vault backup: 2025-01-15 13:46:03
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- copilot-conversation
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---
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**user**: # 2.9 CONFIGURATION CONSTRAINTS
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The configuration of a set S of $\pmb{\nu}$ particles $P_{1},...,P_{\nu}$ in a reference frame A is known whenever the position vector of each particle relative to a point fixed in A is known. Thus, v vector quantities, or, equivalently, $\mathbf{3}\upnu$ scalar quantities, are required for the specification of the configuration of S in $\pmb{A}$
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If the motion of $\boldsymbol{s}$ is affected by the presence of bodies that come into contact with one or more of $P_{1},...,P_{\nu}$ , restrictions are imposed on the positions that the affected particles may occupy, and $\pmb{s}$ is said to be subject to configuration constraints;
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an equationexpressing sucharestrictionis calleda holonomicconstraintequation. $\mathbf{a}_{x}$ $\mathbf{a}_{y}$ ， ${\mathfrak{a}}_{z}$ are mutually perpendicular unit vectors fixed in $A$ and $x_{i},y_{i},z_{i},$ called Cartesiancoordinatesof $P_{i}$ in $A$ , are defined as
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翻译成中文
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[Timestamp: 2025/01/14 16:07:35]
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**ai**: # 2.9 配置约束条件
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一组 $\pmb{\nu}$ 粒子 $P_1, \ldots, P_\nu$ 在参考系 A 中的配置是已知的，当每个粒子相对于固定在 A 中的一个点的位置矢量被知道时。因此，需要 v 个矢量量或等价地 $\mathbf{3}\upnu$ 个标量量来指定 S 在 $\pmb{A}$ 中的配置。
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如果由于与 $P_1, \ldots, P_\nu$ 中一个或多个粒子接触的物体的存在而影响了 $\boldsymbol{s}$ 的运动，就会对这些受影响的粒子可能占据的位置施加限制，并且说 $\pmb{s}$ 受到配置约束；一个表达此类限制的方程被称为齐次约束方程。$\mathbf{a}_x$、$\mathbf{a}_y$ 和 ${\mathfrak{a}}_z$ 是固定在 $A$ 中且互相垂直的单位矢量，而 $x_i, y_i, z_i$，称为 $P_i$ 在 A 中的笛卡尔坐标，则定义如下：
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---
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这段文字描述了粒子系统在参考系中配置的约束条件，并提到了齐次约束方程以及相关的笛卡尔坐标。
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[Timestamp: 2025/01/14 16:08:25]
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**user**: # 2.9 CONFIGURATION CONSTRAINTS
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||||
The configuration of a set S of $\pmb{\nu}$ particles $P_{1},...,P_{\nu}$ in a reference frame A is known whenever the position vector of each particle relative to a point fixed in A is known. Thus, v vector quantities, or, equivalently, $\mathbf{3}\upnu$ scalar quantities, are required for the specification of the configuration of S in $\pmb{A}$
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If the motion of $\boldsymbol{s}$ is affected by the presence of bodies that come into contact with one or more of $P_{1},...,P_{\nu}$ , restrictions are imposed on the positions that the affected particles may occupy, and $\pmb{s}$ is said to be subject to configuration constraints;
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an equation expressing such a restriction is called a holonomic constraint equation.If $\mathbf{a}_{x}$ $\mathbf{a}_{y}$ ， ${\mathfrak{a}}_{z}$ are mutually perpendicular unit vectors fixed in $A$ and $x_{i},y_{i},z_{i},$ called Cartesian coordinates of $P_{i}$ in $A$ , are defined as
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翻译成中文
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[Timestamp: 2025/01/14 16:13:59]
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**ai**: # 2.9 配置约束条件
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一组 $\pmb{\nu}$ 粒子 $P_1, \ldots, P_\nu$ 在参考系 A 中的配置是已知的，当每个粒子相对于固定在 A 中的一个点的位置矢量被知道时。因此，需要 v 个矢量量或等价地 $\mathbf{3}\upnu$ 个标量量来指定 S 在 $\pmb{A}$ 中的配置。
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||||
如果由于与 $P_1, \ldots, P_\nu$ 中一个或多个粒子接触的物体的存在而影响了 $\boldsymbol{s}$ 的运动，就会对这些受影响的粒子可能占据的位置施加限制，并且说 $\pmb{s}$ 受到配置约束；一个表达此类限制的方程被称为齐次约束方程。如果 $\mathbf{a}_x$、$\mathbf{a}_y$ 和 ${\mathfrak{a}}_z$ 是固定在 $A$ 中且互相垂直的单位矢量，而 $x_i, y_i, z_i$，称为 $P_i$ 在 A 中的笛卡尔坐标，则定义如下：
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---
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这段文字解释了粒子系统在参考系中配置的约束条件，并提到了齐次约束方程以及相关的笛卡尔坐标。
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[Timestamp: 2025/01/14 16:14:54]
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**user**: where $t$ is the time. Holonomic constraint equations are classified as rheonomic or scleronomic, according to whether the function $f$ does, or does not, contain t explicitly.
