body number in the multibody system will be omitted for simplicity.  
+在本章中，我们将简要讨论连续介质力学，并介绍许多在后续章节中用于动态分析多变形体系统计算方法开发中重要的概念和定义。首先，讨论变形体的运动学，并引入雅可比矩阵、位移向量的梯度、应变张量和旋转张量等重要定义。然后，利用这些定义来表达应变向量，用位移的导数来表示。在第3节中，将对应变分量的物理意义进行简要讨论，在第4节中，将介绍其他变形量。在第5节中，定义应力分量，并推导重要的柯西应力公式。在第6节中，推导平衡方程的偏微分形式，并利用其证明应力张量的对称性。前六节中建立的运动学和力学关系不依赖于体系的材料性质，因此同样适用于所有材料。在第7节中，将讨论用于区分不同材料的本构关系。最后，在第8节中，将推导一个关于应力和应变分量的弹性力的虚功表达式。由于在本章中，我们将关注体系中单个体的变形分析，为了简化，将省略表示多体体系中物体编号的下标 $i$。
 
-# 4.1 KINEMATICS OF DEFORMABLE BODIES  
+# 4.1 KINEMATICS OF DEFORMABLE BODIES  可变形体的运动学
 
-The deformation, or change of shape, of a body depends on the motion of each particle relative to its neighbors. Therefore, basic to any presentation of deformable body kinematics is the understanding of particle kinematics. We introduce a fixed rectangular Cartesian coordinate system $\mathbf{X}_{1}\mathbf{X}_{2}\mathbf{X}_{3}$ with origin $o$ . Throughout this chapter and the chapters that follow, the global motion will be motion relative to this fixed frame of reference and, unless otherwise stated, all vector and tensor components defined globally are components in the $\mathbf{X}_{1}\mathbf{X}_{2}\mathbf{X}_{3}$ coordinate system. Suppose that at time $t=0$ a deformable body occupies a fixed region of space $B_{o}$ , which may be finite or infinite in extent. Suppose that the body moves so that at a subsequent time $t$ it occupies a new continuous region of space $B$ . An assumption (which is an essential feature of continuum mechanics) will be made that we can identify individual particles of the body; that is, we assume that we can identify a point $P$ with position vector $\xi$ , which is occupied at $t$ by the particle that was at $P_{o}$ at time $t=0$ . Then the final displacement of $P$ can be written as  
+The deformation, or change of shape, of a body depends on the motion of each particle relative to its neighbors. Therefore, basic to any presentation of deformable body kinematics is the understanding of particle kinematics. We introduce a fixed rectangular Cartesian coordinate system $\mathbf{X}_{1}\mathbf{X}_{2}\mathbf{X}_{3}$ with origin $O$ . Throughout this chapter and the chapters that follow, the global motion will be motion relative to this fixed frame of reference and, unless otherwise stated, all vector and tensor components defined globally are components in the $\mathbf{X}_{1}\mathbf{X}_{2}\mathbf{X}_{3}$ coordinate system. Suppose that at time $t=0$ a deformable body occupies a fixed region of space $B_{o}$ , which may be finite or infinite in extent. Suppose that the body moves so that at a subsequent time $t$ it occupies a new continuous region of space $B$ . An assumption (which is an essential feature of continuum mechanics) will be made that we can identify individual particles of the body; that is, we assume that we can identify a point $P$ with position vector $\xi$ , which is occupied at $t$ by the particle that was at $P_{o}$ at time $t=0$ . Then the final displacement of $P$ can be written as  
+
+可变形体的变形，或形状变化，取决于每个粒子相对于其邻居的运动。因此，任何可变形体运动学介绍的基础是对粒子运动学的理解。我们引入一个固定的矩形笛卡尔坐标系 $\mathbf{X}_{1}\mathbf{X}_{2}\mathbf{X}_{3}$，原点为 $O$。在本章以及后续章节中，全局运动将是相对于这个固定参考系的运动，除非另有说明，所有定义的全局矢量和张量分量都是在 $\mathbf{X}_{1}\mathbf{X}_{2}\mathbf{X}_{3}$ 坐标系中表示的。假设在时间 $t=0$ 时，一个可变形体占据空间中的固定区域 $B_{o}$，该区域可以是有限的或无限的。假设该体移动，使得在随后的时间 $t$ 时，它占据一个新的连续空间区域 $B$。我们将做出一个假设（这是连续介质力学的一个重要特征），即我们可以识别可变形体的单个粒子；也就是说，我们假设我们可以将一个点 $P$，其位置矢量为 $\xi$，与在时间 $t$ 时被位于 $P_{o}$ 的粒子占据的位置相关联，其中 $P_{o}$ 位于时间 $t=0$ 时。那么 $P$ 的最终位移可以写为
 
