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

@@ -99,19 +99,21 @@ $$
 for $i=1,2,\dots,N_{D}$ . Here, the first term constitutes the acceleration dependent forces, whereas the other terms are only dependent on the time $t$ and the state­variables, the displacements $\mathbf{q}$ and velocities $\dot{\mathbf{q}}$ . We will now take a closer look at the first four inertia force terms given by the kinetic energy.  
 
 The total kinetic energy is the integral of the kinetic energy of each particle over the entire volume $\mathcal{V}$ of the structure:  
+对于 $i=1,2,\dots,N_{D}$。其中，第一项构成与加速度相关的力，而其他项仅取决于时间 $t$ 和状态变量，即位移 $\mathbf{q}$ 和速度 $\dot{\mathbf{q}}$。我们现在将仔细研究由动能给出的前四项惯性力。
 
+总动能是结构在整个体积 $\mathcal{V}$ 上每个粒子的动能积分：
 $$
 T=\int_{\mathcal{V}}\frac{1}{2}\;\rho\;\dot{\mathbf{r}}^{T}\dot{\mathbf{r}}\;d\mathcal{V}
 $$  
 
-where $()^{T}$ denotes to the transpose of a matrix or a vector (single columned matrix), and r˙ is the velocity vector of the particle given as the time derivative of its position vector $\boldsymbol{\mathsf{r}}=\boldsymbol{\mathsf{r}}(t,\mathbf{q})$ that may be explicit time­dependent e.g. for sub­structures that are rotating with a prescribed average speed. The velocity vector can be expanded to  
-
+where $()^{T}$ denotes to the transpose of a matrix or a vector (single columned matrix), and r˙ is the velocity vector of the particle given as the time derivative of its position vector $\boldsymbol{\mathsf{r}}=\boldsymbol{\mathsf{r}}(t,\mathbf{q})$ that may be explicit time­ dependent e.g. for sub­structures that are rotating with a prescribed average speed. The velocity vector can be expanded to  
+其中 $()^{T}$ 表示矩阵或向量（单列矩阵）的转置，$\dot{\mathbf{r}}$ 是粒子的速度向量，表示为其位置向量 $\boldsymbol{\mathsf{r}}=\boldsymbol{\mathsf{r}}(t,\mathbf{q})$ 的时间导数，该位置向量可能显式地依赖于时间，例如对于以给定平均速度旋转的子结构。速度向量可以展开为
 $$
 \dot{\pmb{r}}=\frac{d\pmb{r}}{d t}=\sum_{j=1}^{N_{D}}\frac{\partial\pmb{r}}{\partial q_{j}}\dot{q}_{j}+\frac{\partial\pmb{r}}{\partial t}
 $$  
 
 from which these properties of the position and velocity vectors can be shown  
-
+从中可以证明位置和速度向量的这些特性
 $$
 {\frac{\partial{\dot{\mathbf{r}}}}{\partial{\dot{q}}_{i}}}={\frac{\partial\mathbf{r}}{\partial q_{i}}}\quad{\mathsf{a n d}}\quad{\frac{\partial^{2}{\dot{\mathbf{r}}}}{\partial{\dot{q}}_{i}\partial q_{j}}}={\frac{\partial\mathbf{r}}{\partial q_{i}\partial q_{j}}}={\frac{\partial^{2}{\dot{\mathbf{r}}}}{\partial q_{i}\partial{\dot{q}}_{j}}}
 $$