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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

@@ -6934,23 +6934,30 @@ $$
 \ddot{\mathbf{r}}_{A}=\mathbf{L}\ddot{\mathbf{q}}+\dot{\mathbf{L}}\dot{\mathbf{q}}={\left[\begin{array}{l}{0.66}\\ {22.32}\end{array}\right]}+{\left[\begin{array}{l}{-46.702}\\ {-4.159}\end{array}\right]}={\left[\begin{array}{l}{-46.042}\\ {18.161}\end{array}\right]}\,\mathrm{m}/\mathrm{sec}^{2}
 $$  
 
-# 5.2 INERTIA OF DEFORMABLE BODIES  
+# 5.2 INERTIA OF DEFORMABLE BODIES 可變形體的慣性  
 
-In this section, we develop the kinetic energy of deformable bodies and point out the differences between the inertia properties of deformable bodies that undergo finite rotations and the inertia properties of both rigid and structural systems. In addition to the inertia tensor and the conventional mass matrix that appear, respectively, in rigid body dynamics and the dynamics of linear structural systems, it will be shown that a set of inertia shape integrals that depend on the assumed displacement field must be evaluated in order to completely define the mass matrix of deformable bodies that undergo large reference rotations. Moreover, the body inertia tensor depends on the elastic deformation and, as a consequence, is time-variant.  
+In this section, we develop the kinetic energy of deformable bodies and point out the differences between the inertia properties of deformable bodies that undergo finite rotations and the inertia properties of both rigid and structural systems. In addition to the inertia tensor and the conventional mass matrix that appear, respectively, in rigid body dynamics and the dynamics of linear structural systems, **it will be shown that a set of inertia shape integrals that depend on the assumed displacement field must be evaluated in order to completely define the mass matrix of deformable bodies that undergo large reference rotations.** Moreover, the body inertia tensor depends on the elastic deformation and, as a consequence, is time-variant.  
+在本节中，我们将推导可变形体的动能，并指出在有限转动下，可变形体的惯性特性与刚体和结构体系的惯性特性之间的差异。除了分别出现在刚体动力学和线性结构体系动力学中的惯性张量和传统质量矩阵之外，我们将展示需要**计算一组取决于假设位移场的惯性形状积分**，才能完全定义在发生大参考转动的可变形体的质量矩阵。 此外，体惯性张量取决于弹性变形，因此是随时间变化的。
 
 Mass Matrix The following definition of the kinetic energy is used:  
+惯性矩阵
+
+采用以下动能定义：
 
 $$
 T^{i}=\frac{1}{2}\int_{V^{i}}\rho^{i}\dot{\mathbf{r}}^{i^{\mathrm{T}}}\dot{\mathbf{r}}^{i}d V^{i}
 $$  
 
-where $T^{i}$ is the kinetic energy of body $i$ in the system; $\rho^{i}$ and $V^{i}$ are, respectively, the mass density and volume of body $i$ ; and ˙riis the global velocity vector of an arbitrary point on the body. Using the expression of the velocity vector of Eq. 21 given in the preceding section, one can write the kinetic energy of Eq. 39 as  
+where $T^{i}$ is the kinetic energy of body $i$ in the system; $\rho^{i}$ and $V^{i}$ are, respectively, the mass density and volume of body $i$ ; and ${\mathbf{r}}^{i}$ is the global velocity vector of an arbitrary point on the body. Using the expression of the velocity vector of Eq. 21 given in the preceding section, one can write the kinetic energy of Eq. 39 as  
+其中，$T^{i}$ 是系统内第 $i$ 个体的动能；$\rho^{i}$ 和 $V^{i}$ 分别是第 $i$ 个体的质量密度和体积；${\mathbf{r}}^{i}$ 是该个体上任意一点的全局速度矢量。利用前一节中公式 21 给出的速度矢量表达式，可以将公式 39 中的动能写为：
+
 
 $$
 T^{i}=\frac{1}{2}\int_{V^{i}}\rho^{i}\dot{\mathbf{q}}^{i^{\mathrm{T}}}\mathbf{L}^{i^{\mathrm{T}}}\mathbf{L}^{i}\dot{\mathbf{q}}^{i}d V^{i}
 $$  
 
