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

@@ -295,8 +295,7 @@ $$
 $$  
 
 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 substructure ­index $b$ are omitted for brevity.  
-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。
@@ -304,6 +303,7 @@ In the following, the substructure­index $b$ are omitted for brevity.
 ### 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  
+将(1.12)代入(1.8a)，展开并按${2\times2}$块矩阵形式对各项进行排序，得到质量矩阵元素
 $$
 {m_{i j}^{00}=\int_{\mathcal{V}}\rho\left(\mathbf{r}_{0,q_{i}}^{T}\mathbf{r}_{0,q_{j}}+\left(\mathbf{r}_{0,q_{i}}^{T}\mathbf{R}_{0,q_{j}}+\mathbf{r}_{0,q_{j}}^{T}\mathbf{R}_{0,q_{i}}\right)\mathbf{r}_{1}+\mathbf{r}_{1}^{T}\mathbf{R}_{0,q_{i}}^{T}\mathbf{R}_{0,q_{j}}\mathbf{r}_{1}\right)d\mathcal{V}}\tag{1.21a}
 $$
@@ -362,39 +362,99 @@ $$
 $$  
 
 where the notation $\begin{array}{r}{\pmb{\mathsf{A}}_{b a s e,i}\;\equiv\;\int_{\mathcal{V}}\!\rho\,\pmb{\mathsf{r}}_{1,q_{i}}\pmb{\mathsf{r}}_{1}^{T}d\mathcal{V}}\end{array}$ is introduced for this volume integral. Finally, the last integral in (1.22c) defined the entries of the local mass matrix of the substructure $\pmb{\mathbb{M}}^{11}$ independent of the base motion.  
-其中引入了符号$\begin{array}{r}{\pmb{\mathsf{A}}_{b a s e,i}\;\equiv\;\int_{\mathcal{V}}\!\rho\,\pmb{\mathsf{r}}_{1,q_{i}}\pmb{\mathsf{r}}_{1}^{T}d\mathcal{V}}\end{array}$来表示这个体积积分。最后，(1.22c)中的最后一个积分定义了子结构局部质量矩阵$\pmb{\mathbb{M}}^{11}$的各项，其与基础运动无关。
+其中引入了符号$\begin{array}{r}{\pmb{\mathsf{A}}_{b a s e,i}\;\equiv\;\int_{\mathcal{V}}\!\rho\,\pmb{\mathsf{r}}_{1,q_{i}}\pmb{\mathsf{r}}_{1}^{T}d\mathcal{V}}\end{array}$来表示这个体积积分。最后，(1.22c)中的最后一个积分定义了子结构局部质量矩阵$\pmb{{M}}^{11}$的各项，其与基础运动无关。
 ### Gyroscopic matrix陀螺矩阵  
 