 ![](edad69f6175d67e753b74c5944c7b3f2ba207c66da0ba94bf946b21a668f5a53.jpg)  
 Figure 4.1 Deformed and undeformed configurations.  
@@ -5323,6 +5329,7 @@ u_{1}=\xi_{1}-x_{1},~~~~u_{2}=\xi_{2}-x_{2},~~~~u_{3}=\xi_{3}-x_{3}
 $$  
 
 In the theory of functions, it is shown that Eq. 1 has a single-valued continuous solution if and only if the following determinant does not vanish:  
+在函数论中，可以证明方程 1 存在唯一连续解，当且仅当以下行列式不为零：
 
 $$
 |\mathbf{J}|=\left|\xi_{1,1}\right.\ \ \xi_{1,2}\ \ \ \xi_{1,3}\right|
@@ -5334,7 +5341,8 @@ $$
 \mathbf{J}=\left[\begin{array}{c c c}{1+u_{1,1}}&{u_{1,2}}&{u_{1,3}}\\ {u_{2,1}}&{1+u_{2,2}}&{u_{2,3}}\\ {u_{3,1}}&{u_{3,2}}&{1+u_{3,3}}\end{array}\right]
 $$  
 
-where $u_{1},u_{2}$ , and $u_{3}$ are the components of the displacement vector and $u_{i,j}=$ $\partial u_{i}/\partial x_{j}$ . If the particles of the body are not displaced at all, the vector $\xi$ is equal to the vector $\mathbf{X}$ , and accordingly, the displacement vector is the zero vector. In this case, the Jacobian matrix $\mathbf{J}$ is the identity matrix. Since an assumption is made that the deformation is a continuous function, the determinant of the Jacobian matrix is expected to be positive for small continuous deformation. Furthermore, the determinant of the Jacobian matrix $\mathbf{J}$ cannot become negative by a continuous deformation of the medium without passing through the excluded value which is zero. Therefore, a necessary and sufficient condition for a continuous deformation to be physically possible is that the determinant of the Jacobian matrix J be greater than zero.  
+where $u_{1},u_{2}$ , and $u_{3}$ are the components of the displacement vector and $u_{i,j}=$ $\partial u_{i}/\partial x_{j}$ . If the particles of the body are not displaced at all, the vector $\xi$ is equal to the vector $\mathbf{X}$ , and accordingly, the displacement vector is the zero vector. In this case, the Jacobian matrix $\mathbf{J}$ is the identity matrix. Since an assumption is made that the deformation is a continuous function, the determinant of the Jacobian matrix is expected to be positive for small continuous deformation. Furthermore, the determinant of the Jacobian matrix $\mathbf{J}$ cannot become negative by a continuous deformation of the medium without passing through the excluded value which is zero. Therefore, a necessary and sufficient condition for a continuous deformation to be physically possible is that the determinant of the Jacobian matrix J be greater than zero. 
+其中，$u_{1}、u_{2}$ 和 $u_{3}$ 分别是位移向量的分量，且 $u_{i,j}=$ $\partial u_{i}/\partial x_{j}$ 。如果物体的粒子完全没有位移，则向量 $\xi$ 等于向量 $\mathbf{X}$，因此位移向量为零向量。在这种情况下，雅可比矩阵 $\mathbf{J}$ 为单位矩阵。由于假设变形是连续函数，因此预计对于小的连续变形，雅可比矩阵的行列式为正。 此外，在连续变形中，雅可比矩阵 $\mathbf{J}$ 的行列式不能在不经过零（排除值）的情况下变为负。因此，连续变形在物理上可行的必要且充分条件是雅可比矩阵 J 的行列式大于零。 
 