 Since the total vector of generalized coordinates ${\bf q}^{i}$ is assumed to be time-dependent and the mass density $\rho^{i}$ may depend on the location of the point, we can write Eq. 40 as  
+由于假设广义坐标向量 ${\bf q}^{i}$ 的总和随时间变化，且质量密度 $\rho^{i}$ 可能取决于点的空间位置，我们可以将公式 40 写为：
 
 $$
 T^{i}=\frac{1}{2}\dot{\mathbf{q}}^{i}\mathbf{\Lambda}^{\!\!\mathsf{T}}\left[\int_{V^{i}}\rho^{i}\mathbf{L}^{i}\mathbf{\Lambda}^{\!\!\mathrm{T}}\mathbf{L}^{i}d V^{i}\right]\dot{\mathbf{q}}^{i}
@@ -6969,26 +6976,33 @@ $$
 $$  
 
 Using the definition of ${\bf L}^{i}$ of Eq. 22, one can write the mass matrix of body $i$ in a more explicit form as  
+使用 ${\bf L}^{i}$ 的定义（见公式 22），可以将第 $i$ 个体的质量矩阵写成更明确的形式如下：
 
 $$
-\begin{array}{r l}&{\mathbf{M}^{i}=\int_{V^{i}}\rho^{i}\left[\begin{array}{c}{\mathbf{I}}\\ {\mathbf{B}^{i^{\mathrm{T}}}}\\ {(\mathbf{A}^{i}\mathbf{S}^{i})^{\mathrm{T}}}\end{array}\right]\mathbf{I}\quad\mathbf{B}^{i}\quad\mathbf{A}^{i}\mathbf{S}^{i}]d V^{i}}\\ &{\quad=\int_{V^{i}}\rho^{i}\left[\begin{array}{c c c}{\mathbf{I}}&{\mathbf{B}^{i}}&{\mathbf{A}^{i}\mathbf{S}^{i}}\\ {\mathbf{S}^{i^{\mathrm{T}}}\mathbf{B}^{i}}&{\mathbf{B}^{i^{\mathrm{T}}}\mathbf{A}^{i}\mathbf{S}^{i}}\\ {\mathrm{symmetric}}&{\mathbf{S}^{i^{\mathrm{T}}}\mathbf{S}^{i}}\end{array}\right]d V^{i}}\end{array}
+\begin{array}{r l}&{\mathbf{M}^{i}=\int_{V^{i}}\rho^{i}\left[\begin{array}{c}{\mathbf{I}}\\ {\mathbf{B}^{i^{\mathrm{T}}}}\\ {(\mathbf{A}^{i}\mathbf{S}^{i})^{\mathrm{T}}}\end{array}\right][\mathbf{I}\quad\mathbf{B}^{i}\quad\mathbf{A}^{i}\mathbf{S}^{i}]d V^{i}}\\ &{\quad=\int_{V^{i}}\rho^{i}\left[\begin{array}{c c c}{\mathbf{I}}&{\mathbf{B}^{i}}&{\mathbf{A}^{i}\mathbf{S}^{i}}\\ 
+{}& {\mathbf{B}^{i^{\mathrm{T}}}\mathbf{B}^{i}}&{\mathbf{B}^{i^{\mathrm{T}}}\mathbf{A}^{i}\mathbf{S}^{i}}\\ {\mathrm{symmetric}}&{}&{\mathbf{S}^{i^{\mathrm{T}}}\mathbf{S}^{i}}\end{array}\right]d V^{i}}\end{array}
 $$  
 
 where the orthogonality of the transformation matrix, that is, $\mathbf{A}^{i^{\mathrm{T}}}\mathbf{A}^{i}=\mathbf{I}$ , is used in order to simplify the submatrix in the lower right-hand corner of Eq. 44. The mass matrix of Eq. 44 can also be written as  
+其中，变换矩阵的正交性，即 $\mathbf{A}^{i^{\mathrm{T}}}\mathbf{A}^{i}=\mathbf{I}$ ，被用于简化公式44的右下角子矩阵。公式44中的质量矩阵也可以写成：
+
 
 $$
-\mathbf{M}^{i}=\left[\begin{array}{c c c}{\mathbf{m}_{R R}^{i}}&{\mathbf{m}_{R\theta}^{i}}&{\mathbf{m}_{R f}^{i}}\\ &{\mathbf{m}_{\theta\theta}^{i}}&{\mathbf{m}_{\theta f}^{i}}\\ {\mathrm{symmetric}}&{\mathbf{m}_{f f}^{i}}\end{array}\right]
+\mathbf{M}^{i}=\left[\begin{array}{c c c}{\mathbf{m}_{R R}^{i}}&{\mathbf{m}_{R\theta}^{i}}&{\mathbf{m}_{R f}^{i}}\\ &{\mathbf{m}_{\theta\theta}^{i}}&{\mathbf{m}_{\theta f}^{i}}\\ {\mathrm{symmetric}} &&{\mathbf{m}_{f f}^{i}}\end{array}\right]
 $$  
 
