diff --git a/书籍/力学书籍/CASEstab_theory_manual/auto/CASEstab_theory_manual.md b/书籍/力学书籍/CASEstab_theory_manual/auto/CASEstab_theory_manual.md
index 30ffe18..1b479fc 100644
--- a/书籍/力学书籍/CASEstab_theory_manual/auto/CASEstab_theory_manual.md
+++ b/书籍/力学书籍/CASEstab_theory_manual/auto/CASEstab_theory_manual.md
@@ -72,7 +72,7 @@ E.3 Miscellaneous formulas 39
 
 This chapter contains the derivations of the nonlinear equations of motion based on the kinematic formulation of articulated substructures to form the entire turbine structure.  
 本章包含基于铰接子结构运动学公式的非线性运动方程的推导，以形成整个机组结构。
-# 1.1 Lagrange equations of articulated substructures  
+## 1.1 Lagrange equations of articulated substructures  
 
 The Lagrangian of a structure ${{L}}=T-V$ is given by its total kinetic energy $T$ and the total potential energy $V$ of the conservative forces acting on the structure, e.g. gravity and elastic forces. Using Lagrange’s equations, the nonlinear equations of motion can be derived as  
 结构的拉格朗日量 ${{L}}=T-V$ 由其总动能 $T$ 和作用在结构上的保守力（例如重力和弹力）的总势能 $V$ 给出。利用拉格朗日方程，可以推导出非线性运动方程，如下所示：
@@ -147,7 +147,7 @@ The fifth term of (1.7) describes the conservative forces such as gravity and el
 
 (1.7)式的第五项描述了保守力，例如重力和弹力，这些力可以通过势能$V$来定义。由标量势能函数导数给出的力函数取决于运动学公式和所应用的弹性理论。(1.7)式的第六项描述了来自结构阻尼机制（例如材料和摩擦）的纯耗散力，我们将使用在第...节中描述的模态阻尼方法对其进行建模。(1.7)式的最后一项描述了作用在结构上的广义非保守力，这些力将在第...章中讨论。. .  
 
-# 1.1.1 Generic topology of structure  
+### 1.1.1 Generic topology of structure  
 
 The turbine structure is divided into a number of articulated substructures where one substructure may be connected to one or more substructures through its connection points. Let $b$ be the index of a substructure then we herein assume that it is connected to substructure $b-1$ which is connected to substructure $b-2$ and so on until substructure 0. In practice, this numbering will not be continuous (e.g. the blades will each have their own number but connected to the same hub which can have one number), but we can always create an intermediate index list for each substructure in which the numbering is continuous.  
 
@@ -176,35 +176,86 @@ $$
 $$  
 
 where the translations and rotations of the substructure base are described by  
-
+其中，子结构基础的平动和转动由
 $$
-\mathbf{r}_{0,b}=\sum_{k=0}^{b-1}\mathbf{R}_{0,k}\mathbf{r}_{k}=\mathbf{r}_{0,b-1}+\mathbf{R}_{0,b-1}\mathbf{r}_{b-1}\quad{\mathrm{and}}\quad\mathbf{R}_{0,b}=\left(\prod_{l=0}^{b-1}\mathbf{O}_{l}\right)\mathbf{S}_{b}\mathbf{B}_{b}=\mathbf{R}_{0,b-1}\mathbf{R}_{b-1}\mathbf{S}_{b-1}^{T}\mathbf{S}_{b}\mathbf{B}_{b}
+\mathbf{r}_{0,b}=\sum_{k=0}^{b-1}\mathbf{R}_{0,k}\mathbf{r}_{k}=\mathbf{r}_{0,b-1}+\mathbf{R}_{0,b-1}\mathbf{r}_{b-1}\quad{\mathrm{and}}\quad\mathbf{R}_{0,b}=\left(\prod_{l=0}^{b-1}\mathbf{O}_{l}\right)\mathbf{S}_{b}\mathbf{B}_{b}=\mathbf{R}_{0,b-1}\mathbf{R}_{b-1}\mathbf{S}_{b-1}^{T}\mathbf{S}_{b}\mathbf{B}_{b}\tag{1.13}
 $$  
 