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Example Figure2.9.1showstwosmall blocks, $P_{1}$ and $P_{2}$ ,connected by a thin rod $R$ of length $L$ ,and constrained to remain between two parallel panes of glass that are attached to each other, forming a rigid body $B_{\cdot}$ This body is made to rotate at a constant rate $\omega$ about a line $\boldsymbol{Y}$ fixed both in $B$ and in a reference frame $A$ Treating $P_{1}$ and $P_{2}$ as a set ${\cal{S}}$ of two particles, and letting $\mathbf{p}_{1}$ and ${\bf p}_{2}$ be their position vectors relative to the point $o$ shown in Fig.2.9.1, one can express $\mathbf{p}_{1}$ and $\mathfrak{g}_{2}$ as
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翻译成中文
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[Timestamp: 2025/01/14 16:18:18]
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**ai**: 齐次约束方程根据函数 \( f \) 是否显式包含时间 \( t \)，被分类为流体动力学的（rheonomic）或刚性的（scleronomic）。
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|
||||
示例图 2.9.1 展示了两个小块，\( P_1 \) 和 \( P_2 \)，它们通过一根长度为 \( L \) 的细棒 \( R \) 连接，并被限制在两块平行玻璃板之间移动。这些玻璃板相互连接，形成一个刚体 \( B_{\cdot} \)。该刚体以恒定速率 \( \omega \) 旋转于一条线 \( \boldsymbol{Y} \)，此线固定在 \( B \) 和参考系 \( A \) 中。将 \( P_1 \) 和 \( P_2 \) 视为两个粒子的集合 \( {\cal{S}} \)，并设其相对于图 2.9.1 所示点 \( o \) 的位置矢量分别为 \( \mathbf{p}_1 \) 和 \( \mathfrak{g}_2 \)，则可以表达出 \( \mathbf{p}_1 \) 和 \( \mathfrak{g}_2 \)。
|
||||
|
||||
---
|
||||
|
||||
这段文字描述了一个示例，其中两个粒子通过一根细棒连接，并受到约束条件的限制，在旋转刚体中移动。齐次约束方程根据是否包含时间因素进行分类。
|
||||
[Timestamp: 2025/01/14 16:19:06]
|
||||
|
||||
**user**: When a set $s$ of $v$ particles $P_{1},\dots,P_{v}$ is subject to constraints (see Sec.2.9) represented by $M$ holonomic constraint equations, only
|
||||
|
||||
$$
|
||||
\boldsymbol{n}\triangleq\boldsymbol{3}\nu-\boldsymbol{M}
|
||||
$$
|
||||
|
||||
of the $3v$ Cartesian coordinates $x_{i},y_{i},z_{i}\,(i=1,\ldots,\nu)$ of S in a reference frame $\pmb{A}$ are independent of each other. Under these circumstances one can express each of $x_{i},y_{i},z_{i}\,(i=1,\ldots,\nu)$ as a single-valued function of the time $t$ and $\pmb{n}$ functions of $t$ say, $q_{1}(t),\ldots,q_{n}(t),$ in such a way that the constraint equations are satisfied identically for all values of $t$ and $q_{1},\ldots,q_{n}$ in a given domain. The quantities $q_{1},\ldots,q_{n}$ are called generalized coordinates for $s$ in $\pmb{A}$
|
||||
|
||||
Example For the set $s$ in the example in Sec. 2.9, $\nu=2$ and $M=3.$ Hence $n=3$ . Three generalized coordinates for $s$ in $\pmb{A}$ may be introduced by expressing $x_{i},y_{i},z_{i}\,(i=1,2)$ as
|
||||
翻译成中文
|
||||
[Timestamp: 2025/01/14 16:37:34]
|
||||
|
||||
**ai**: 当一个由 \( v \) 个粒子组成的集合 \( s = \{P_1, \dots, P_v\} \) 受到约束（见第 2.9 节）并由 \( M \) 个齐次约束方程表示时，参考系 \( \pmb{A} \) 中的 \( S \) 的 \( 3v \) 个笛卡尔坐标 \( x_i, y_i, z_i\ (i=1,\ldots,v) \) 中只有
|
||||
|
||||
$$
|
||||
\boldsymbol{n} \triangleq 3\nu - M
|
||||
$$
|
||||
|
||||
个是相互独立的。在这种情况下，可以将每一个 \( x_i, y_i, z_i\ (i=1,\ldots,v) \) 表示为时间 \( t \) 和 \( n \) 个函数 \( q_1(t), \ldots, q_n(t) \) 的单值函数，使得约束方程对于给定域中的所有 \( t \) 和 \( q_1, \ldots, q_n \) 值都被恒等满足。这些量 \( q_1, \ldots, q_n \) 被称为 \( s \) 在 \( \pmb{A} \) 中的广义坐标。
|
||||
|
||||
示例：对于第 2.9 节中的集合 \( s \)，有 \( v = 2 \) 和 \( M = 3 \)。因此，\( n = 3 \)。可以引入三个广义坐标来表示 \( S \) 在 \( \pmb{A} \) 中的 \( x_i, y_i, z_i\ (i=1,2) \)，使得
|
||||
|
||||
---
|
||||
|
||||
这段文字解释了在约束条件下，如何通过引入广义坐标来简化粒子系统中独立变量的数量。示例说明了具体情况下的应用。
|
||||
[Timestamp: 2025/01/14 16:38:51]
|
||||
@ -0,0 +1,81 @@
|