 Inserting (1.12) into (1.8b), expanding and sorting the terms into the $_{2\times2}$ block matrix form yield the gyroscopic matrix elements  
 将 (1.12) 代入 (1.8b)，展开并整理各项成 $2\times2$ 块矩阵形式，得到陀螺矩阵元素。
+
 $$
-\begin{array}{l}{{g_{i j}^{00}=2\displaystyle\int_{\mathcal{V}}\rho\left(\mathbf{r}_{0,q_{i}}^{T}\dot{\mathbf{r}}_{0,q_{j}}+\mathbf{r}_{0,q_{i}}^{T}\dot{\mathbf{R}}_{0,q_{j}}\mathbf{r}_{1}+\dot{\mathbf{r}}_{0,q_{j}}^{T}\mathbf{R}_{0,q_{i}}\mathbf{r}_{1}+\mathbf{r}_{1}^{T}\mathbf{R}_{0,q_{i}}^{T}\dot{\mathbf{R}}_{0,q_{j}}\mathbf{r}_{1}\right)d\mathcal{V}}}\\ {{g_{i j}^{01}=2\displaystyle\int_{\mathcal{V}}\rho\left(\dot{\mathbf{r}}_{0,q_{j}}^{T}\mathbf{R}_{0}\mathbf{r}_{1,q_{i}}+\mathbf{r}_{0,q_{i}}^{T}\dot{\mathbf{R}}_{0}\mathbf{r}_{1,q_{j}}+\mathbf{r}_{1}^{T}\mathbf{R}_{0,q_{i}}^{T}\dot{\mathbf{R}}_{0}\mathbf{r}_{1,q_{j}}+\mathbf{r}_{1}^{T}\dot{\mathbf{R}}_{0,q_{j}}^{T}\mathbf{R}_{0}\mathbf{r}_{1,q_{i}}\right)d\mathcal{V}}}\\ {{g_{i j}^{11}=2\displaystyle\int_{\mathcal{V}}\rho\mathbf{r}_{1,q_{i}}^{T}\mathbf{R}_{0}^{T}\dot{\mathbf{R}}_{0}\mathbf{r}_{1,q_{j}}d\mathcal{V}}}\end{array}
+{g_{i j}^{00}=2\displaystyle\int_{\mathcal{V}}\rho\left(\mathbf{r}_{0,q_{i}}^{T}\dot{\mathbf{r}}_{0,q_{j}}+\mathbf{r}_{0,q_{i}}^{T}\dot{\mathbf{R}}_{0,q_{j}}\mathbf{r}_{1}+\dot{\mathbf{r}}_{0,q_{j}}^{T}\mathbf{R}_{0,q_{i}}\mathbf{r}_{1}+\mathbf{r}_{1}^{T}\mathbf{R}_{0,q_{i}}^{T}\dot{\mathbf{R}}_{0,q_{j}}\mathbf{r}_{1}\right)d\mathcal{V}}\tag{1.28a}
-$$  
+$$
+
+$$
+{{g_{i j}^{01}=2\displaystyle\int_{\mathcal{V}}\rho\left(\dot{\mathbf{r}}_{0,q_{j}}^{T}\mathbf{R}_{0}\mathbf{r}_{1,q_{i}}+\mathbf{r}_{0,q_{i}}^{T}\dot{\mathbf{R}}_{0}\mathbf{r}_{1,q_{j}}+\mathbf{r}_{1}^{T}\mathbf{R}_{0,q_{i}}^{T}\dot{\mathbf{R}}_{0}\mathbf{r}_{1,q_{j}}+\mathbf{r}_{1}^{T}\dot{\mathbf{R}}_{0,q_{j}}^{T}\mathbf{R}_{0}\mathbf{r}_{1,q_{i}}\right)d\mathcal{V}}}\tag{1.28b}
+$$
+$$
+{{g_{i j}^{11}=2\displaystyle\int_{\mathcal{V}}\rho\mathbf{r}_{1,q_{i}}^{T}\mathbf{R}_{0}^{T}\dot{\mathbf{R}}_{0}\mathbf{r}_{1,q_{j}}d\mathcal{V}}}\tag{1.28c}
+$$
+  
 
 and $g_{i j}^{10}$ is simply given by switching the indices $i$ and $j$ in the expression (1.28b) for $g_{i j}^{01}$ . We isolate the integration over the entire substructure volume to the local deformation vector and its derivatives by using (E.7), whereby the gyroscopic matrix elements are written as  
+$g_{i j}^{10}$ 简单地通过在 $g_{i j}^{01}$ 的表达式 (1.28b) 中切换指标 $i$ 和 $j$ 来给出。我们通过使用 (E.7) 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，从而将陀螺矩阵元素写为
 