 The Jacobian matrix of Eq. 5 can be written as  
 
@@ -5343,12 +5351,14 @@ $$
 $$  
 
 where $\mathbf{I}$ is a $3\times3$ identity matrix and $\bar{\mathbf{J}}$ is the gradient of the displacement vector defined as  
+其中 $\mathbf{I}$ 是一个 $3\times3$ 单位矩阵，而 $\bar{\mathbf{J}}$ 定义为位移向量的梯度，即
 
 $$
 \bar{\mathbf{J}}=\left[\begin{array}{r r r}{\frac{\partial u_{1}}{\partial x_{1}}}&{\frac{\partial u_{1}}{\partial x_{2}}}&{\frac{\partial u_{1}}{\partial x_{3}}}\\ {\frac{\partial u_{2}}{\partial x_{1}}}&{\frac{\partial u_{2}}{\partial x_{2}}}&{\frac{\partial u_{2}}{\partial x_{3}}}\\ {\frac{\partial u_{3}}{\partial x_{1}}}&{\frac{\partial u_{3}}{\partial x_{2}}}&{\frac{\partial u_{3}}{\partial x_{3}}}\end{array}\right]=\left[\begin{array}{r r r}{u_{1,1}}&{u_{1,2}}&{u_{1,3}}\\ {u_{2,1}}&{u_{2,2}}&{u_{2,3}}\\ {u_{3,1}}&{u_{3,2}}&{u_{3,3}}\end{array}\right]
 $$  
 
-The gradient of the displacement vector is a second-order tensor and can be represented as the sum of a symmetric tensor and antisymmetric tensor, that is,  
+The gradient of the displacement vector is a second-order tensor and can be represented as the sum of a symmetric tensor and antisymmetric tensor, that is, 
+位移向量的梯度是一个二阶张量，可以表示为一个对称张量和一个反对称张量的和，即： 
 
 $$
 \bar{\mathbf{J}}=\bar{\mathbf{J}}_{s}+\bar{\mathbf{J}}_{r}
@@ -5361,14 +5371,17 @@ $$
 $$  
 
 in which $2e_{i j}=2e_{j i}=u_{i,j}+u_{j,i}$ , $2\omega_{i j}=u_{i,j}-u_{j,i}=-2\omega_{j i}$ , and the subscript $(,i)$ denotes the differentiation with respect to $x_{i}$ . For small deformation, it will be shown later that $\bar{\mathbf{J}}_{s}$ describes the strain components at a point in the deformable body, whereas $\bar{\mathbf{J}}_{r}$ characterizes the mean rotation of a volume element.  
+其中 $2e_{ij}=2e_{ji}=u_{i,j}+u_{j,i}$，$2\omega_{ij}=u_{i,j}-u_{j,i}=-2\omega_{ji}$，且下标 $(,i)$ 表示对 $x_{i}$ 的求导。对于小变形，后续将证明 $\bar{\mathbf{J}}_{s}$ 描述了变形体中一点处的应变分量，而 $\bar{\mathbf{J}}_{r}$ 则表征了体积单元的平均旋转。
 
 Example 4.1 The displacement of a body is described in terms of the undeformed rectangular coordinates $(x_{1},x_{2},x_{3})$ as  
+例 4.1 刚体的位移，用未变形的矩形坐标 $(x_{1},x_{2},x_{3})$ 来描述，为
 
 $$
 u_{1}=k_{1}+k_{2}x_{1},\ u_{2}=k_{3}+k_{4}x_{1}+k_{5}(x_{1})^{2}+k_{6}(x_{1})^{3},\ u_{3}=0
 $$  
 
-where $k_{i}$ $_{\cdot i},(i=1,\ldots,6)$ are constants. In this case, the spatial derivatives of the vector $\mathbf{u}$ are defined as  
+where $k_{i}$ $,(i=1,\ldots,6)$ are constants. In this case, the spatial derivatives of the vector $\mathbf{u}$ are defined as  
+其中，$k_{i}$ $,(i=1,\ldots,6)$ 为常数。 此时，向量 $\mathbf{u}$ 的空间导数定义如下：
 