||||
---
|
||||
epoch: 1736904015261
|
||||
modelKey: phi4:latest|ollama
|
||||
tags:
|
||||
- copilot-conversation
|
||||
---
|
||||
|
||||
**user**: The number n of generalized coordinates of a set S of $\pmb{\nu}$ particles in a reference frame $\pmb{A}$ (see Sec.2.10) is the smallest number of scalar quantities such that to every assignment of values to these quantities and the time $t$ (within a domain of interest) there corresponds a definite admissible configuration of $s$ in $\pmb{A}$ . Frequently, one can find $\pmb{n}$ by inspection rather than by determining the number $M$ of holonomic constraint equations (see Sec. 2.9) and then subtracting $M$ from $\,\mathfrak{z}_{v}$ For example, suppose that S consists of $\upnu$ particles $P_{1},...,P_{\nu}$ forming a rigid body $\pmb{B}$ that is free to move in A. Then there corresponds a definite admissible configuration of Sin $\pmb{A}$ to every assignment of values to three Cartesian coordinates o one particle of $\pmb{B}$ and three angles that characterize the orientation of $\pmb{B}$ in $\pmb{A}$ Hence, $\pmb{n}=6.$ The same conclusion is obtained formally by letting $\mathfrak{p}_{1},\ldots,\mathfrak{p}_{\nu}$ be the position vectors from a point fixed in $\pmb{A}$ to $P_{1},...,P_{\nu}$ , respectively, and noting that rigidity can be ensured by letting $P_{1},P_{2}$ , and $\boldsymbol{P_{3}}$ be noncollinear particles and requiring (1) that the distances between $\boldsymbol{P}_{1}$ and $P_{2},P_{2}$ and $\boldsymbol{P_{3}}$ , and $\boldsymbol{P_{3}}$ and $\boldsymbol{P}_{1}$ remain constant, so that
|
||||
翻译成中文
|
||||
[Timestamp: 2025/01/15 09:20:15]
|
||||
|
||||
**ai**: 在参考系 \( \pmb{A} \) 中，一组 \( \pmb{\nu} \) 粒子的广义坐标数 \( n \) 是最小数量的标量量，使得对于这些量和时间 \( t \) 的任何分配（在感兴趣的域内），都对应于参考系 \( \pmb{A} \) 中粒子集合 \( S \) 的一个明确可接受的配置。通常，可以通过观察而非确定 \( M \) 个齐次约束方程的数量（见第2.9节）然后从 \( \,\mathfrak{z}_{v} \) 中减去 \( M \)，来找到 \( n \)。例如，假设集合 \( S \) 包含 \( \upnu \) 个粒子 \( P_{1},...,P_{\nu} \)，它们形成一个在参考系 \( A \) 中自由移动的刚体 \( \pmb{B} \)。那么，对于给定的三个坐标和三个角度（用以描述 \( \pmb{B} \) 在 \( \pmb{A} \) 中的方向），就对应于参考系 \( A \) 中粒子集合 \( S \) 的一个明确可接受的配置。因此，\( n=6 \)。通过正式方法得出相同结论：设 \( \mathfrak{p}_{1},\ldots,\mathfrak{p}_{\nu} \) 是从参考系 \( A \) 中固定点到 \( P_{1},...,P_{\nu} \) 的位置矢量。注意，通过让 \( P_{1},P_{2} \)，和 \( \boldsymbol{P_{3}} \) 为非共线粒子，并要求（1）\( P_{1} \) 和 \( P_{2} \), \( P_{2} \) 和 \( \boldsymbol{P_{3}} \), 以及 \( \boldsymbol{P_{3}} \) 和 \( \boldsymbol{P}_{1} \) 之间的距离保持不变，可以确保刚体性。
|
||||
[Timestamp: 2025/01/15 09:21:27]
|
||||
|
||||
**user**: # 2.12 GENERALIZED SPEEDS
|
||||
|
||||
As will be seen presently, expressions for angular velocities of rigid bodies and velocities of points of a system S whose configuration in a reference frame $\pmb{A}$ is characterized by $\pmb{n}$ generalized coordinates $q_{1},\ldots,q_{n}$ (see Sec. 2.10) can be brought into particularly advantageous forms through the introduction of $\pmb{n}$ quantities $u_{1},\ldots,u_{n}$ ,called generalized speeds for $s$ in $\pmb{A}$ , these being quantities defined by equations of the form
|
||||
翻译成中文
|
||||
[Timestamp: 2025/01/15 09:35:56]
|
||||
|
||||
**ai**: # 2.12 广义速度
|
||||
|
||||
如将要看到的，通过引入 \( n \) 个量 \( u_{1},\ldots,u_{n} \)，称为参考系 \( \pmb{A} \) 中系统 \( S \) 的广义速度，可以使刚体的角速度和系统 \( S \) 点在参考系 \( \pmb{A} \) 中配置由 \( n \) 个广义坐标 \( q_{1},\ldots,q_{n} \)（见第2.10节）所特征化时的速度表达式形式更为有利。这些量是通过以下形式的方程定义的：
|
||||
|
||||
---
|
||||
|
||||
在此段落中，广义速度 \( u_1, \ldots, u_n \) 被引入以优化刚体角速度和系统点速度的表达方式，当其配置由广义坐标 \( q_1, \ldots, q_n \) 描述时。