 where $\mathsf{r}_{k}$ is the local position vector in the frame of substructure $k$ to which substructure $k+1$ is connected; this vector is independent of time because there are no bearings involved. We use the index “1” for the local substructure deformation given by $\boldsymbol{\mathsf{r}}_{1,b}$ and $^{\ast}0^{\ast}$ for the deformation and orientation of its base given by $\boldsymbol{\mathsf{r}}_{0,b}$ and $\pmb{\mathsf{R}}_{0,b}$ . Note that these position vector and orientation matrix are functions of DOFs of substructures supporting the substructure $b$ , but not of DOFs of other substructures or substructure $b$ itself.  
+其中 $\mathsf{r}_{k}$ 是子结构 $k$ 坐标系中的局部位置矢量，子结构 $k+1$ 连接到该子结构 $k$；该矢量与时间无关，因为不涉及轴承。我们使用索引“1”表示由 $\boldsymbol{\mathsf{r}}_{1,b}$ 给出的局部子结构变形，使用 $^{\ast}0^{\ast}$ 表示由 $\boldsymbol{\mathsf{r}}_{0,b}$ 和 $\pmb{\mathsf{R}}_{0,b}$ 给出的其基座的变形和方向。请注意，这些位置矢量和方向矩阵是支撑子结构 $b$ 的自由度的函数，而不是其他子结构或子结构 $b$ 本身自由度的函数。
 
 In the subsequent derivations, we will need the time derivatives:  
 
