Merge remote-tracking branch 'origin/master'

书籍/力学书籍/CASEstab_theory_manual/auto/CASEstab_theory_manual.md

@@ -280,62 +280,79 @@ $$
 
 ### 1.1.2 Inertia forces on and from a substructure  
 
-The inertia forces from a substructure is “felt” by all substructures supporting it, i.e., there may only be entries in rows and columns of the mass, gyroscopic, and centrifugal stiffness matrices for DOFs of the supporting substructures that appear in the vector and matrix functions $\boldsymbol{\mathsf{r}}_{0,b}$ and $\mathbf{R}_{0,b}$ . We subdivide the mass, gyroscopic, and centrifugal stiffness matrices into $_{2\times2}$ block matrices, e.g. the mass matrix as  
+The inertia forces from a substructure is “felt” by all substructures supporting it, i.e., there may only be entries in rows and columns of the mass, gyroscopic, and centrifugal stiffness matrices for DOFs of the supporting substructures that appear in the vector and matrix functions $\boldsymbol{\mathsf{r}}_{0,b}$ and $\mathbf{R}_{0,b}$ . We subdivide the mass, gyroscopic, and centrifugal stiffness matrices into ${2\times2}$ block matrices, e.g. the mass matrix as  
-
+子结构产生的惯性力被所有支撑它的子结构“感知”，即，质量、陀螺和离心刚度矩阵的行和列中可能只存在支撑子结构自由度的条目，这些自由度出现在向量和矩阵函数 $\boldsymbol{\mathsf{r}}_{0,b}$ 和 $\mathbf{R}_{0,b}$ 中。我们将质量、陀螺和离心刚度矩阵细分为 ${2\times2}$ 块矩阵，例如质量矩阵为
 $$
-\boldsymbol{\mathsf{M}}=\left[\begin{array}{l l}{\boldsymbol{\mathsf{M}}^{00}}&{\boldsymbol{\mathsf{M}}^{01}}\\ {\boldsymbol{\mathsf{M}}^{10}}&{\boldsymbol{\mathsf{M}}^{11}}\end{array}\right]
+\boldsymbol{\mathsf{M}}=\left[\begin{array}{l l}{\boldsymbol{\mathsf{M}}^{00}}&{\boldsymbol{\mathsf{M}}^{01}}\\ {\boldsymbol{\mathsf{M}}^{10}}&{\boldsymbol{\mathsf{M}}^{11}}\end{array}\right]\tag{1.19}
 $$  
 
-where the diagonal matrices $\mathbb{M}^{00}$ and $\mathbb{M}^{11}$ are the mass matrix contributions for the DOFs of the supporting sub­ structures and the DOFs of the substructure, respectively, and $\mathbb{M}^{10}$ and $\pmb{\mathbb{M}}^{01}$ are the mass matrix contributions that coupled these DOFs. Remember that $\mathbb{M}^{10}=\mathbb{M}^{01^{T}}$ because the mass matrix is symmetric; the gyroscopic and centrifugal stiffness matrices are not symmetric.  
+where the diagonal matrices $\boldsymbol{\mathsf{M}}^{00}$ and $\boldsymbol{\mathsf{M}}^{11}$ are the mass matrix contributions for the DOFs of the supporting sub­ structures and the DOFs of the substructure, respectively, and $\boldsymbol{\mathsf{M}}^{10}$ and $\boldsymbol{\mathsf{M}}^{01}$ are the mass matrix contributions that coupled these DOFs. Remember that $\boldsymbol{\mathsf{M}}^{10}=\boldsymbol{\mathsf{M}}^{01^{T}}$ because the mass matrix is symmetric; the gyroscopic and centrifugal stiffness matrices are not symmetric.  
-
+其中对角矩阵 $\boldsymbol{\mathsf{M}}^{00}$ 和 $\boldsymbol{\mathsf{M}}^{11}$ 分别是支撑子结构自由度和子结构自由度的质量矩阵贡献，而 $\boldsymbol{\mathsf{M}}^{10}$ 和 $\boldsymbol{\mathsf{M}}^{01}$ 是耦合这些自由度的质量矩阵贡献。请记住，由于质量矩阵是对称的，所以 $\boldsymbol{\mathsf{M}}^{10}=\boldsymbol{\mathsf{M}}^{01^{T}}$；陀螺和离心刚度矩阵则不对称。
 Similar, we subdivide the nonlinear three­dimensional gyroscopic matrix into two $_{2\times2}$ block matrices:  
-
+类似地，我们将非线性三维陀螺矩阵细分为两个 ${2\times2}$ 块矩阵：
 $$
-\mathsf{H}_{i}^{0}=\left[\begin{array}{l l}{\mathsf{H}^{000}}&{\mathsf{H}^{001}}\\ {\mathsf{H}^{010}}&{\mathsf{H}^{011}}\end{array}\right]\ \mathsf{a n d}\ \mathsf{H}_{i}^{1}=\left[\begin{array}{l l}{\mathsf{H}^{100}}&{\mathsf{H}^{101}}\\ {\mathsf{H}^{110}}&{\mathsf{H}^{111}}\end{array}\right]
+\mathsf{H}_{i}^{0}=\left[\begin{array}{l l}{\mathsf{H}^{000}}&{\mathsf{H}^{001}}\\ {\mathsf{H}^{010}}&{\mathsf{H}^{011}}\end{array}\right]\ \mathsf{a n d}\ \mathsf{H}_{i}^{1}=\left[\begin{array}{l l}{\mathsf{H}^{100}}&{\mathsf{H}^{101}}\\ {\mathsf{H}^{110}}&{\mathsf{H}^{111}}\end{array}\right]\tag{1.20}
 $$  
 