 $$
-\begin{array}{r l}&{g_{i j}^{00}=2M\mathbf{r}_{0,q_{i}}^{T}\dot{\mathbf{t}}_{0,q_{j}}+2\left(\mathbf{r}_{0,q_{i}}^{T}\dot{\mathbf{R}}_{0,q_{j}}+\dot{\mathbf{r}}_{0,q_{j}}^{T}\mathbf{R}_{0,q_{i}}\right)M\mathbf{r}_{c g}+2\left(\mathbf{R}_{0,q_{i}}^{T}\dot{\mathbf{R}}_{0,q_{j}}\right):\mathbf{l}_{b a s e}}\\ &{g_{i j}^{01}=2\dot{\mathbf{r}}_{0,q_{j}}^{T}\mathbf{R}_{0}M\mathbf{r}_{c g,q_{i}}+2\mathbf{r}_{0,q_{i}}^{T}\dot{\mathbf{R}}_{0}M\mathbf{r}_{c g,q_{j}}+2\left(\mathbf{R}_{0,q_{i}}^{T}\dot{\mathbf{R}}_{0}\right):\mathbf{A}_{b a s e,j}+2\left(\dot{\mathbf{R}}_{0,q_{j}}^{T}\mathbf{R}_{0}\right):\mathbf{A}_{b a s e,i}}\\ &{g_{i j}^{11}=2\left(\mathbf{R}_{0}^{T}\dot{\mathbf{R}}_{0}\right):\mathsf{A}_{b a s e,1,i j}}\end{array}
+{g_{i j}^{00}=2M\mathbf{r}_{0,q_{i}}^{T}\dot{\mathbf{r}}_{0,q_{j}}+2\left(\mathbf{r}_{0,q_{i}}^{T}\dot{\mathbf{R}}_{0,q_{j}}+\dot{\mathbf{r}}_{0,q_{j}}^{T}\mathbf{R}_{0,q_{i}}\right)M\mathbf{r}_{c g}+2\left(\mathbf{R}_{0,q_{i}}^{T}\dot{\mathbf{R}}_{0,q_{j}}\right):\mathbf{I}_{b a s e}}\tag{1.29a}
-$$  
+$$
+$$
+{g_{i j}^{01}=2\dot{\mathbf{r}}_{0,q_{j}}^{T}\mathbf{R}_{0}M\mathbf{r}_{c g,q_{i}}+2\mathbf{r}_{0,q_{i}}^{T}\dot{\mathbf{R}}_{0}M\mathbf{r}_{c g,q_{j}}+2\left(\mathbf{R}_{0,q_{i}}^{T}\dot{\mathbf{R}}_{0}\right):\mathbf{A}_{b a s e,j}+2\left(\dot{\mathbf{R}}_{0,q_{j}}^{T}\mathbf{R}_{0}\right):\mathbf{A}_{b a s e,i}}\tag{1.29b}
+$$
+$$
+{g_{i j}^{11}=2\left(\mathbf{R}_{0}^{T}\dot{\mathbf{R}}_{0}\right):\mathsf{A}_{b a s e,1,i j}}\tag{1.29c}
+$$
 
+  
 where $\begin{array}{r}{\mathsf{A}_{b a s e,1,i j}\;\equiv\;\int_{\mathcal{V}}\!\rho\,\mathsf{r}_{1,q_{j}}\mathsf{r}_{1,q_{i}}^{T}d\mathcal{V}}\end{array}$ is introduced for the volume integral over the substructure of the matrix defined by the first derivatives of the local deformation vector with respect to the local DOFs.  
+其中 $\begin{array}{r}{\mathsf{A}_{base,1,ij}\;\equiv\;\int_{\mathcal{V}}\!\rho\,\mathsf{r}_{1,q_{j}}\mathsf{r}_{1,q_{i}}^{T}d\mathcal{V}}\end{array}$ 是对由局部变形向量相对于局部自由度的第一导数定义的矩阵子结构的体积积分。
 
-# Nonlinear gyroscopic matrix  
+# Nonlinear gyroscopic matrix非线性陀螺矩阵  
 
 Inserting (1.12) into (1.8c), expanding and sorting the terms into the two $_{2\times2}$ block matrix forms (1.20) yield the nonlinear gyroscopic matrix elements  
+将(1.12)代入(1.8c)，展开并按两个$_{2\times2}$块矩阵形式(1.20)对各项进行排序，得到非线性陀螺矩阵元素
 