 $$
-\begin{array}{r l}&{\tilde{F}_{\mathrm{0,A}}=\displaystyle\sum_{k=1}^{k-1}\mathbb{E}_{\boldsymbol{u},\boldsymbol{u}^{k}}}\\ &{\tilde{\mathbf{R}}_{\mathrm{0,A}}=\displaystyle\sum_{k=1}^{k-1}\left(\prod_{i=0}^{n-1}\mathbf{\sigma}_{0}\right)\le\mathbf{1}_{\mathcal{B}_{i}\setminus\mathbb{R}_{i}}\mathbf{1}_{\mathcal{S}_{i}^{\top}}\left(\prod_{i=1}^{k-1}\mathbf{\sigma}_{0}\right)\le\mathbf{1}_{\mathcal{B}_{i}\setminus+}\left(\prod_{i=0}^{n-1}\mathbf{\sigma}_{0}\right)\le\mathbf{1}_{\mathcal{B}_{i}},}\\ &{\tilde{F}_{\mathrm{0,A}}=\displaystyle\sum_{k=0}^{k-1}\mathbb{E}_{\boldsymbol{u},\boldsymbol{u}^{k}}}\\ &{\tilde{F}_{\mathrm{0,A}}=\displaystyle\sum_{k=1}^{k-1}\mathbb{E}_{\boldsymbol{u},\boldsymbol{u}^{k}}}\\ &{\tilde{\mathbf{R}}_{\mathrm{0,B}}=\displaystyle\sum_{k=0}^{k-1}\left(\sum_{i=0}^{k-1}\bigg(\prod_{i=1}^{n}\mathbf{\sigma}_{i}\bigg)\le\mathbf{1}_{\mathcal{B}_{i}\setminus\mathbb{R}_{i}}\mathbf{1}_{\mathcal{S}_{i}^{\top}}\left(\prod_{i=1}^{k-1}\mathbf{\sigma}_{i}\right)\right)\le\mathbf{1}_{\mathcal{B}_{i}\setminus\mathbb{R}_{i}}\mathbf{1}_{\mathcal{S}_{i}^{\top}}\left(\prod_{i=1}^{k-1}\mathbf{\sigma}_{i}\right)\le\mathbf{1}_{\mathcal{B}_{i}},}\\ &{\qquad+\displaystyle\sum_{k=1}^{k-1}\left(\prod_{i=0}^{k-1}\mathbf{\sigma}_{i}\right)\le\mathbf{1}_{\mathcal{B}_{i}\setminus\mathbb{R}_{i}}\mathbf{1}_{\mathcal{S}_{i}^{\top}}\left(\begin{array}{c c c}{\displaystyle\sum_{i=1}^{k-1}\left(\prod_{i=1}^{n}\mathbf{\sigma}_{i}\right)}&{\mathbf{S}_{i}\mathbf{1}_{\mathcal{B}_{i}}\mathbf{1}_{\mathcal{S}_{i}^{\top}}
+\mathbf{r}_{0,b} = \sum_{k=0}^{b-1} \mathbf{R}_{0,k} \mathbf{r}_k\tag{1.14a}
+$$
+
+$$
+\dot{\mathbf{R}}_{0,b} = \sum_{k=0}^{b-1} \left( \prod_{l=0}^{k-1} \mathbf{O}_l \right) s_k \dot{\mathbf{B}}_k \mathbf{R}_k \mathbf{S}_k^T \left( \prod_{l=k+1}^{b-1} \mathbf{O}_l \right) s_b \mathbf{B}_b + \left( \prod_{l=0}^{b-1} \mathbf{O}_l \right) s_b \dot{\mathbf{B}}_b\tag{1.14b}
+$$
+
+$$
+\ddot{\mathbf{r}}_{0,b} = \sum_{k=0}^{b-1} \ddot{\mathbf{R}}_{0,k} \mathbf{r}_k\tag{1.14c}
+$$
+
+$$
+\begin{aligned}
+\ddot{\mathbf{R}}_{0,b} = & \sum_{k=0}^{b-1} \left( \sum_{m=0}^{k-1} \left( \prod_{l=0}^{m-1} \mathbf{O}_l \right) s_m \ddot{\mathbf{B}}_m \mathbf{R}_m \mathbf{S}_m^T \left( \prod_{l=m+1}^{k-1} \mathbf{O}_l \right) \right) s_k \dot{\mathbf{B}}_k \mathbf{R}_k \mathbf{S}_k^T \left( \prod_{l=k+1}^{b-1} \mathbf{O}_l \right) s_b \mathbf{B}_b \\
+& + \sum_{k=0}^{b-1} \left( \prod_{l=0}^{k-1} \mathbf{O}_l \right) s_k \dot{\mathbf{B}}_k \mathbf{R}_k \mathbf{S}_k^T \left( \sum_{m=k+1}^{b-1} \left( \prod_{l=k+1}^{m-1} \mathbf{O}_l \right) s_m \ddot{\mathbf{B}}_m \mathbf{R}_m \mathbf{S}_m^T \left( \prod_{l=m+1}^{b-1} \mathbf{O}_l \right) \right) s_b \mathbf{B}_b \\
+& + \sum_{k=0}^{b-1} \left( \prod_{l=0}^{k-1} \mathbf{O}_l \right) s_k \ddot{\mathbf{B}}_k \mathbf{R}_k \mathbf{S}_k^T \left( \prod_{l=k+1}^{b-1} \mathbf{O}_l \right) s_b \mathbf{B}_b \\
+& + 2 \sum_{k=0}^{b-1} \left( \prod_{l=0}^{k-1} \mathbf{O}_l \right) s_k \dot{\mathbf{B}}_k \mathbf{R}_k \mathbf{S}_k^T \left( \prod_{l=k+1}^{b-1} \mathbf{O}_l \right) s_b \dot{\mathbf{B}}_b + \left( \prod_{l=0}^{b-1} \mathbf{O}_l \right) s_b \ddot{\mathbf{B}}_b
+\end{aligned} \tag{1.14d}
 $$  
 
 where we have used that the bearing matrices $\mathbf{B}_{k}$ represent the only explicit time dependency. Note that some bearings have a constant angle or are function of a free bearing DOF, whereby $\dot{\ B_{k}}=\ddot{\ B}_{k}=\pmb{0}$ . In most applications, there will only be a single bearing with an explicit time dependency in terms of a constant speed of $\Omega$ . Let this single bearing be for substructure number $k$ then the time derivatives (1.14) reduce to  
+其中我们使用了轴承矩阵 $\mathbf{B}_{k}$ 代表了唯一的显式时间依赖性。注意，有些轴承具有恒定角度或是一个自由轴承自由度的函数，因此 $\dot{\ B_{k}}=\ddot{\ B}_{k}=\pmb{0}$ 。在大多数应用中，只有一个轴承具有以恒定速度 $\Omega$ 表示的显式时间依赖性。假设这个单一轴承对应于子结构编号 $k$，则时间导数 (1.14) 简化为
 