|
||||
[Timestamp: 2025/01/15 09:36:41]
|
||||
|
||||
**user**: where $Y_{r s}$ and $Z_{\nu}$ are functions of $q_{1},\ldots,q_{n}$ , and the time t. These functions must be chosen such that Eqs. (1) can be solved uniquely for ${\dot{q}}_{1},\dots,{\dot{q}}_{n}$ . Equations (1) are called kinematical differential equations for $\boldsymbol{s}$ in $\pmb{A}$
|
||||
|
||||
Example Letting $\boldsymbol{s}$ be the set of two particles considered in the example in Sec. 2.9, and using as generalized coordinates the quantities $q_{1},q_{2},q_{3}$ indicated in Fig. 2.10.1, one may define three generalized speeds as
|
||||
翻译成中文
|
||||
[Timestamp: 2025/01/15 09:38:31]
|
||||
|
||||
**ai**: 其中 \( Y_{rs} \) 和 \( Z_{\nu} \) 是 \( q_1,\ldots,q_n \) 和时间 \( t \) 的函数。这些函数必须选择得当，以便方程（1）能够唯一地解出 \( {\dot{q}}_{1},\dots,{\dot{q}}_{n} \)。方程（1）被称为参考系 \( \pmb{A} \) 中系统 \( \boldsymbol{s} \) 的运动学微分方程。
|
||||
|
||||
例子：设 \( \boldsymbol{s} \) 是第2.9节中讨论的两个粒子集合，使用图2.10.1中指示的广义坐标量 \( q_1, q_2, q_3 \)，可以定义三个广义速度为：
|
||||
|
||||
---
|
||||
|
||||
在这段描述中，\( Y_{rs} \) 和 \( Z_{\nu} \) 是依赖于广义坐标和时间的函数，它们需要被选择以确保能够唯一解出广义坐标的导数。方程（1）是系统在参考系中的运动学微分方程。接下来提供了一个例子，说明如何为特定粒子集合定义广义速度。
|
||||
[Timestamp: 2025/01/15 09:39:10]
|
||||
|
||||
**user**: Here, the third component is a bit more complicated than in Eqs. (13) and (14), but the introduction of generalized speeds has led to a noticeable simplification, nevertheless. The guiding idea in writing Eqs. (7) and (2) was thus to enable one to replace Eqs. (11) and (12), respectively, with expressins having, as nearly as possible, the same simple form as Eq. (10). As for Eq<. (9), their use does not lead to any simplifications since Eq. (10) cannot be simplified further, but they were included to show that the concept of generalized speeds remains applicable even under these circumstances.
|
||||
|
||||
The simplification of an angular velocity expression through the use of generalized speeds can be illustrated by returning to the example in Sec. 2.4. The angular velocity expression recorded in Eq. (2.4.10), while simple in form, is unsuitable for certain purposes because $\mathbf{k}_{2},\,\mathbf{k}_{7}$ ,and ${\bf k}_{3}$ are not mutually perpendicular. To overcome this difficulty, one can let $\bf{k_{4}}$ be a unit vector directed as shown in Fig. 2.4.1 and note that ${\bf k}_{3}$ then is given by
|
||||
翻译成中文
|
||||
[Timestamp: 2025/01/15 09:43:17]
|
||||
|
||||
**ai**: 在这里，第三个分量比方程（13）和（14）更复杂，但引入广义速度仍然导致了显著的简化。写出方程（7）和（2）的指导思想是为了能够用与方程（10）形式尽可能相似的表达式替代方程（11）和（12）。至于方程（9），它们的使用并不会带来简化，因为方程（10）无法进一步简化，但它们被包括在内是为了展示即使在这些情况下，广义速度的概念仍然适用。
|
||||
|
||||
通过使用广义速度来简化角速度表达式可以通过回到第2.4节中的例子进行说明。方程（2.4.10）记录的角速度表达式形式上是简单的，但对于某些目的而言不适用，因为 \( \mathbf{k}_{2},\,\mathbf{k}_{7} \) 和 \( {\bf k}_{3} \) 不互相垂直。要克服这个困难，可以让 \( \bf{k_{4}} \) 是一个指向图2.4.1中所示方向的单位矢量，并注意到此时 \( {\bf k}_{3} \) 则给出为：
|
||||
|
||||
---
|
||||
|
||||
在这段文字中，尽管引入广义速度后第三个分量变得更复杂，但整体上仍然实现了简化。方程（7）和（2）的目标是用与方程（10）形式相似的表达式替代其他方程。对于方程（9），虽然它们没有带来进一步的简化，但展示了广义速度概念在不同情况下的适用性。接着通过一个例子说明如何使用广义速度简化角速度表达式。
|
||||
[Timestamp: 2025/01/15 09:44:35]
|
||||
|
||||
**user**: # 2.13 MOTION CONSTRAINTS
|
||||
|
||||
It can occur that, for physical reasons, the generalized speeds $u_{1},\ldots,u_{n}$ for a system S in a reference frame $A$ (see Sec. 2.4) are not independent of each other. In that event, $s$ is said to be subject to motion constraints, and an equation that relates $u_{1},\ldots,u_{n}$ to each other is called a nonholonomic constraint equation.