 $$
 \begin{array}{l l l}{{u_{1,1}=\displaystyle\frac{\partial u_{1}}{\partial x_{1}}=k_{2},\ \ }}&{{u_{1,2}=\displaystyle\frac{\partial u_{1}}{\partial x_{2}}=0,\ \ }}&{{u_{1,3}=\displaystyle\frac{\partial u_{1}}{\partial x_{3}}=0}}\\ {{u_{2,1}=\displaystyle\frac{\partial u_{2}}{\partial x_{1}}=k_{4}+2k_{5}x_{1}+3k_{6}(x_{1})^{2}}}&{{}}\\ {{u_{2,2}=\displaystyle\frac{\partial u_{2}}{\partial x_{2}}=0,\ \ }}&{{u_{2,3}=\displaystyle\frac{\partial u_{2}}{\partial x_{3}}=0}}\\ {{u_{3,1}=\displaystyle\frac{\partial u_{3}}{\partial x_{1}}=0,\ \ }}&{{u_{3,2}=\displaystyle\frac{\partial u_{3}}{\partial x_{2}}=0,\ \ }}&{{u_{3,3}=\displaystyle\frac{\partial u_{3}}{\partial x_{3}}=0}}\end{array}
@@ -5399,41 +5412,46 @@ $$
 $$  
 
 Gradient Transformation In Chapter 7 of this book, a nonlinear finite element formulation for the large deformation analysis of flexible multibody systems is presented. In this formulation, position vector gradients are used as nodal coordinates. It is, therefore, important to understand the rules that govern the transformation of the position vector gradients. In order to develop this transformation, we consider a deformable body whose material points are defined before displacements (original or reference configuration) by the position vector $\mathbf{X}$ in the coordinate system $\mathbf{X}_{1}\mathbf{X}_{2}\mathbf{X}_{3}$ and by the vector $\bar{\bf x}$ in another coordinate system $\bar{\bf X}_{1}\bar{\bf X}_{2}\bar{\bf X}_{3}$ . Let $\xi$ and $\bar{\xi}$ be, respectively, the position vectors of the material points in the $\mathbf{X}_{1}\mathbf{X}_{2}\mathbf{X}_{3}$ and $\bar{\bf X}_{1}\bar{\bf X}_{2}\bar{\bf X}_{3}$ coordinate systems after displacements (current configuration). It follows that  
+梯度变换 在本书第7章中，介绍了一种用于柔性多体系统大变形分析的非线性有限元公式。在该公式中，位置向量梯度被用作节点坐标。因此，理解控制位置向量梯度变换的规则至关重要。为了发展这种变换，我们考虑一个变形体，其材料点在位移之前（原始或参考构型）由位置向量 $\mathbf{X}$ 在坐标系 $\mathbf{X}_{1}\mathbf{X}_{2}\mathbf{X}_{3}$ 中定义，以及由向量 $\bar{\bf x}$ 在另一个坐标系 $\bar{\bf X}_{1}\bar{\bf X}_{2}\bar{\bf X}_{3}$ 中定义。设 $\xi$ 和 $\bar{\xi}$ 分别为位移后（当前构型）材料点在 $\mathbf{X}_{1}\mathbf{X}_{2}\mathbf{X}_{3}$ 和 $\bar{\bf X}_{1}\bar{\bf X}_{2}\bar{\bf X}_{3}$ 坐标系中的位置向量。由此可见，
 
 $$
 \mathbf{x}=\mathbf{A}\bar{\mathbf{x}}\,,\quad\xi(\mathbf{x})=\mathbf{A}\bar{\xi}=\xi(\bar{\mathbf{x}})
 $$  
 
-where $\mathbf{A}$ is the orthogonal transformation matrix that defines the orientation of the coordinate system $\bar{\mathbf{X}}_{1}\bar{\mathbf{X}}_{2}\bar{\mathbf{X}}_{3}$ with respect to the coordinate system $\mathbf{X}_{1}\mathbf{X}_{2}\mathbf{X}_{3}$ . The matrix  
+where $\mathbf{A}$ is the orthogonal transformation matrix that defines the orientation of the coordinate system $\bar{\mathbf{X}}_{1}\bar{\mathbf{X}}_{2}\bar{\mathbf{X}}_{3}$ with respect to the coordinate system $\mathbf{X}_{1}\mathbf{X}_{2}\mathbf{X}_{3}$ . The matrix of the position vector gradients (Jacobian matrix) can then be written as  
+其中 $\mathbf{A}$ 是正交变换矩阵，它定义了坐标系 $\bar{\mathbf{X}}_{1}\bar{\mathbf{X}}_{2}\bar{\mathbf{X}}_{3}$ 相对于坐标系 $\mathbf{X}_{1}\mathbf{X}_{2}\mathbf{X}_{3}$ 的方向。位置矢量梯度矩阵（雅可比矩阵）可以写成：
 