 $$
-\begin{array}{r l}&{\dot{\mathbf{r}}_{0,b}=\displaystyle\sum_{l=k}^{b-1}\dot{\mathbf{R}}_{0,l}\mathbf{r}_{l}}\\ &{\dot{\mathbf{R}}_{0,b}=\Omega\left(\prod_{l=0}^{k-1}\mathbf{O}_{l}\right)\,\mathbf{S}_{k}\mathbf{B}_{k}\mathbf{N}_{k}\mathbf{R}_{k}\mathbf{S}_{k}^{T}\,\left(\displaystyle\prod_{l=k+1}^{b-1}\mathbf{O}_{l}\right)\mathbf{S}_{b}\mathbf{B}_{b}=\Omega\mathbf{R}_{0,k}\mathbf{N}_{k}\mathbf{R}_{0,k}^{T}\mathbf{R}_{0,b}}\\ &{\ddot{\mathbf{r}}_{0,b}=\displaystyle\sum_{l=k}^{b-1}\dot{\mathbf{R}}_{0,l}\mathbf{r}_{l}}\\ &{\ddot{\mathbf{R}}_{0,b}=\Omega^{2}\left(\prod_{l=0}^{k-1}\mathbf{O}_{l}\right)\,\mathbf{S}_{k}\mathbf{B}_{k}\mathbf{N}_{k}^{2}\mathbf{R}_{k}\mathbf{S}_{k}^{T}\,\left(\displaystyle\prod_{l=k+1}^{b-1}\mathbf{O}_{l}\right)\,\mathbf{S}_{b}\mathbf{B}_{b}=\Omega^{2}\mathbf{R}_{0,k}\mathbf{N}_{k}^{2}\mathbf{R}_{0,k}^{T}\mathbf{R}_{0,l}}\end{array}
-$$  
+\dot{\mathbf{r}}_{0,b} = \sum_{l=k}^{b-1} \dot{\mathbf{R}}_{0,l} \mathbf{r}_l \tag{1.15a}
+$$
+
+$$
+\dot{\mathbf{R}}_{0,b} = \Omega \left( \prod_{l=0}^{k-1} \mathbf{O}_l \right) s_k \mathbf{B}_k \mathbf{N}_k \mathbf{R}_k \mathbf{S}_k^T \left( \prod_{l=k+1}^{b-1} \mathbf{O}_l \right) s_b \mathbf{B}_b = \Omega \mathbf{R}_{0,k} \mathbf{N}_k \mathbf{R}_{0,k}^T \mathbf{R}_{0,b} \tag{1.15b}
+$$
+
+$$
+\ddot{\mathbf{r}}_{0,b} = \sum_{l=k}^{b-1} \ddot{\mathbf{R}}_{0,l} \mathbf{r}_l \tag{1.15c}
+$$
+
+$$
+\ddot{\mathbf{R}}_{0,b} = \Omega^2 \left( \prod_{l=0}^{k-1} \mathbf{O}_l \right) s_k \mathbf{B}_k \mathbf{N}_k^2 \mathbf{R}_k \mathbf{S}_k^T \left( \prod_{l=k+1}^{b-1} \mathbf{O}_l \right) s_b \mathbf{B}_b = \Omega^2 \mathbf{R}_{0,k} \mathbf{N}_k^2 \mathbf{R}_{0,k}^T \mathbf{R}_{0,b} \quad \tag{1.15d}
+$$
 