|
||||
|
||||
When a system S is not subject to motion constraints, then $s$ is said to be a holonomic system possessing n degrees of freedom in $\pmb{A}$ .If $s$ is subject to motion constraints, $s$ is called a nonholonomic system.
|
||||
|
||||
When all nonholonomic constraint equations can be expressed as the $m$ relationships
|
||||
翻译成中文
|
||||
[Timestamp: 2025/01/15 10:07:25]
|
||||
|
||||
**ai**: # 2.13 运动约束条件
|
||||
|
||||
有时，由于物理原因，系统 \( S \) 在参考系 \( A \) 中的广义速度 \( u_1, \ldots, u_n \)（见第2.4节）之间不是独立的。在这种情况下，\( s \) 被称为受到运动约束条件，并且将 \( u_1, \ldots, u_n \) 互相联系起来的方程被称为非全微分（nonholonomic）约束方程。
|
||||
|
||||
当系统 \( S \) 不受运动约束时，\( s \) 被称为具有 \( n \) 自由度在参考系 \( \pmb{A} \) 中的全微分（holonomic）系统。如果 \( s \) 受到运动约束，则被称为非全微分（nonholonomic）系统。
|
||||
|
||||
当所有非全微分约束方程都可以表示为以下 \( m \) 个关系时：
|
||||
|
||||
---
|
||||
|
||||
在这段文字中，讨论了系统的广义速度可能因物理原因而不是独立的情况。如果存在这样的依赖性，则称系统受到运动约束，并且相关方程被称为非全微分约束方程。若没有这种约束，系统则是全微分系统。对于受到运动约束的系统，所有约束方程可以用一组关系来表示。
|
||||
[Timestamp: 2025/01/15 10:08:27]
|
||||
@ -679,7 +679,7 @@ $$
|
||||
|
||||
and verify that the resulting equation is precisely Eq. (1.4.11). The point here is not that use of Eq. (1) facilitates the evaluation of ordinary derivatives; indeed, it may complicate matters. What is important is to realize that one may treat the same vector in a variety of ways, that the formalism one uses to construct the ordinary derivative of the vector in a given reference frame depends on the functional character one attributes to the vector, but that the result one obtains is independent of the approach taken. In the sequel, Eq. (1) will be used primarily in the course of certain derivations, rather than for the actual evaluation of ordinary derivatives.
|
||||
|
||||
KINEMATICS
|
||||
# Chapter 2 KINEMATICS
|
||||
|
||||
Considerations of kinematics play a central role in dynamics Indeed, one's efectiveness in formulating equations of motion depends primarily on one's ability to construct corrct mathematical expressions for kinematical quantities such as angular velocities of rigid bodies, velocities of points, and so forth. Therefore, mastery of the material in this chapter is essential.
|
||||
|
||||
@ -687,56 +687,54 @@ The sections that follow can be divided into four groups. Sections 2.1-2.5, Whic
|
||||
|
||||
# 2.1 ANGULAR VELOCITY
|
||||
|
||||
The use of angular velocities greatly facilitates the analysis of motions of systems containing rigid bodies. We begin our discussion of this topic with a formal
|
||||
The use of angular velocities greatly facilitates the analysis of motions of systems containing rigid bodies. We begin our discussion of this topic with a formal definition of angular velocity; while it is abstract, this definition provides a sound basis for the derivation of theorems [see, for example, Eq. (2)] used to solve physical problems. $^{\dagger}$
|
||||
|
||||
definition of angular velocity; while it is abstract, this definition provides a sound basis for the derivation of theorems [see, for example, Eq. (2)] used to solve physical problems. $^{\dagger}$
|
||||
|
||||