 where  
 
 $$
-\left.\begin{array}{l}{{{\bf m}}_{R R}^{i}=\int_{V^{i}}\rho^{i}{\bf I}\,d V^{i},\qquad{\bf m}_{R\theta}^{i}=\int_{V^{i}}\rho^{i}{\bf B}^{i}d V^{i}}\\ {{\bf m}_{R f}^{i}={\bf A}^{i}\int_{V^{i}}\rho^{i}{\bf S}^{i}d V^{i},\qquad{\bf m}_{\theta\theta}^{i}=\int_{V^{i}}\rho^{i}{\bf B}^{i}{\bf\Gamma}{\bf B}^{i}d V^{i}}\\ {{\bf m}_{\theta f}^{i}=\int_{V^{i}}\rho^{i}{\bf B}^{i}{\bf\Gamma}{\bf A}^{i}{\bf S}^{i}d V^{i},\qquad{\bf m}_{f f}^{i}=\int_{V^{i}}\rho^{i}{\bf S}^{i}{\bf\Gamma}{\bf S}^{i}d V^{i}}\end{array}\right\}
+\left.\begin{array}{l}{{{\bf m}}_{R R}^{i}=\int_{V^{i}}\rho^{i}{\bf I}\,d V^{i},\qquad{\bf m}_{R\theta}^{i}=\int_{V^{i}}\rho^{i}{\bf B}^{i}d V^{i}}\\ {{\bf m}_{R f}^{i}={\bf A}^{i}\int_{V^{i}}\rho^{i}{\bf S}^{i}d V^{i},\qquad{\bf m}_{\theta\theta}^{i}=\int_{V^{i}}\rho^{i}{\mathbf{B}^{i^{\mathrm{T}}}\mathbf{B}^{i}}^{i}d V^{i}}\\ {{\bf m}_{\theta f}^{i}=\int_{V^{i}}\rho^{i}{\mathbf{B}^{i^{\mathrm{T}}}{\bf A}^{i}{\bf S}^{i}}d V^{i},\qquad{\bf m}_{f f}^{i}=\int_{V^{i}}\rho^{i}{\mathbf{S}^{i^{\mathrm{T}}}{\bf S}^{i}d V^{i}}}\end{array}\right\}
 $$  
 
 Note that the two submatrices ${\bf{m}}_{R R}^{i}$ and $\mathbf{m}_{f f}^{i}$ associated, respectively, with the translational reference and elastic coordinates, are constant. Other matrices, however,  
+请注意，与平移参考坐标和弹性坐标分别相关的两个子矩阵 ${\bf{m}}_{R R}^{i}$ 和 $\mathbf{m}_{f f}^{i}$ 是恒定的。然而，其他矩阵则不然，
 
 depend on the system generalized coordinates, and as a result they are implicit functions of time. In terms of the submatrices defined in Eq. 46, one can write the kinetic energy of the deformable body $i$ as  
+依赖于系统广义坐标，因此它们是时间的隐函数。 按照公式 46 中定义的子矩阵，可以将变形体 $i$ 的动能写为：
+
 
 $$
 \begin{array}{r l}&{T^{i}=\cfrac{1}{2}\big(\dot{\mathbf{R}}^{i^{\mathrm{T}}}\mathbf{m}_{R R}^{i}\dot{\mathbf{R}}^{i}+2\dot{\mathbf{R}}^{i^{\mathrm{T}}}\mathbf{m}_{R\theta}^{i}\dot{\mathbf{\Gamma}}\dot{\mathbf{\Gamma}}+2\dot{\mathbf{R}}^{i^{\mathrm{T}}}\mathbf{m}_{R f}^{i}\dot{\mathbf{q}}_{f}^{i}+\dot{\mathbf{\Gamma}}\dot{\mathbf{\Gamma}}^{\mathrm{\dagger}}\mathbf{m}_{\theta\theta}^{i}\dot{\mathbf{\Gamma}}}\\ &{\qquad+\,2\dot{\mathbf{\Theta}}^{i^{\mathrm{T}}}\mathbf{m}_{\theta f}^{i}\dot{\mathbf{q}}_{f}^{i}+\dot{\mathbf{q}}_{f}^{i^{\mathrm{T}}}\mathbf{m}_{f f}^{i}\dot{\mathbf{q}}_{f}^{i}\big)}\end{array}