 where $\mathsf{H}_{i}^{0}$ contains the nonlinear gyroscopic matrix elements for DOF $q_{i}$ on a supporting substructure, and $\mathsf{H}_{i}^{1}$ contains the elements for DOF $q_{i}$ on the substructure itself. Thus, the element $h_{i j k}^{000}$ describe the coefficient of the gyroscopic force from velocities ${\dot{q}}_{j}$ and $\dot{q}_{k}$ of one or two different supporting substructures on the DOF $q_{i}$ of the same or different supporting substructure.  
 
 In the following, the substructure­index $b$ are omitted for brevity.  
+其中 $\mathsf{H}_{i}^{0}$ 包含支撑子结构上对应自由度 $q_{i}$ 的非线性陀螺矩阵元素，且 $\mathsf{H}_{i}^{1}$ 包含子结构本身上对应自由度 $q_{i}$ 的元素。因此，元素 $h_{i j k}^{000}$ 描述了来自一个或两个不同支撑子结构的速度 ${\dot{q}}_{j}$ 和 $\dot{q}_{k}$ 对相同或不同支撑子结构上自由度 $q_{i}$ 产生的陀螺力系数。
+
+在下文中，为了简洁，子结构下标 $b$ 被省略。
 
 # Mass matrix  
 
-Inserting (1.12) into (1.8a), expanding and sorting the terms into the $_{2\times2}$ block matrix form yield the mass matrix elements  
+Inserting (1.12) into (1.8a), expanding and sorting the terms into the ${2\times2}$ block matrix form yield the mass matrix elements  
-
+$$
+{m_{i j}^{00}=\int_{\mathcal{V}}\rho\left(\mathbf{r}_{0,q_{i}}^{T}\mathbf{r}_{0,q_{j}}+\left(\mathbf{r}_{0,q_{i}}^{T}\mathbf{R}_{0,q_{j}}+\mathbf{r}_{0,q_{j}}^{T}\mathbf{R}_{0,q_{i}}\right)\mathbf{r}_{1}+\mathbf{r}_{1}^{T}\mathbf{R}_{0,q_{i}}^{T}\mathbf{R}_{0,q_{j}}\mathbf{r}_{1}\right)d\mathcal{V}}\tag{1.21a}
+$$
+$$
+{m_{i j}^{01}=m_{j i}^{10}=\int_{\mathcal{V}}\rho\left(\mathbf{r}_{0,q_{i}}^{T}\mathbf{R}_{0}\mathbf{r}_{1,q_{j}}+\mathbf{r}_{0,q_{j}}^{T}\mathbf{R}_{0}\mathbf{r}_{1,q_{i}}+\mathbf{r}_{1}^{T}\mathbf{R}_{0,q_{i}}^{T}\mathbf{R}_{0}\mathbf{r}_{1,q_{j}}+\mathbf{r}_{1}^{T}\mathbf{R}_{0,q_{j}}^{T}\mathbf{R}_{0}\mathbf{r}_{1,q_{i}}\right)d\mathcal{V}}\tag{1.21b}
+$$
+$$
+{m_{i j}^{11}=\int_{\mathcal{V}}\rho\left(\mathbf{r}_{1,q_{i}}^{T}\mathbf{r}_{1,q_{j}}\right)d\mathcal{V}}\tag{1.21c}
 $$
-\begin{array}{r}{m_{i j}^{00}=\int_{\mathcal{V}}\rho\left(\mathbf{r}_{0,q_{i}}^{T}\mathbf{r}_{0,q_{j}}+\left(\mathbf{r}_{0,q_{i}}^{T}\mathbf{R}_{0,q_{j}}+\mathbf{r}_{0,q_{j}}^{T}\mathbf{R}_{0,q_{i}}\right)\mathbf{r}_{1}+\mathbf{r}_{1}^{T}\mathbf{R}_{0,q_{i}}^{T}\mathbf{R}_{0,q_{j}}\mathbf{r}_{1}\right)d\mathcal{V}}\\ {m_{i j}^{01}=m_{j i}^{10}=\int_{\mathcal{V}}\rho\left(\mathbf{r}_{0,q_{i}}^{T}\mathbf{R}_{0}\mathbf{r}_{1,q_{j}}+\mathbf{r}_{0,q_{j}}^{T}\mathbf{R}_{0}\mathbf{r}_{1,q_{i}}+\mathbf{r}_{1}^{T}\mathbf{R}_{0,q_{i}}^{T}\mathbf{R}_{0}\mathbf{r}_{1,q_{j}}+\mathbf{r}_{1}^{T}\mathbf{R}_{0,q_{j}}^{T}\mathbf{R}_{0}\mathbf{r}_{1,q_{i}}\right)d\mathcal{V}}\\ {m_{i j}^{11}=\int_{\mathcal{V}}\rho\left(\mathbf{r}_{1,q_{i}}^{T}\mathbf{r}_{1,q_{j}}\right)d\mathcal{V}}\end{array}
-$$  
 
 where we have used the notation $()_{,q_{i}}\equiv\partial/\partial q_{i}$ for the first derivatives with respect to DOF $q_{i}$ . It is possible to isolate the integration over the entire substructure volume to the local deformation vector and its derivatives by using the formula (E.7) for the matrix dot product $\equiv\mathsf{A}:\mathsf{B}$ . The mass matrix elements can thereby be written as  
+其中我们使用符号 $()_{,q_{i}}\equiv\partial/\partial q_{i}$ 表示对自由度 $q_{i}$ 的一阶导数。通过使用矩阵点积 $\equiv\mathsf{A}:\mathsf{B}$ 的公式 (E.7)，可以将整个子结构体积上的积分分离到局部变形向量及其导数。因此，质量矩阵元素可以写为
 