 where $\mathbf{N}_{k}$ is the skew symmetric matrix (E.3) defined by the unit­vectors of the bearing axis. In this case, the following products that occur in the subsequent derivations can be written as  
+其中 $\mathbf{N}_{k}$ 是由轴承轴的单位向量定义的斜对称矩阵 (E.3)。在这种情况下，后续推导中出现的以下乘积可以写成
+$$
+\mathbf{r}_{0,b}^T \dot{\mathbf{r}}_{0,b} = \tag{1.16a}
+$$
 
 $$
-\begin{array}{r l}&{\mathbf{r}_{0,b}^{T}\mathbf{\Phi}_{0,b}^{\dagger}=}\\ &{\mathbf{R}_{0,b}^{T}\mathbf{\Phi}_{0,b}^{\dagger}=\Omega\mathbf{R}_{0,b}^{T}\mathbf{R}_{0,i}\mathbf{N}_{k}^{\mathbf{R}}\mathbf{R}_{0,b}^{T}\sum_{l=k}^{b-1}\mathbf{R}_{0,l}\mathbf{r}_{l}=\Omega\left(\mathbf{R}_{0,k}^{T}\mathbf{R}_{0,b}\right)^{T}\mathbf{N}_{k}\mathbf{R}_{0,k}^{T}\left(\mathbf{r}_{0,b}-\mathbf{r}_{0,k}\right)}\\ &{\mathbf{R}_{0,b}^{T}\mathbf{\Phi}_{0,b}^{\dagger}=\Omega\left(\mathbf{R}_{0,k}^{T}\mathbf{R}_{0,b}\right)^{T}\mathbf{N}_{k}\mathbf{\Phi}_{0,b}^{T}\mathbf{R}_{0,b}}\\ &{\mathbf{r}_{0,b}^{T}\mathbf{\Phi}_{0,b}^{\dagger}=}\\ &{\mathbf{R}_{0,b}^{T}\mathbf{\Phi}_{0,b}^{\dagger}=\Omega^{2}\mathbf{R}_{0,b}^{T}\mathbf{R}_{0,b}^{2}\sum_{l=k}^{b-1}\mathbf{R}_{0,l}\mathbf{r}_{l}=\Omega^{2}\left(\mathbf{R}_{0,k}^{T}\mathbf{R}_{0,b}\right)^{T}\mathbf{N}_{k}^{2}\mathbf{R}_{0,k}^{T}\left(\mathbf{r}_{0,b}-\mathbf{r}_{0,k}\right)}\\ &{\mathbf{R}_{0,b}^{T}\mathbf{\Phi}_{0,b}^{\dagger}=\Omega^{2}\left(\mathbf{R}_{0,b}^{T}\mathbf{R}_{0,b}\right)^{T}\mathbf{R}_{k}^{2}\mathbf{R}_{0,k}^{T}\mathbf{R}_{0,b}}\end{array}
+\mathbf{R}_{0,b}^T \dot{\mathbf{r}}_{0,b} = \Omega \mathbf{R}_{0,b}^T \mathbf{R}_{0,k} \mathbf{N}_k \mathbf{R}_{0,k}^T \sum_{l=k}^{b-1} \mathbf{R}_{0,l} \mathbf{r}_l = \Omega \left( \mathbf{R}_{0,k}^T \mathbf{R}_{0,b} \right)^T \mathbf{N}_k \mathbf{R}_{0,k}^T (\mathbf{r}_{0,b} - \mathbf{r}_{0,k}) \tag{1.16b}
+$$
+
+$$
+\mathbf{R}_{0,b}^T \dot{\mathbf{R}}_{0,b} = \Omega \left( \mathbf{R}_{0,k}^T \mathbf{R}_{0,b} \right)^T \mathbf{N}_k \mathbf{R}_{0,k}^T \mathbf{R}_{0,b} \tag{1.16c}
+$$
+
+$$
+\mathbf{r}_{0,b}^T \ddot{\mathbf{r}}_{0,b} = \tag{1.16d}
+$$
+
+$$
+\mathbf{R}_{0,b}^T \ddot{\mathbf{r}}_{0,b} = \Omega^2 \mathbf{R}_{0,b}^T \mathbf{R}_{0,k} \mathbf{N}_k^2 \mathbf{R}_{0,k}^T \sum_{l=k}^{b-1} \mathbf{R}_{0,l} \mathbf{r}_l = \Omega^2 \left( \mathbf{R}_{0,k}^T \mathbf{R}_{0,b} \right)^T \mathbf{N}_k^2 \mathbf{R}_{0,k}^T (\mathbf{r}_{0,b} - \mathbf{r}_{0,k}) \tag{1.16e}
+$$
+
+$$
+\mathbf{R}_{0,b}^T \ddot{\mathbf{R}}_{0,b} = \Omega^2 \left( \mathbf{R}_{0,k}^T \mathbf{R}_{0,b} \right)^T \mathbf{N}_k^2 \mathbf{R}_{0,k}^T \mathbf{R}_{0,b} \tag{1.16f}
 $$  
 