 $$
-\begin{array}{r l}&{\tilde{h}_{i j k}^{\mathrm{OO}}=\int_{\gamma}\left(\tau_{\mathrm{O}_{i}}^{T}\mathbf{R}_{\mathrm{O}_{j},\mathrm{f}_{i}+1}+\tau_{\mathrm{O}_{j}}^{T}\mathbf{R}_{\mathrm{O}_{i},\mathrm{f}_{i}+1}+\tau_{\mathrm{O}_{j},\mathrm{f}_{i}+1}^{T}\mathbf{R}_{\mathrm{O}_{j},\mathrm{f}_{i}+1}+\tau_{\mathrm{P}_{i}}^{T}\mathbf{R}_{\mathrm{O}_{j},\mathrm{f}_{i}+1}^{T}\right)d\gamma}\\ &{\tilde{h}_{i j k}^{\mathrm{O}}=\int_{\gamma}\rho\left(\tau_{\mathrm{O}_{i}}^{T}\mathbf{R}_{\mathrm{O}_{j},\mathrm{f}_{i}+1}+\tau_{\mathrm{P}_{i}}^{T}\mathbf{R}_{\mathrm{O}_{j},\mathrm{f}_{i}+1}^{T}\right)d\gamma}\\ &{\tilde{h}_{i j k}^{\mathrm{O}}=\int_{\gamma}\rho\left(\tau_{\mathrm{O}_{i}}^{T}\mathbf{R}_{\mathrm{O}_{j},\mathrm{f}_{i}+1}+\tau_{\mathrm{P}_{i}}^{T}\mathbf{R}_{\mathrm{O}_{j},\mathrm{f}_{i}+1}^{T}\right)d\gamma}\\ &{\tilde{h}_{i j k}^{\mathrm{O}}=\int_{\gamma}\rho\left(\tau_{\mathrm{O}_{i}}^{T}\mathbf{R}_{\mathrm{O}_{j},\mathrm{f}_{i}+1}+\tau_{\mathrm{P}_{i}}^{T}\mathbf{R}_{\mathrm{O}_{j},\mathrm{f}_{i}+1}^{T}\right)d\gamma}\\ &{\tilde{h}_{i j k}^{\mathrm{O}}=\int_{\gamma}\rho\left(\tau_{\mathrm{O}_{i},\mathrm{f}_{i}+1}^{T}\mathbf{R}_{\mathrm{O}_{j},\mathrm{f}_{i}+1}+\tau_{\mathrm{P}_{i}}^{T}\mathbf{R}_{\mathrm{O}_{j},\mathrm{f}_{i}+1}^{T}\right)d\gamma}\\ &{\tilde{h}_{i j k}^{\mathrm{IO}}=\int_{\gamma}\rho\left(\tau_{\mathrm{O}_{i},\mathrm{f}_{i}+1}^{T}\mathbf{R}_{ 
+h_{ijk}^{000} = \int_{\mathcal{V}} \rho \left( \mathbf{r}_{0,q_i}^T \mathbf{r}_{0,q_j,q_k} + \mathbf{r}_{0,q_i}^T \mathbf{R}_{0,q_j,q_k} \mathbf{r}_1 + \mathbf{r}_{0,q_j,q_k}^T \mathbf{R}_{0,q_i} \mathbf{r}_1 + \mathbf{r}_1^T \mathbf{R}_{0,q_i}^T \mathbf{R}_{0,q_j,q_k} \mathbf{r}_1 \right) d\mathcal{V} \tag{1.30a}
-$$  
+$$
+$$
+h_{ijk}^{001} = \int_{\mathcal{V}} \rho \left( \mathbf{r}_{0,q_i}^T \mathbf{R}_{0,q_j} \mathbf{r}_{1,q_k} + \mathbf{r}_1^T \mathbf{R}_{0,q_i}^T \mathbf{R}_{0,q_j} \mathbf{r}_{1,q_k} \right) d\mathcal{V} \tag{1.30b}
+$$
+$$
+h_{ijk}^{010} = \int_{\mathcal{V}} \rho \left( \mathbf{r}_{0,q_i}^T \mathbf{R}_{0,q_k} \mathbf{r}_{1,q_j} + \mathbf{r}_1^T \mathbf{R}_{0,q_i}^T \mathbf{R}_{0,q_k} \mathbf{r}_{1,q_j} \right) d\mathcal{V} \tag{1.30c}
+$$
+$$
+h_{ijk}^{011} = \int_{\mathcal{V}} \rho \left( \mathbf{r}_{0,q_i}^T \mathbf{R}_{0} \mathbf{r}_{1,q_j,q_k} + \mathbf{r}_1^T \mathbf{R}_{0,q_i}^T \mathbf{R}_{0} \mathbf{r}_{1,q_j,q_k} \right) d\mathcal{V} \tag{1.30d}
+$$
+$$
+h_{ijk}^{100} = \int_{\mathcal{V}} \rho \left( \mathbf{r}_{0,q_j,q_k}^T \mathbf{R}_{0} \mathbf{r}_{1,q_i} + \mathbf{r}_1^T \mathbf{R}_{0,q_j,q_k}^T \mathbf{R}_{0} \mathbf{r}_{1,q_i} \right) d\mathcal{V} \tag{1.30e}
+$$
+$$
+h_{ijk}^{101} = \int_{\mathcal{V}} \rho \, \mathbf{r}_{1,q_i}^T \mathbf{R}_{0}^T \mathbf{R}_{0,q_j} \mathbf{r}_{1,q_k} \, d\mathcal{V} \tag{1.30f}
+$$
+$$
+h_{ijk}^{110} = \int_{\mathcal{V}} \rho \, \mathbf{r}_{1,q_i}^T \mathbf{R}_{0}^T \mathbf{R}_{0,q_k} \mathbf{r}_{1,q_j} \, d\mathcal{V} \tag{1.30g}
+$$
+$$
+h_{ijk}^{111} = \int_{\mathcal{V}} \rho \, \mathbf{r}_{1,q_i}^T \mathbf{r}_{1,q_j,q_k} \, d\mathcal{V} \tag{1.30h}
+$$
 