 $$
-\begin{array}{r l}&{\quad m_{i j}^{00}=\!\!\Gamma_{0,q_{i}}^{T}\mathbf{r}_{0,q_{j}}\!\int_{\gamma}\rho d\gamma+\left(\mathbf{r}_{0,q_{i}}^{T}\mathbf{R}_{0,q_{j}}+\mathbf{r}_{0,q_{j}}^{T}\mathbf{R}_{0,q_{i}}\right)\int_{\gamma}\rho\mathbf{r}_{1}d\gamma+\left(\mathbf{R}_{0,q_{i}}^{T}\mathbf{R}_{0,q_{j}}\right):\int_{\gamma}\rho\mathbf{r}_{1}\mathbf{r}_{1}^{T}d\gamma}\\ &{\quad m_{i j}^{01}=m_{j i}^{10}=\!\!\Gamma_{0,q_{i}}^{T}\mathbf{R}_{0}\!\int_{\gamma}\rho\mathbf{r}_{1,q_{j}}d\gamma+\mathbf{r}_{0,q_{j}}^{T}\mathbf{R}_{0}\!\int_{\gamma}\rho\mathbf{r}_{1,q_{i}}d\gamma}\\ &{\quad\quad\quad\quad\quad+\left(\mathbf{R}_{0,q_{i}}^{T}\mathbf{R}_{0}\right):\int_{\gamma}\rho\mathbf{r}_{1,q_{j}}\mathbf{r}_{1}^{T}d\gamma+\left(\mathbf{R}_{0,q_{j}}^{T}\mathbf{R}_{0}\right):\int_{\gamma}\rho\mathbf{r}_{1,q_{i}}\mathbf{r}_{1}^{T}d\gamma}\\ &{\quad\quad\quad\quad\quad\quad\quad\quad m_{i j}^{11}=\!\!\int_{\gamma}\rho\left(\mathbf{r}_{1,q_{i}}^{T}\mathbf{r}_{1,q_{j}}\right)d\gamma}\end{array}
+{\quad m_{i j}^{00}=\mathbf{r}_{0,q_{i}}^{T}\mathbf{r}_{0,q_{j}}\int_{\mathcal{V}}\rho d\mathcal{V}+\left(\mathbf{r}_{0,q_{i}}^{T}\mathbf{R}_{0,q_{j}}+\mathbf{r}_{0,q_{j}}^{T}\mathbf{R}_{0,q_{i}}\right)\int_{\mathcal{V}}\rho\mathbf{r}_{1}d\mathcal{V}+\left(\mathbf{R}_{0,q_{i}}^{T}\mathbf{R}_{0,q_{j}}\right):\int_{\mathcal{V}}\rho\mathbf{r}_{1}\mathbf{r}_{1}^{T}d\mathcal{V}}\tag{1.22a}
+$$
+
+$$
+\begin{align*}
+m_{i j}^{01}=m_{j i}^{10} ={} & \mathbf{r}_{0,q_{i}}^{T}\mathbf{R}_{0}\int_{\mathcal{V}}\rho\mathbf{r}_{1,q_{j}}d\mathcal{V}+\mathbf{r}_{0,q_{j}}^{T}\mathbf{R}_{0}\int_{\mathcal{V}}\rho\mathbf{r}_{1,q_{i}}d\mathcal{V} \\