 ![](a9cc8fd36b363fcb1dfb3d5f7b3b12bcfda5d1450984ad244ee024ad36c5da87.jpg)  
 Undeformed configuration  
 
-Figure 4.2 Strain components.  
+Figure 4.2 Strain components. 图 4.2 应力分量。
+ 
+
 
-of the position vector gradients (Jacobian matrix) can then be written as  
 
 $$
 \mathbf{J}=\frac{\partial\boldsymbol{\xi}}{\partial\mathbf{x}}=\frac{\partial\boldsymbol{\xi}}{\partial\bar{\mathbf{x}}}\frac{\partial\bar{\mathbf{x}}}{\partial\mathbf{x}}=\frac{\partial\boldsymbol{\xi}}{\partial\bar{\mathbf{x}}}\mathbf{A}^{\mathrm{T}}
 $$  
 
 That is, the gradients of the vectors $\xi$ defined with respect to the coordinate system $\bar{\mathbf{X}}_{1}\bar{\mathbf{X}}_{2}\bar{\mathbf{X}}_{3}$ are defined as  
-
+即，关于坐标系 $\bar{\mathbf{X}}_{1}\bar{\mathbf{X}}_{2}\bar{\mathbf{X}}_{3}$ 定义的向量 $\xi$ 的梯度被定义如下：
 $$
 \frac{\partial\xi}{\partial\bar{\bf x}}={\bf J}{\bf A}=\frac{\partial\xi}{\partial{\bf x}}{\bf A}
 $$  
 
 This rule of position vector gradient transformation is crucial in developing the finite element formulation presented in Chapter 7. Note that in the preceding equation, $\xi$ is still the vector that defines the position vector of the material points in the $\mathbf{X}_{1}\mathbf{X}_{2}\mathbf{X}_{3}$ coordinate system.  
-
+位置向量梯度变换这一规则对于发展第七章中介绍的有限元公式至关重要。请注意，在上述方程中，$\xi$ 仍然是定义材料点在 $\mathbf{X}_{1}\mathbf{X}_{2}\mathbf{X}_{3}$ 坐标系中位置向量的矢量。
 # 4.2 STRAIN COMPONENTS  
 
 In this section, we introduce the strain components that arise naturally in the kinematic analysis of deformable bodies. We define $\delta l_{o}$ to be the distance between two points $P_{o}$ and $Q_{o}$ in the undeformed state as shown in Fig. 4.2 and $\delta l$ to be the distance between these two points in the deformed state. Since the coordinates of $P_{o}$ in the undeformed state are $(x_{1},x_{2},x_{3})$ , we denote the coordinates of $Q_{o}$ as $(x_{1}+d x_{1},x_{2}+d x_{2},x_{3}+d x_{3})$ . Similarly, in the deformed state we denote the coordinates of $P$ and $Q$ as $(\xi_{1},\xi_{2},\xi_{3})$ and $(\xi_{1}+d\xi_{1},\xi_{2}+d\xi_{2},\xi_{3}+d\xi_{3})$ , respectively. Therefore, the distances $\delta l_{o}$ and $\delta l$ can be determined according to  
+在本节中，我们将介绍在变形体运动学分析中自然产生的应变分量。如图4.2所示，我们定义在未变形状态下，两点 $P_{o}$ 和 $Q_{o}$ 之间的距离为 $\delta l_{o}$，在变形状态下，这两点之间的距离为 $\delta l$。由于在未变形状态下，$P_{o}$ 的坐标为 $(x_{1},x_{2},x_{3})$，我们记 $Q_{o}$ 的坐标为 $(x_{1}+d x_{1},x_{2}+d x_{2},x_{3}+d x_{3})$。 类似地，在变形状态下，我们分别记 $P$ 和 $Q$ 的坐标为 $(\xi_{1},\xi_{2},\xi_{3})$ 和 $(\xi_{1}+d\xi_{1},\xi_{2}+d\xi_{2},\xi_{3}+d\xi_{3})$。 因此，可以根据以下公式确定距离 $\delta l_{o}$ 和 $\delta l$。
 