Let $\ b_{1},\ b_{2}$ ， ${\bf b}_{3}$ form a right-handed set of mutually perpendicular unit vectors fixed in a rigid body $\pmb{B}$ moving in a reference frame $\pmb{A}$ .The angular velocity of $\pmb{B}$ in $\pmb{A}$ , denoted by $\mathbf{\omega_{op}^{B}}$ , is defined as
|
||||
Let $\ b_{1},\ b_{2}$ ， ${\bf b}_{3}$ form a right-handed set of mutually perpendicular unit vectors fixed in a rigid body $\pmb{B}$ moving in a reference frame $\pmb{A}$ .The angular velocity of $\pmb{B}$ in $\pmb{A}$ , denoted by ${\bf{ }}^{A}{\bf{\omega}}^{B}$ , is defined as
|
||||
|
||||
$$
|
||||
{\bf{\nabla}}^{A}{\bf{\Theta}}^{B}\triangleq{\bf{b}}_{1}\,\frac{{{\bf{\Theta}}}^{A}d{\bf{b}}_{2}}{d t}\cdot{\bf{b}}_{3}\,+\,{\bf{b}}_{2}\,\frac{{{\bf{\Theta}}}^{A}d{\bf{b}}_{3}}{d t}\,\cdot{\bf{b}}_{1}\,+\,{\bf{b}}_{3}\,\frac{{{\bf{\Theta}}}^{A}d{\bf{b}}_{1}}{d t}\cdot{\bf{b}}_{2}
|
||||
{\bf{ }}^{A}{\bf{\omega}}^{B}\triangleq{\bf{b}}_{1}\,\frac{{{\bf{ }}}^{A}d{\bf{b}}_{2}}{d t}\cdot{\bf{b}}_{3}\,+\,{\bf{b}}_{2}\,\frac{{{\bf{ }}}^{A}d{\bf{b}}_{3}}{d t}\,\cdot{\bf{b}}_{1}\,+\,{\bf{b}}_{3}\,\frac{{{\bf{ }}}^{A}d{\bf{b}}_{1}}{d t}\cdot{\bf{b}}_{2} \tag{1}
|
||||
$$
|
||||
|
||||
One task facilitated by the use of angular velocity vectors is the time-differentiation of vectors fixed in a rigid body, for it enables one to obtain the first timederivative of such a vector by performing a cross-multiplication. Specifically, if $\upbeta$ is any vector fixed in $\pmb{B}.$ then
|
||||
One task facilitated by the use of angular velocity vectors is the time-differentiation of vectors fixed in a rigid body, for it enables one to obtain the first timederivative of such a vector by performing a cross-multiplication. Specifically, if $\beta$ is any vector fixed in $\pmb{B}.$ then
|
||||
|
||||
$$
|
||||
{\frac{{\mathbf{\nabla}}^{A}d{\mathfrak{B}}}{d t}}={\mathbf{\nabla}}^{A}\upomega^{B}\times{\mathfrak{B}}
|
||||
{\frac{{\mathbf{ }}^{A}d{\beta}}{d t}}={\mathbf{}}^{A}\omega^{B}\times{{\beta}}\tag{2}
|
||||
$$
|
||||
|
||||
Derivation Using dots to denote time-differentiation in $\pmb{A}$ , one can rewrite Eq. (1)as
|
||||
|
||||
$$
|
||||
{\cal A}_{0}{\bf0}^{B}\triangleq{\bf b}_{1}\dot{\bf b}_{2}\cdot{\bf b}_{3}+{\bf b}_{2}\dot{\bf b}_{3}\cdot{\bf b}_{1}+{\bf b}_{3}\dot{\bf b}_{1}\cdot{\bf b}_{2}
|
||||
{\bf{ }}^{A}{\bf{\omega}}^{B}\triangleq{\bf b}_{1}\dot{\bf b}_{2}\cdot{\bf b}_{3}+{\bf b}_{2}\dot{\bf b}_{3}\cdot{\bf b}_{1}+{\bf b}_{3}\dot{\bf b}_{1}\cdot{\bf b}_{2}\tag{3}
|
||||
$$
|
||||
|
||||
and cross-multiplication of Eq. (3) with ${\bf b}_{1}$ gives
|
||||
|
||||
$$
|
||||
{\bf{\nabla}}^{A}\!\Theta^{B}\times\,{\bf{b}}_{1}={\bf{b}}_{2}\times{\bf{\nabla}}{\bf{b}}_{1}{\bf{\dot{b}}}_{3}\cdot{\bf{b}}_{1}+{\bf{b}}_{3}\times{\bf{\nabla}}{\bf{b}}_{1}{\bf{\dot{b}}}_{1}\cdot{\bf{b}}_{2}
|
||||
{\bf{ }}^{A}{\bf{\omega}}^{B}\times\,{\bf{b}}_{1}={\bf{b}}_{2}\times{\bf}{\bf{b}}_{1}{\bf{\dot{b}}}_{3}\cdot{\bf{b}}_{1}+{\bf{b}}_{3}\times{\bf}{\bf{b}}_{1}{\bf{\dot{b}}}_{1}\cdot{\bf{b}}_{2}\tag{4}
|
||||
$$
|
||||
|
||||
Now,since $\mathbf{b}_{1},\,\mathbf{b}_{2},\,\mathbf{b}_{3}$ form a right-handed set of mutually perpendicular unit vectors, each can be expressed as a cross-product involving the remaining two. For example,