+& +\left(\mathbf{R}_{0,q_{i}}^{T}\mathbf{R}_{0}\right):\int_{\mathcal{V}}\rho\mathbf{r}_{1,q_{j}}\mathbf{r}_{1}^{T}d\mathcal{V}+\left(\mathbf{R}_{0,q_{j}}^{T}\mathbf{R}_{0}\right):\int_{\mathcal{V}}\rho\mathbf{r}_{1,q_{i}}\mathbf{r}_{1}^{T}d\mathcal{V}
+\end{align*}\tag{1.21b}
+$$
+$$
+m_{ij}^{11}=\int_{\mathcal{V}}\rho\left(\mathbf{r}_{1,q_{i}}^{T}\mathbf{r}_{1,q_{j}}\right)d\mathcal{V}\tag{1.22c}
 $$  
 
 where the blue colored integrals of these expressions can computed in the code object of the substructure, independent of its base motion. Some of these integrals over the volume of the substructure have physical meanings as  
-
+这些表达式中蓝色积分可以在子结构的code object中计算，独立于其基础运动。其中一些在子结构体积上的积分具有物理意义，例如
 $$
-\begin{array}{r l}&{\quad\quad\displaystyle\int_{\mathcal{V}}\rho d\mathcal{V}=M}\\ &{\quad\quad\displaystyle\int_{\mathcal{V}}\rho\,\mathbf{r}_{1}d\mathcal{V}=M\left\{\begin{array}{l}{x_{c g}}\\ {y_{c g}}\\ {z_{c g}}\end{array}\right\}=M\mathbf{r}_{c g}}\\ &{\quad\displaystyle\int_{\mathcal{V}}\rho\,\mathbf{r}_{1}\mathbf{r}_{1}^{T}d\mathcal{V}=\left[\begin{array}{l l l}{i_{x x}}&{i_{x y}}&{i_{x z}}\\ {i_{x y}}&{i_{y y}}&{i_{y z}}\\ {i_{x z}}&{i_{y z}}&{i_{z z}}\end{array}\right]=\mathbf{l}_{b a s e}}\end{array}
+\begin{array}{r l}&{\quad\quad\displaystyle\int_{\mathcal{V}}\rho d\mathcal{V}=M}\\ &{\quad\quad\displaystyle\int_{\mathcal{V}}\rho\,\mathbf{r}_{1}d\mathcal{V}=M\left\{\begin{array}{l}{x_{c g}}\\ {y_{c g}}\\ {z_{c g}}\end{array}\right\}=M\mathbf{r}_{c g}}\\ &{\quad\displaystyle\int_{\mathcal{V}}\rho\,\mathbf{r}_{1}\mathbf{r}_{1}^{T}d\mathcal{V}=\left[\begin{array}{l l l}{i_{x x}}&{i_{x y}}&{i_{x z}}\\ {i_{x y}}&{i_{y y}}&{i_{y z}}\\ {i_{x z}}&{i_{y z}}&{i_{z z}}\end{array}\right]=\mathbf{I}_{b a s e}}\end{array}\tag{1.23}
 $$  
 