 where we have used the notation $()_{,q_{i},q_{j}}\equiv\partial^{2}/\partial q_{i}\partial q_{j}$ for the second derivatives with respect to DOFs $q_{i}$ and $q_{j}$ . Again, the integration over the entire substructure volume is isolated to the local deformation vector and its derivatives by using (E.7), whereby the nonlinear gyroscopic matrix elements are written as  
-
+其中我们使用了符号 $()_{,q_{i},q_{j}}\equiv\partial^{2}/\partial q_{i}\partial q_{j}$ 表示对自由度 $q_{i}$ 和 $q_{j}$ 的二阶导数。同样，通过使用 (E.7)，对整个子结构体积的积分被分离到局部变形向量及其导数，从而非线性陀螺矩阵元素被写为
 $$
-\begin{array}{r l}&{h_{i j k}^{00}=M_{0,q_{i}}^{T}\mathbf{\Delta}\mathbf{f}_{0,q_{j},q_{k}}+\left(\mathbf{r}_{0,q_{i}}^{T}\mathbf{R}_{0,q_{j},q_{k}}+\mathbf{r}_{0,q_{i},q_{k}}^{T}\mathbf{R}_{0,q_{i}}\right)M_{\mathbf{r},q}+\left(\mathbf{R}_{0,q_{i}}^{T}\mathbf{R}_{0,q_{j},q_{k}}\right):\mathbf{\Delta}_{\mathbf{b}_{\mathrm{asc}}}}\\ &{h_{i j k}^{00}=\mathbf{r}_{0,q_{i}}^{T}\mathbf{R}_{0,q_{j}}M_{\mathbf{r},q_{k}}+\left(\mathbf{R}_{0,q_{i}}^{T}\mathbf{R}_{0,q_{j}}\right):\mathbf{\Delta}\mathbf{\hat{A}}_{\mathbf{b}_{\mathrm{asc}},k}}\\ &{h_{i j k}^{01}=\mathbf{r}_{0,q_{i}}^{T}\mathbf{R}_{0,q_{k}}M_{\mathbf{r},q_{j},q_{k}}+\left(\mathbf{R}_{0,q_{i}}^{T}\mathbf{R}_{0,q_{k}}\right):\mathbf{\Delta}\mathbf{\hat{A}}_{\mathbf{b}_{\mathrm{asc}},j}}\\ &{h_{i j k}^{01}=\mathbf{r}_{0,q_{i}}^{T}\mathbf{R}_{0}M_{\mathbf{r},q_{j},q_{k}}+\left(\mathbf{R}_{0,q_{i}}^{T}\mathbf{R}_{0}\right):\mathbf{\Delta}\mathbf{\hat{A}}_{\mathbf{basc},j,j}}\\ &{h_{i j k}^{100}=\mathbf{r}_{0,q_{i},q_{k}}^{T}\mathbf{R}_{0}M\mathbf{r}_{\mathbf{c},q_{i}}+\left(\mathbf{R}_{0,q_{i},q_{k}}^{T}\mathbf{R}_{0}\right):\mathbf{\Delta}\mathbf{\hat{A}}_{\mathbf{basc},i,j}}\\ &{h_{i j k}^{101}=\left(\mathbf{R}_{0}^{T}\mathbf{R}_{0,q_{j}}\right):\mathbf{\Delta}\mathbf{\hat{A}}_{\mathbf{basc},1,i}}\\ &{h_{i j k}^{11}=\left(