|
||||
|
||||
$$
|
||||
\mathbf{b}_{2}=\mathbf{b}_{3}\times\mathbf{b}_{1}\qquad\mathbf{b}_{3}=\mathbf{b}_{1}\times\mathbf{b}_{2}
|
||||
\mathbf{b}_{2}=\mathbf{b}_{3}\times\mathbf{b}_{1}\qquad\mathbf{b}_{3}=\mathbf{b}_{1}\times\mathbf{b}_{2}\tag{5}
|
||||
$$
|
||||
|
||||
and substitution into Eq. (4) yields
|
||||
|
||||
$$
|
||||
{\bf\nabla}^{A}\Theta^{B}\times{\bf\Phi}\mathbf{b}_{1}=-{\bf\Phi}{\bf b}_{3}\,{\bf\dot{b}}_{3}\cdot{\bf b}_{1}+{\bf b}_{2}\,{\bf\dot{b}}_{1}\cdot{\bf b}_{2}
|
||||
{\bf{ }}^{A}{\bf{\omega}}^{B}\times\mathbf{b}_{1}=-{\bf b}_{3}\,{\bf\dot{b}}_{3}\cdot{\bf b}_{1}+{\bf b}_{2}\,{\bf\dot{b}}_{1}\cdot{\bf b}_{2}\tag{6}
|
||||
$$
|
||||
|
||||
Moreover, time-differentiation of the equations ${\mathfrak{b}}_{1}\cdot{\mathfrak{b}}_{1}=1$ and ${\bf b}_{3}\cdot{\bf b}_{1}=0$ produces
|
||||
Moreover, time-differentiation of the equations ${{b}}_{1}\cdot{{b}}_{1}=1$ and ${\bf b}_{3}\cdot{\bf b}_{1}=0$ produces
|
||||
|
||||
$$
|
||||
\mathbf{\dot{b}}_{1}\cdot\mathbf{b}_{1}=0\qquad\mathbf{\dot{b}}_{3}\cdot\mathbf{b}_{1}=\mathbf{\dot{\Phi}}-\mathbf{\dot{b}}_{1}\cdot\mathbf{b}_{3}
|
||||
\mathbf{\dot{b}}_{1}\cdot\mathbf{b}_{1}=0\qquad\mathbf{\dot{b}}_{3}\cdot\mathbf{b}_{1}=\mathbf{}-\mathbf{\dot{b}}_{1}\cdot\mathbf{b}_{3}\tag{7}
|
||||
$$
|
||||
|
||||
and with the aid of these one can rewrite Eq. (6) as
|
||||
|
||||
$$
|
||||
{\bf\nabla}^{A}\!\Theta^{B}\times{\bf\dot{b}}_{1}={\bf b}_{1}{\bf\dot{b}}_{1}\cdot{\bf b}_{1}+{\bf b}_{2}{\bf\dot{b}}_{1}\cdot{\bf b}_{2}+{\bf b}_{3}{\bf\dot{b}}_{1}\cdot{\bf b}_{3}
|
||||
{\bf{ }}^{A}{\bf{\omega}}^{B}\times{\bf\dot{b}}_{1}={\bf b}_{1}{\bf\dot{b}}_{1}\cdot{\bf b}_{1}+{\bf b}_{2}{\bf\dot{b}}_{1}\cdot{\bf b}_{2}+{\bf b}_{3}{\bf\dot{b}}_{1}\cdot{\bf b}_{3}\tag{8}
|
||||
$$
|
||||
|
||||
$^\dagger$ The frequently employed definition of angular velocity as the limit of $\Delta\pmb{\theta}/\Delta t$ as $\pmb{\Delta t}$ approaches zero is deficient in this regard.
|
||||
@ -744,34 +742,34 @@ $^\dagger$ The frequently employed definition of angular velocity as the limit o
|
||||
But the right-hand member of this equation is simply a way of writing ${\dot{\mathbf{b}}}_{1}$ [see Eq. (1.3.3)]. Consequently,
|
||||
|
||||
$$
|
||||
{}^{A}{\bf60}^{B}\times{\bf b}_{1}={\bf\dot{b}}_{1}
|
||||
{}^{A}{\omega}^{B}\times{\bf b}_{1}={\bf\dot{b}}_{1}\tag{9}
|
||||
$$
|
||||
|
||||
Similarly,
|
||||
|
||||
$$
|
||||
{\bf\nabla}^{A}\odot{\bf^{B}}\times{\bf\dot{b}}_{2}={\bf\dot{b}}_{2}\qquad{\bf\nabla}^{A}\odot{\bf^{B}}\times{\bf\dot{b}}_{3}={\bf\dot{b}}_{3}
|
||||
{}^{A}{\omega}^{B}\times{\bf\dot{b}}_{2}={\bf\dot{b}}_{2}, \qquad{}^{A}{\omega}^{B}\times{\bf\dot{b}}_{3}={\bf\dot{b}}_{3}\tag{10}
|
||||
$$
|
||||
|
||||
and, after expressing any vector $\upbeta$ fixed in $\pmb{B}$ as
|
||||
|
||||
$$
|
||||
\beta=\beta_{1}\mathbf{b}_{1}+\beta_{2}\mathbf{b}_{2}+\beta_{3}\mathbf{b}_{3}
|
||||
\beta=\beta_{1}\mathbf{b}_{1}+\beta_{2}\mathbf{b}_{2}+\beta_{3}\mathbf{b}_{3}\tag{11}
|
||||
$$
|
||||
|
||||
where $\beta_{1},\,\beta_{2},\,\beta_{3}$ are constants, so that