 $$
 \left.\begin{array}{l}{(\delta l_{o})^{2}=(d{\bf x})^{\mathrm{T}}(d{\bf x})=(d x_{1})^{2}+(d x_{2})^{2}+(d x_{3})^{2}}\\ {(\delta l)^{2}=(d\xi)^{\mathrm{T}}(d\xi)=(d\xi_{1})^{2}+(d\xi_{2})^{2}+(d\xi_{3})^{2}}\end{array}\right\}
 $$  
 
-where $d\mathbf{x}=[d x_{1}\ d x_{2}\ d x_{3}]^{\mathrm{T}}$ and $d\xi=[d\xi_{1}~d\xi_{2}~d\xi_{3}]^{\mathrm{T}}$ . One may write Eq. 1 as $\boldsymbol{\xi}=\mathbf{x}+\mathbf{u}$ , from which  
+where $d\mathbf{x}=[d x_{1}\ d x_{2}\ d x_{3}]^{\mathrm{T}}$ and $d\xi=[d\xi_{1}~d\xi_{2}~d\xi_{3}]^{\mathrm{T}}$ . One may write Eq. 1 as $\boldsymbol{\xi}=\mathbf{x}+\mathbf{u}$ , from which 
+其中 $\mathbf{d}\mathbf{x}=[d x_{1}\ d x_{2}\ d x_{3}]^{\mathrm{T}}$ 且 $d\xi=[d\xi_{1}~d\xi_{2}~d\xi_{3}]^{\mathrm{T}}$ 。 可以将公式1写成 $\boldsymbol{\xi}=\mathbf{x}+\mathbf{u}$ ，由此 
 
 $$
 d\xi=d\mathbf{x}+{\frac{\partial\mathbf{u}}{\partial\mathbf{x}}}d\mathbf{x}=\bigg(\mathbf{I}+{\frac{\partial\mathbf{u}}{\partial\mathbf{x}}}\bigg)d\mathbf{x}

力学书籍/力学/多体系统动力学（第2版）＝DYNAMICS OF MULTIBODY SYSTEMS(SECOND EDITION) (刘延柱 戈新生 潘振宽) (Z-Library)/auto/多体系统动力学（第2版）＝DYNAMICS OF MULTIBODY SYSTEMS(SECOND EDITION) (刘延柱 戈新生 潘振宽) (Z-Library).md

@@ -9902,19 +9902,19 @@ $$
 $o$ 点的矢径 $r_{\mathrm{0}}$ 在惯性基 $(\mathbf{\nabla}O_{0}\,,\underline{{e}}^{(0)}$ )中的笛卡儿坐标 $\underline{{r}}_{0}^{(0)}\,=\,\bigl(\,x_{0}\,\quad y_{0}\,\quad z_{0}\,\bigr)^{\,\top}$ 确定浮动基的基点位置， $(\,O\,,\underline{{e}}\,)$ 相对 $(\,O_{\!\circ}\,,\underline{{e}}^{\,(0)}$ )的角度坐标 $\underline{{\theta}}=\left(\begin{array}{l l l}{\theta_{1}}&{\theta_{2}}&{\theta_{3}}\end{array}\right)^{\top}$ 确定浮动基的姿态,组合为浮动基的广义坐标 $\underline{{q_{f}}}$  
 
 $$
-\underline{{q}}_{f}\ =\ \big(\ \underline{{r}}_{0}^{(\ 0)\,\Gamma}\qquad\underline{{\theta}}^{\,\Gamma}\big)^{\,\intercal}
+\underline{{q}}_{f}\ =\ \big(\ \underline{{r}}_{0}^{(\ 0)\,T}\qquad\underline{{\theta}}^{\,T}\big)^{\,T}
 $$  
 
 若改用欧拉参数代替绝对坐标描述浮动基的姿态，则 $q_{f}$ 中的 $\underline{{\theta}}$ 应以欧拉参数的坐标阵 $\underline{{\boldsymbol{\Lambda}}}$ 代替。变形位移 $\underline{{\boldsymbol{u}}}$ 可利用瑞利－里兹法、有限单元法或模态分析及综合法离散为有限自由度，表示为  
 
 $$
-\underline{{u}}\ =\ \underline{{\psi}}\ q_{\ d}
+\underline{{u}}\ =\ \underline{{\psi}}\ \underline{q_{d}}
 $$  
 