-where $M$ is the total mass of the substructure, $\mathbf{r}_{c g}=\{x_{c g},y_{c g},z_{c g}\}^{T}$ is the center of gravity of the substructure measured from its base in the ground­fixed inertia frame, and $\mid_{b a s e}$ is a matrix related to the rotational inertia of  
+where $M$ is the total mass of the substructure, $\mathbf{r}_{c g}=\{x_{c g},y_{c g},z_{c g}\}^{T}$ is the center of gravity of the substructure measured from its base in the ground­fixed inertia frame, and $\mathbf{I}_{b a s e}$ is a matrix related to the rotational inertia of the substructure about its base given by the integrals defined as  
-
+其中$M$是子结构的质量，$\mathbf{r}_{c g}=\{x_{c g},y_{c g},z_{c g}\}^{T}$是子结构相对于其基础在地面固定惯性系中测量的重心，$\mathbf{I}_{b a s e}$是与子结构绕其基础的转动惯量相关的矩阵，由定义的积分给出
-the substructure about its base given by the integrals defined as  
-
 $$
-i_{\alpha\beta}=\int_{\mathcal{V}}\rho\,r_{1,\alpha}r_{1,\beta}d\mathcal{V}
+i_{\alpha\beta}=\int_{\mathcal{V}}\rho\,r_{1,\alpha}r_{1,\beta}d\mathcal{V}\tag{1.24}
 $$  
 
 where the subscripts $(\alpha,\beta)\in\left\{x,y,z\right\}$ denote the coordinate component of the position vector. These integrals can be used to compute the moments and products of inertia [1] for the substructure about its base defined in the ground­fixed frame as  
-
+其中下标 $(\alpha,\beta)\in\left\{x,y,z\right\}$ 表示位置向量的坐标分量。这些积分可用于计算子结构相对于其在地面固定坐标系中定义的基座的惯性矩和惯性积。
 $$
-\begin{array}{c}{I_{x x}=i_{y y}+i_{z z}\ ,\ I_{y y}=i_{x x}+i_{z z}\ ,\ I_{z z}=i_{x x}+i_{y y}\ ,}\\ {I_{x y}=I_{y x}=-i_{x y}\ ,\ I_{x z}=I_{z x}=-i_{x z}\ ,\mathsf{a n d}\ I_{y z}=I_{z y}=-i_{y z}}\end{array}
+\begin{array}{c}{I_{x x}=i_{y y}+i_{z z}\ ,\ I_{y y}=i_{x x}+i_{z z}\ ,\ I_{z z}=i_{x x}+i_{y y}\ ,}\\ {I_{x y}=I_{y x}=-i_{x y}\ ,\ I_{x z}=I_{z x}=-i_{x z}\ ,\mathsf{a n d}\ I_{y z}=I_{z y}=-i_{y z}}\end{array}\tag{1.25}
 $$  
 
 The remaining integrals over the substructure volume of (1.22) involve the derivatives of the local position vector with respect to the DOFs of the substructure. One is the derivative of the center of gravity position:  
-
+剩余的(1.22)子结构体积积分涉及局部位置矢量相对于子结构自由度的导数。其中一个是重心位置的导数：
 $$
-\int_{\mathcal{V}}\rho\,\mathbf{r}_{1,q_{i}}d\mathcal{V}=M\mathbf{r}_{c g,q_{i}}
+\int_{\mathcal{V}}\rho\,\mathbf{r}_{1,q_{i}}d\mathcal{V}=M\mathbf{r}_{c g,q_{i}}\tag{1.26}
 $$  
 
 Another volume integral is the “asymmetric half” of the derivative of the symmetric inertia matrix