 which are time independent scalers, vectors and matrices because  
-
+它们是时不变的标量、矢量和矩阵，因为
 $$
-\mathbf{\mathfrak{I}}_{0,k}^{T}\mathbf{R}_{0,b}=\mathbf{B}_{k}^{T}\mathbf{S}_{k}^{T}\left(\prod_{l=k-1}^{0}\mathbf{O}_{l}^{T}\right)\left(\prod_{l=0}^{b-1}\mathbf{O}_{l}\right)\mathbf{S}_{b}\mathbf{B}_{b}=\mathbf{B}_{k}^{T}\mathbf{S}_{k}^{T}\left(\prod_{l=k}^{b-1}\mathbf{O}_{l}\right)\mathbf{S}_{b}\mathbf{B}_{b}=\mathbf{R}_{k}\mathbf{S}_{k}^{T}\left(\prod_{l=k+1}^{b-1}\mathbf{O}_{l}\right)\mathbf{S}_{b}\mathbf{B}_{b}
+\mathbf{{R}}_{0,k}^{T}\mathbf{R}_{0,b}=\mathbf{B}_{k}^{T}\mathbf{S}_{k}^{T}\left(\prod_{l=k-1}^{0}\mathbf{O}_{l}^{T}\right)\left(\prod_{l=0}^{b-1}\mathbf{O}_{l}\right)\mathbf{S}_{b}\mathbf{B}_{b}=\mathbf{B}_{k}^{T}\mathbf{S}_{k}^{T}\left(\prod_{l=k}^{b-1}\mathbf{O}_{l}\right)\mathbf{S}_{b}\mathbf{B}_{b}=\mathbf{R}_{k}\mathbf{S}_{k}^{T}\left(\prod_{l=k+1}^{b-1}\mathbf{O}_{l}\right)\mathbf{S}_{b}\mathbf{B}_{b}\tag{1.17}
 $$  
 
 if $b\geq k$  
@@ -212,10 +263,22 @@ if $b\geq k$
 We also need the DOF derivatives:  
 
 $$
-\begin{array}{r l}{\displaystyle\mathbf{r}_{0,b,q_{i}}=\sum_{k=0}^{b-1}\big(\mathbf{R}_{0,k,q_{i}}\mathbf{r}_{k}+\mathbf{R}_{0,k}\mathbf{r}_{k,q_{i}}\big)}&{}\\ {\displaystyle\mathbf{R}_{0,b,q_{i}}=}&{}\\ {\displaystyle\mathbf{r}_{0,b,q_{i},q_{j}}=}&{}\\ {\displaystyle\mathbf{R}_{0,b,q_{i},q_{j}}=}&{}\end{array}
+\mathbf{r}_{0,b,q_i} = \sum_{k=0}^{b-1} (\mathbf{R}_{0,k,q_i} \mathbf{r}_k + \mathbf{R}_{0,k} \mathbf{r}_{k,q_i}) \tag{1.18a}
+$$
+
+$$
+\mathbf{R}_{0,b,q_i} = \tag{1.18b}
+$$
+
+$$
+\mathbf{r}_{0,b,q_i,q_j} = \tag{1.18c}
+$$
+
+$$
+\mathbf{R}_{0,b,q_i,q_j} = \tag{1.18d}
 $$  
 
-# 1.1.2 Inertia forces on and from a substructure  
+### 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