+h_{ijk}^{000} = M \mathbf{r}_{0,q_i}^T \mathbf{r}_{0,q_j,q_k} + \left( \mathbf{r}_{0,q_i}^T \mathbf{R}_{0,q_j,q_k} + \mathbf{r}_{0,q_j,q_k}^T \mathbf{R}_{0,q_i} \right) M \mathbf{r}_{cg} + \left( \mathbf{R}_{0,q_i}^T \mathbf{R}_{0,q_j,q_k} \right) : \mathbf{I}_{base} \tag{1.31a}
+$$
+$$
+h_{ijk}^{001} = \mathbf{r}_{0,q_i}^T \mathbf{R}_{0,q_j} M \mathbf{r}_{cg,q_k} + \left( \mathbf{R}_{0,q_i}^T \mathbf{R}_{0,q_j} \right) : \mathbf{A}_{base,k} \tag{1.31b}
+$$
+$$
+h_{ijk}^{010} = \mathbf{r}_{0,q_i}^T \mathbf{R}_{0,q_k} M \mathbf{r}_{cg,q_j} + \left( \mathbf{R}_{0,q_i}^T \mathbf{R}_{0,q_k} \right) : \mathbf{A}_{base,j} \tag{1.31c}
+$$
+$$
+h_{ijk}^{011} = \mathbf{r}_{0,q_i}^T \mathbf{R}_{0} M \mathbf{r}_{cg,q_j,q_k} + \left( \mathbf{R}_{0,q_i}^T \mathbf{R}_{0} \right) : \mathbf{A}_{base,2,jk} \tag{1.31d}
+$$
+$$
+h_{ijk}^{100} = \mathbf{r}_{0,q_j,q_k}^T \mathbf{R}_{0} M \mathbf{r}_{cg,q_i} + \left( \mathbf{R}_{0,q_j,q_k}^T \mathbf{R}_{0} \right) : \mathbf{A}_{base,i} \tag{1.31e}
+$$
+$$
+h_{ijk}^{101} = \left( \mathbf{R}_{0}^T \mathbf{R}_{0,q_j} \right) : \mathbf{A}_{base,1,ik} \tag{1.31f}
+$$
+$$
+h_{ijk}^{110} = \left( \mathbf{R}_{0}^T \mathbf{R}_{0,q_k} \right) : \mathbf{A}_{base,1,ij} \tag{1.31g}
+$$
+$$
+h_{ijk}^{111} = \int_{\mathcal{V}} \rho \, \mathbf{r}_{1,q_i}^T \mathbf{r}_{1,q_j,q_k} \, d\mathcal{V} \tag{1.31h}
 $$  
-
 where $\begin{array}{r}{\mathsf{A}_{b a s e,2,i j}\;\equiv\;\int_{\mathcal{V}}\rho\,\mathsf{r}_{1,q_{i},q_{j}}\,\mathsf{r}_{1}^{T}d\mathcal{V}}\end{array}$ is introduced for the volume integral over the substructure of the matrix defined by the local deformation vector and its second derivatives with respect to the local DOFs.  
-
+其中 $\begin{array}{r}{\mathsf{A}_{base,2,ij}\;\equiv\;\int_{\mathcal{V}}\rho\,\mathsf{r}_{1,q_{i},q_{j}}\,\mathsf{r}_{1}^{T}d\mathcal{V}}\end{array}$ 是对由局部变形向量及其对局部自由度的二阶导数定义的矩阵子结构上的体积积分。
 # Centrifugal force vector  
 
 Inserting (1.12) into (1.8d) and expanding lead to these acceleration (centrifugal) forces on the supporting DOFs and the local substructure DOFs: