vault backup: 2025-04-15 16:53:12

2025-04-15 16:53:13 +08:00 · 2025-04-15 16:53:13 +08:00 · 8e18d5e689
commit 8e18d5e689
parent 25c5cf713a
6 changed files with 2400 additions and 2116 deletions
--- a/test.md
+++ b/test.md
--- a/力学书籍/力学/Kane-Dynamics-Theory-Applications/auto/Kane-Dynamics-Theory-Applications.md
+++ b/力学书籍/力学/Kane-Dynamics-Theory-Applications/auto/Kane-Dynamics-Theory-Applications.md
@ -2318,17 +2318,22 @@ $$

 The motion that results when forces act on a material system depends not only on the forces, but also on the constitution of the system. In particular, the manner in which mass is distributed throughout a system generally affects the behavior of the system. For example, suppose that a rod is supported at one end by a fixed horizontal pin and that a relatively heavy particle is attached at a point of the rod, so that together the rod and the particle form a pendulum. The frequency of the oscillations that ensue when the pendulum is released from rest after having been displaced from the vertical depends on the location of the particle along the rod, that is, on the manner in which mass is distributed throughout the pendulum.  

-For the purpose of certain analyses, it is unnecessary to know in detail how mass is distributed throughout each of the bodies forming a system; all one needs to know for each body is the location of the mass center, as well as the values of six quantities called inertia scalars. The subject of mass center location is considered in Secs. 3.1 and 3.2. Products of inertia and moments of inertia, which are inertia scalars, are defined in Sec. 3.3 in terms of quantities called inertia vectors. Sections 3.4-3.7 deal with the evaluation of inertia scalars, in which connection inertia matrices and inertia dyadics are discussed. A special kind of moment of inertia, called a principal moment of inertia, is introduced in Sec. 3.8. The chapter concludes with an examination of the relationship between principal moments of inertia, on the one hand, and maximum and minimum moments of inertia, on the Other hand.  
+**For the purpose of certain analyses, it is unnecessary to know in detail how mass is distributed throughout each of the bodies forming a system; all one needs to know for each body is the location of the mass center, as well as the values of six quantities called inertia scalars**. The subject of mass center location is considered in Secs. 3.1 and 3.2. Products of inertia and moments of inertia, which are inertia scalars, are defined in Sec. 3.3 in terms of quantities called inertia vectors. Sections 3.4-3.7 deal with the evaluation of inertia scalars, in which connection inertia matrices and inertia dyadics are discussed. A special kind of moment of inertia, called a principal moment of inertia, is introduced in Sec. 3.8. The chapter concludes with an examination of the relationship between principal moments of inertia, on the one hand, and maximum and minimum moments of inertia, on the Other hand.  
+当力作用于一个物质系统时，产生的运动不仅取决于这些力，也取决于系统的构成。特别是，质量在系统中的分布方式通常会影响系统的行为。例如，假设一根杆在一端由一个固定的水平销支撑，并且在杆的某个点上连接了一个相对较重的粒子，使得杆和粒子共同构成一个单摆。当单摆从垂直位置偏离后释放，产生的振荡频率取决于粒子沿杆的位置，也就是说，取决于质量在单摆中的分布方式。
+
+为了某些分析的目的，不必详细了解构成系统的每个物体的质量分布情况；对于每个物体，只需要知道其质量中心的位置，以及六个称为惯性标量的数值即可。质量中心的位置将在第3.1和3.2节中进行讨论。惯性标量的惯性矩和惯性积将在第3.3节中，用惯性矢量的概念来定义。第3.4-3.7节讨论惯性标量的计算，其中涉及惯性矩阵和惯性二阶张量。一种特殊的惯性矩，称为主惯性矩，将在第3.8节中介绍。本章最后将考察主惯性矩与最大和最小惯性矩之间的关系。

 # 3.1 MASS CENTER  

-$\boldsymbol{s}$ is a set of partiles $P_{1},...,P_{v}$ of masses $m_{1},\ldots,m_{\nu}$ respetivly,there exists a unique point $s^{*}$ such that  
+If $S$ is a set of particles $P_{1},...,P_{v}$ of masses $m_{1},\ldots,m_{\nu}$ respectively, there exists a unique point $S^{*}$ such that  
+如果 $S$ 是由粒子 $P_{1},...,P_{v}$ 组成的集合，它们的质量分别为 $m_{1},\ldots,m_{\nu}$，那么存在一个唯一的点 $S^{*}$，使得

 $$
 \sum_{i=1}^{\nu}\,m_{i}\mathbf{r}_{i}=0
 $$  

-where $\mathbf{r}_{i}$ is the position vector from $s^{*}$ to $P_{i}\;(i=1,\ldots,\nu).\;S^{*},$ , called the mass center of S, can be located as follows. Let $^o$ be any point whatsoever, and let $\pmb{{\mathsf{p}}_{i}}$ be the position vector from $^o$ to $P_{i}\;(i=1,\ldots,\nu)$ Then $\mathfrak{p}^{\ast}$ , the position vector from $^o$ to $s^{*}$ , is given by  
+where $\mathbf{r}_{i}$ is the position vector from $S^{*}$ to $P_{i}\;(i=1,\ldots,\nu).\;S^{*},$ , called the mass center of S, can be located as follows. Let $O$ be any point whatsoever, and let ${{\mathsf{p}}_{i}}$ be the position vector from $O$ to $P_{i}\;(i=1,\ldots,\nu)$ Then ${{\mathsf{p}}_{i}}$ , the position vector from $O$ to $S^{*}$ , is given by 
+其中 $\mathbf{r}_{i}$ 是从 $S^{*}$ 到 $P_{i}\;(i=1,\ldots,\nu)$ 的位置向量。$S^{*}$，称为 S 的质心，其位置可以如下确定。令 $O$ 为任意一点，并且 ${\mathsf{p}}_{i}$ 是从 $O$ 到 $P_{i}\;(i=1,\ldots,\nu)$ 的位置向量。那么 ${\mathsf{p}}_{i}$，即从 $O$ 到 $S^{*}$ 的位置向量，由

 $$
 \mathbf{p}^{*}={\frac{\displaystyle\sum_{i=1}^{\nu}m_{i}\mathbf{p}_{i}}{\displaystyle\sum_{i=1}^{\nu}m_{i}}}
--- a/力学书籍/力学/多体系统动力学（第2版）＝DYNAMICS
+++ b/力学书籍/力学/多体系统动力学（第2版）＝DYNAMICS
@ -9696,17 +9696,17 @@ $$

 # 第九章 柔性多体系统动力学  

-当多体系统中物体变形对运动的影响不容忽略时，必须采用柔性多体模型替代多刚体模型。与多刚体系统相比，柔性多体系统建模的主要问题是如何描述变形体的位形。在小变形假设的前提下，浮动坐标系方法将物体的运动视为浮动坐标系的大范围运动与小弹性变形的叠加。用多刚体系统动力学方法描述大范围运动，同时采用瑞利一里茨法、模态分析法或有限单元法实现小变形的离散化。对于大变形问题，则必须在惯性坐标系中直接描述变形体的位形，如近年提出的绝对节点坐标方法。本章首先介绍浮动坐标系的基本概念，叙述基于瑞利一里茨法及有限单元法的浮动坐标系方法。最后叙述绝对节点坐标方法。  
+当多体系统中物体变形对运动的影响不容忽略时，必须采用柔性多体模型替代多刚体模型。与多刚体系统相比，**柔性多体系统建模的主要问题是如何描述变形体的位形**。**在小变形假设的前提下，浮动坐标系方法将物体的运动视为浮动坐标系的大范围运动与小弹性变形的叠加**。**用多刚体系统动力学方法描述大范围运动，同时采用瑞利一里茨法、模态分析法或有限单元法实现小变形的离散化**。对于大变形问题，则必须在惯性坐标系中直接描述变形体的位形，如近年提出的绝对节点坐标方法。本章首先介绍浮动坐标系的基本概念，叙述基于瑞利一里茨法及有限单元法的浮动坐标系方法。最后叙述绝对节点坐标方法。  

 # 9.1 浮动坐标系方法  

 # 9.1.1 浮动坐标系  

-第一章中对刚体运动学的分析是利用与刚体固结的参考坐标系，即刚体的连体基确定刚体的位置和姿态。变形体与刚体的情况不同，由于变形体在运动过程中各质点之间有相对位移，以致任何参考系都不可能与变形体完全固结。为确定变形体的位置和姿态，仍需要建立适当的参考坐标系。但这坐标系不可能与变形体固结，而只是“浮动”在变形体内，称为浮动坐标系。以浮动坐标系为参考系，可将变形体的实际运动理解为浮动坐标系的大范围刚体运动与相对浮动坐标系的变形运动的合成。  
+第一章中对刚体运动学的分析是利用与刚体固结的参考坐标系，即刚体的连体基确定刚体的位置和姿态。**变形体与刚体的情况不同，由于变形体在运动过程中各质点之间有相对位移，以致任何参考系都不可能与变形体完全固结。为确定变形体的位置和姿态，仍需要建立适当的参考坐标系。但这坐标系不可能与变形体固结，而只是“浮动”在变形体内，称为浮动坐标系**。以浮动坐标系为参考系，可将变形体的实际运动理解为浮动坐标系的大范围刚体运动与相对浮动坐标系的变形运动的合成。  

 浮动坐标系有多种选取方法。选取的原则是要使所建立的动力学方程尽量避免大范围刚体运动与弹性小变形运动的耦合，有利于对变形体运动的数值计算。  

-设变形体内任意点 $P$ 在变形前的位置为 $\boldsymbol{P}_{0}$ ，$\pmb{\rho}$ 和 ${\pmb\rho}_{0}$ 分别为 $P$ 和 $\boldsymbol{P}_{0}$ 相对变形体内任选的参考点 $o$ 的矢径， $\pmb{u}$ 为 $P$ 点的位移矢量， $_r$ 和 $r_{0}$ 为 $P$ 和$o$ 相对惯性参考系中固定点 $O_{0}$ 的矢径（图9.1），  
+设变形体内任意点 $P$ 在变形前的位置为 $\boldsymbol{P}_{0}$ ，$\pmb{\rho}$ 和 ${\pmb\rho}_{0}$ 分别为 $P$ 和 $\boldsymbol{P}_{0}$ 相对变形体内任选的参考点 $O$ 的矢径， $\pmb{u}$ 为 $P$ 点的位移矢量， $r$ 和 $r_{0}$ 为 $P$ 和$O$ 相对惯性参考系中固定点 $O_{0}$ 的矢径（图9.1），  

 ![](images/c3a2b4a0373ef750c1e3814eaf31a8d14437497156fd6d6b9f6cddcf46e89f8a.jpg)  
 图9.1变形体  
@ -9714,16 +9714,16 @@ $$
 则有  

 $$
-{\textbf{\em r}}=\,{\pmb r}_{0}\,+\,{\pmb\rho}\,,\quad{\pmb\rho}\,=\,{\pmb\rho}_{0}\,+\,{\pmb u}
+{{r}}=\,{\pmb r}_{0}\,+\,{\pmb\rho}\,,\quad{\pmb\rho}\,=\,{\pmb\rho}_{0}\,+\,{\pmb u}
 $$  

-柔性体相对 $o$ 点的动量矩为  
+柔性体相对 $O$ 点的动量矩为  

 $$
-L\;=\;\int\!\pmb{\rho}\,\times\,\dot{r}\,\mathrm{d}m\;=\;\int\!\pmb{\rho}\,\times\,\left(\,\dot{r}_{\!\!\!0}\;+\frac{}{}\dot{\pmb{\rho}}\,\right)\mathrm{d}m
+L\;=\;\int\!\pmb{\rho}\,\times\,\dot{r}\,\mathrm{d}m\;=\;\int\!\pmb{\rho}\,\times\,\left(\,\dot{{\pmb r}_{0}}\,+\,\dot{{\pmb\rho}}\,\right)\mathrm{d}m
 $$  

-在变形体内建立以 $o$ 为基点的浮动坐标系 $(\,O\,,\underline{{e}}\,)$ ,设此坐标系的转动角速度为 $\omega$ ,将式（9.1.2）中 $\dot{\pmb\rho}$ 的求导过程改为相对浮动坐标系进行，利用式(1.2.23）表示为  
+在变形体内建立以$O$为基点的浮动坐标系 $(\,O\,,\underline{{e}}\,)$ ,设此坐标系的转动角速度为 $\omega$ ,将式（9.1.2）中 $\dot{\pmb\rho}$ 的求导过程改为相对浮动坐标系进行，利用式(1.2.23）表示为  

 $$
 \dot{\pmb{\rho}}\ =\ \stackrel{\circ}{\pmb{\rho}}\ +\ \pmb{\omega}\times\pmb{\rho}
@ -9732,25 +9732,25 @@ $$
 空心点表示浮动坐标系中的相对导数。将上式代人式（9.1.2），且利用式（A.4.9)化作  

 $$
-L~=~J\cdot\omega\,+\,\left(\int\!\rho\mathrm{d}m\right)\times\dot{r}_{\!\circ}~+\,\int\!\rho\,\times\,\!\rho\mathrm{d}m
+L~=~J\cdot\omega\,+\,\left(\int\!\rho\mathrm{d}m\right)\times\dot{r}_{0}~+\,\int\!\rho\,\times\,\!\stackrel{\circ}\rho\mathrm{d}m
 $$  

-其中， $\boldsymbol{J}$ 为变形体相对 $o$ 点的惯量张量  
+其中， $\boldsymbol{J}$ 为变形体相对$O$点的惯量张量  

 $$
 J\,=\,\int(\rho^{2}E\,-\,\pmb{\rho}\pmb{\rho})\,\mathrm{d}m
 $$  

-与刚体不同，变形体的惯量张量J并非常值，而是随物体的变形而改变。如将0 点选择为变形体的质心，则满足  
+与刚体不同，变形体的惯量张量 $\boldsymbol{J}$ 并非常值，而是随物体的变形而改变。如将$O$点选择为变形体的质心，则满足  

 $$
 \int\!\!\pmb{\rho}\mathrm{d}m\ =\ \int\!\!\pmb{\rho}_{0}\mathrm{d}m\ =\ \int\!\!\pmb{u}\,\mathrm{d}m\ =\ \mathbf{0}
 $$  

-变形体的质心在变形前后视为同一0点，但并非变形体内确定的物质点。将式(9.1.6）代人式(9.1.4），简化为  
+变形体的质心在变形前后视为同一$O$点，但并非变形体内确定的物质点。将式(9.1.6）代人式(9.1.4），简化为  

 $$
-\textbf{\em L}=J\cdot\omega\,+\,\int_{\rho}\times\mathring{\rho}\,\mathrm{d}m
+{L}=J\cdot\omega\,+\,\int{\rho}\times\mathring{\rho}\,\mathrm{d}m
 $$  

 1889年梯塞朗（Tisserand）提出浮动坐标系的建立方法是令式（9.1.7）右边第二项等于零  
@ -9759,7 +9759,7 @@ $$
 \int\!\!\pmb{\rho}\times\mathring{\pmb{\rho}}\,\mathrm{d}m\;=\;\mathbf{0}
 $$  

-于是变形体的动量矩有与刚体相同的计算公式L=J·∞。用此方法定义的浮动坐标系称为梯塞朗坐标系，其位置由式（9.1.8）的3个投影式确定。利用梯塞朗坐标系计算的变形体相对运动的动能为最小值。为证明此结论，令相对运动动能的变分为零  
+于是变形体的动量矩有与刚体相同的计算公式${L}=J\cdot\omega$。用此方法定义的浮动坐标系称为梯塞朗坐标系，其位置由式（9.1.8）的3个投影式确定。利用梯塞朗坐标系计算的变形体相对运动的动能为最小值。为证明此结论，令相对运动动能的变分为零  

 $$
 \delta\Big(\frac{1}{2}\!\!\int\!\!\bar{\pmb{\rho}}\,\cdot\,\bar{\pmb{\rho}}\,\mathrm{d}m\Big)\,=\,\int\!\!\bar{\pmb{\rho}}\,\cdot\,\delta\bar{\pmb{\rho}}\,\mathrm{d}m\,\,=\,0
--- a/多体+耦合求解器/风机模型.excalidraw.md
+++ b/多体+耦合求解器/风机模型.excalidraw.md
@ -13,6 +13,8 @@ tags: [excalidraw]
 Q1： yaw 主动力为何如此计算？
 由控制计算，作为一个外力直接施加 ^enHSE5Ed

+刚体：platform 机舱 高速轴 轮毂 ^fklhUVEK
+
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 ```
 %%
--- a/角动力学.excalidraw.md
+++ b/角动力学.excalidraw.md
--- a/补课/多体动力学/框架.canvas
+++ b/补课/多体动力学/框架.canvas
@ -11,19 +11,22 @@
 		{"id":"67a07cb33040e073","type":"text","text":"线加速度 求法","x":510,"y":700,"width":250,"height":60},
 		{"id":"eeb7df4b945bff86","type":"text","text":"各个部件inertia dyadic求法","x":155,"y":820,"width":250,"height":60},
 		{"id":"931f7a20403882f5","type":"text","text":"塔架、叶片的GAF GIF求法","x":155,"y":940,"width":250,"height":60},
-		{"id":"9ba9cf03738bfda2","type":"text","text":"Sympy优势：\n- linear acc、angular acc内置方法\n- GIF求解，按照公式，清晰明了\n\n劣势：\n- 不支持柔性体\n- 主动力没有好办法","x":155,"y":1060,"width":355,"height":240},
+		{"id":"9ba9cf03738bfda2","type":"text","text":"Sympy优势：\n- linear acc、angular acc内置方法\n- GIF求解，按照公式，清晰明了\n\n劣势：\n- 不支持柔性体\n- 主动力没有好办法，主动力都比较复杂，普遍问题","x":155,"y":1060,"width":355,"height":260},
 		{"id":"869b7f96937e4202","type":"text","text":"低速轴、高速轴、发电机、摩擦力等求法","x":510,"y":820,"width":250,"height":60},
-		{"id":"690b6cebbb1e52ad","x":840,"y":700,"width":250,"height":60,"type":"text","text":"是否正确？"},
-		{"id":"50d5c2753f1f2ec3","x":840,"y":790,"width":250,"height":90,"type":"text","text":"广义主动力还可以怎么求？yaw、低速轴、高速轴"},
-		{"id":"9fc02c3e78a69a7a","x":510,"y":940,"width":250,"height":60,"type":"text","text":"坐标系定义好之后，原点无所谓，有方向即可"},
-		{"id":"f13fc730aad4e78c","x":840,"y":940,"width":250,"height":180,"type":"text","text":" 偏速度$\\pmb{v}_{\\nu}^{(\\,r\\,)}$ 或 $\\pmb{\\omega}_{i}^{(\\prime)}$ 是将**标量形式的广义速率**赋予**方向性的矢量系数**。从具体算例可以看出， $\\pmb{v}_{\\nu}^{(r)}$ 或 $\\pmb{\\omega}_{i}^{(r)}$ 实际上就是某些基矢量或基矢量的线性组合。"},
-		{"id":"01e5d049c040e822","x":840,"y":1180,"width":250,"height":220,"type":"text","text":"所谓偏速度 $\\pmb{v}_{\\nu}^{(\\textrm{r})}\\left(\\,r=1\\,,2\\,,\\cdots,f\\right)$ 实际上是某些特定的基矢量或基矢量的线性组合，因此，广义主动力或广义惯性力就是系统内全部主动力或惯性力沿这些特定基矢量方向的投影。"},
-		{"id":"8867bfcfd58ae90b","x":1180,"y":940,"width":250,"height":120,"type":"text","text":"对于完整系统，凯恩方法中的广义主动力 ${\\boldsymbol{F}}^{(r)}$ 等同于拉格朗日方程中的广义力 $Q_{r}$ 。  "}
+		{"id":"690b6cebbb1e52ad","type":"text","text":"是否正确？","x":840,"y":700,"width":250,"height":60},
+		{"id":"50d5c2753f1f2ec3","type":"text","text":"广义主动力还可以怎么求？yaw、低速轴、高速轴","x":840,"y":790,"width":250,"height":90},
+		{"id":"9fc02c3e78a69a7a","type":"text","text":"坐标系定义好之后，原点无所谓，有方向即可  但是是与刚体固接的","x":510,"y":940,"width":250,"height":87},
+		{"id":"f13fc730aad4e78c","type":"text","text":" 偏速度$\\pmb{v}_{\\nu}^{(\\,r\\,)}$ 或 $\\pmb{\\omega}_{i}^{(\\prime)}$ 是将**标量形式的广义速率**赋予**方向性的矢量系数**。从具体算例可以看出， $\\pmb{v}_{\\nu}^{(r)}$ 或 $\\pmb{\\omega}_{i}^{(r)}$ 实际上就是某些基矢量或基矢量的线性组合。","x":840,"y":940,"width":250,"height":180},
+		{"id":"01e5d049c040e822","type":"text","text":"所谓偏速度 $\\pmb{v}_{\\nu}^{(\\textrm{r})}\\left(\\,r=1\\,,2\\,,\\cdots,f\\right)$ 实际上是某些特定的基矢量或基矢量的线性组合，因此，广义主动力或广义惯性力就是系统内全部主动力或惯性力沿这些特定基矢量方向的投影。","x":840,"y":1180,"width":250,"height":220},
+		{"id":"8867bfcfd58ae90b","type":"text","text":"对于完整系统，凯恩方法中的广义主动力 ${\\boldsymbol{F}}^{(r)}$ 等同于拉格朗日方程中的广义力 $Q_{r}$ 。  ","x":1180,"y":940,"width":250,"height":120},
+		{"id":"bff8e414fcb04560","x":240,"y":620,"width":444,"height":60,"type":"text","text":"速度、角速度 = 偏速度/偏角速度 * 广义速率 + remain项 对时间t求导"}
 	],
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 	]
 }