-其中， $\underline{{\boldsymbol{\psi}}}$ 为形函数矩阵,即模态分析法中的模态函数矩阵， $\boldsymbol{q}_{\textit{d}}$ 为 $n_{\mathrm{d}}$ 个模态坐标组成的列阵  
+其中， $\underline{{\boldsymbol{\psi}}}$ 为形函数矩阵,即模态分析法中的模态函数矩阵， $\underline{q_{d}}$ 为 $n_{\mathrm{d}}$ 个模态坐标组成的列阵  
 
 $$
-\underline{{\boldsymbol{\psi}}}\;=\;\left(\begin{array}{l l l l}{\psi_{11}}&{\psi_{12}}&{\cdots}&{\psi_{1n_{d}}}\\ {\psi_{21}}&{\psi_{22}}&{\cdots}&{\psi_{2n_{d}}}\\ {\psi_{31}}&{\psi_{32}}&{\cdots}&{\psi_{3n_{d}}}\end{array}\right),\quad\underline{{\boldsymbol{q}}}_{\textit{d}}\;=\;\left(\begin{array}{l l l l}{q_{d1}}&{q_{d2}}&{\cdots}&{q_{d n_{d}}\right)^{\top}\;\left(9,1,32\right),}\end{array}
+\underline{{\boldsymbol{\psi}}}\;=\;\left(\begin{array}{l l l l}{\psi_{11}}&{\psi_{12}}&{\cdots}&{\psi_{1n_{d}}}\\ {\psi_{21}}&{\psi_{22}}&{\cdots}&{\psi_{2n_{d}}}\\ {\psi_{31}}&{\psi_{32}}&{\cdots}&{\psi_{3n_{d}}}\end{array}\right),\underline{{q}}_{d}\ =\ \big(\ {q}_{d1}\qquad\ {q}_{d2}\qquad{\cdots}\qquad\ {q}_{dn}\big)^{\,T}
 $$  
 
 则式（9.1.28)可写作  
@@ -9923,10 +9923,10 @@ $$
 \underline{{\dot{r}}}^{(\;0)}\;=\;\left(\:\underline{{E}}\,\:\:\:\:\:\underline{{A}}^{(\;01\rangle}\,\,\underline{{\tilde{\rho}}}^{\;\mathrm{T}}\underline{{D}}\,\right)\;\underline{{\dot{q}}}_{f}\;+\;\underline{{A}}^{(\;01\rangle}\,\underline{{\Psi}}\,\underline{{\dot{q}}}_{d}
 $$  
 
-将 $q_{j}$ 和 $q_{d}$ 组成变形体的 $6+n_{\mathrm{d}}$ 阶广义坐标阵 $\underline{{\boldsymbol{q}}}$  
+将 $q_{f}$ 和 $q_{d}$ 组成变形体的 $6+n_{\mathrm{d}}$ 阶广义坐标阵 $\underline{{q}}$  
 
 $$
-\underline{{q}}\ =\ \big(\,\underline{{q}}_{f}^{\operatorname{T}}\qquad\underline{{q}}_{\ }^{\operatorname{T}}\big)^{\operatorname{T}}\ =\ \big(\,\underline{{r}}_{0}^{\operatorname{(0)T}}\qquad\underline{{\theta}}^{\operatorname{T}}\qquad\underline{{q}}_{\ }^{\operatorname{T}}\big)^{\operatorname{T}}
+\underline{{q}}\ =\ \big(\,\underline{{q}}_{f}^{\operatorname{T}}\qquad\underline{{q}}_{d}{\ }^{\operatorname{T}}\big)^{\operatorname{T}}\ =\ \big(\,\underline{{r}}_{0}^{\operatorname{(0)T}}\qquad\underline{{\theta}}^{\operatorname{T}}\qquad\underline{{q}}_{d}{\ }^{\operatorname{T}}\big)^{\operatorname{T}}
 $$  
 
 则式（9.1.33)可用广义速度 $\dot{q}$ 表示为