|
||||
|
||||
$$
|
||||
\dot{\pmb{\beta}}\,=\,\beta_{1}\dot{\bf b}_{1}+\beta_{2}\dot{\bf b}_{2}+\beta_{3}\dot{\bf b}_{3}
|
||||
\dot{\pmb{\beta}}\,=\,\beta_{1}\dot{\bf b}_{1}+\beta_{2}\dot{\bf b}_{2}+\beta_{3}\dot{\bf b}_{3}\tag{12}
|
||||
$$
|
||||
|
||||
one arrives at
|
||||
|
||||
$$
|
||||
\begin{array}{r l r}&{}&{\dot{\mathfrak{b}}\;=\;\beta_{1}{}^{A}\mathfrak{upomega}^{B}\times\,\mathfrak{b}_{1}+\beta_{2}{}^{A}\mathfrak{\omega}^{B}\times\,\mathfrak{b}_{2}+\beta_{3}{}^{A}\mathfrak{\omega}^{B}\times\,\mathfrak{b}_{3}}\\ &{}&{(10)\qquad\qquad\qquad\qquad\qquad\qquad\qquad(10)}\\ &{}&{=\;{\mathfrak{a}}\mathfrak{\omega}^{B}\times(\beta_{1}\mathfrak{b}_{1}+\beta_{2}\mathfrak{b}_{2}+\beta_{3}\mathfrak{b}_{3})\;=\;{}^{A}\mathfrak{\omega}^{B}\times\,\mathfrak{b}}\end{array}
|
||||
\begin{array}{r l r}&{}&{\dot{\beta}\;=\;\beta_{1}{}^{A}\mathfrak{\omega}^{B}\times\,b_{1}+\beta_{2}{}^{A}\mathfrak{\omega}^{B}\times\,b_{2}+\beta_{3}{}^{A}\mathfrak{\omega}^{B}\times\,b_{3}}\\ &{}&{(10)\qquad\qquad\qquad\qquad\qquad\qquad\qquad(10)}\\ &{}&{=\;{}^{A}\mathfrak{\omega}^{B}\times(\beta_{1}{b}_{1}+\beta_{2}{b}_{2}+\beta_{3}{b}_{3})\;=\;{}^{A}\mathfrak{\omega}^{B}\times\,{b}}\end{array}
|
||||
$$
|
||||
|
||||
Examples Figure 2.1.1 shows a rigid satellite $\pmb{B}$ in orbit about the Earth $\pmb{A}$ A dextral set of mutually perpendicular unit vectors $\mathbf{b}_{1},\,\mathbf{b}_{2},\,\mathbf{b}_{3}$ is fixed in $\pmb{B}$ and a similar such set, ${\mathfrak{a}}_{1},\,{\mathfrak{a}}_{2},\,{\mathfrak{a}}_{3}$ , is fixed in $\pmb{A}$ Measurements are made to determine the time-histories of $\alpha_{i},\beta_{i},\gamma_{i}$ , defined as
|
||||
Examples Figure 2.1.1 shows a rigid satellite $\pmb{B}$ in orbit about the Earth $\pmb{A}$ A dextral set of mutually perpendicular unit vectors $\mathbf{b}_{1},\,\mathbf{b}_{2},\,\mathbf{b}_{3}$ is fixed in $\pmb{B}$ and a similar such set, ${{a}}_{1},\,{{a}}_{2},\,{{a}}_{3}$ , is fixed in $\pmb{A}$ Measurements are made to determine the time-histories of $\alpha_{i},\beta_{i},\gamma_{i}$ , defined as
|
||||
|
||||
$$
|
||||
\alpha_{i}\triangleq\mathbf{b}_{1}\cdot\mathbf{a}_{i}\qquad\beta_{i}\triangleq\mathbf{b}_{2}\cdot\mathbf{a}_{i}\qquad\gamma_{i}\triangleq\mathbf{b}_{3}\cdot\mathbf{a}_{i}\qquad(i=1,2,3)
|
||||
@ -916,7 +914,7 @@ with $\omega$ defined as
|
||||
|
||||
Figure 2.2.1
|
||||
|
||||
where $\theta$ is theradianmeasure of the anglebetween aline $L_{A}$ whose orientationis fixed in $A$ andaline $L_{B}$ similarlyfixed in $B$ (seeFig.2.2.1),both lines are perpendicularto $\mathbf{k}$ and $\theta$ isregarded aspositivewhen the anglecanbegeneratedbyarotation of $B$ relative to $A$ duringwhich aright-handed screwrigidly attachedto $B$ and parallel to $\mathbf{k}$ advances in the direction of $\mathbf{k}$ The scalarquantity $\omega$ is called an angularspeedof $B$ in $A$ [The indefinite article“an”is usedhere because,if $\tilde{\mathbf{k}}$ and $\bar{\omega}$ are defined as ${\dot{\mathbf{k}}}\ {\triangleq}\ -\mathbf{k}$ and $\tilde{\omega}\triangleq-\omega$ then Eq. (1) can be written ${}^{A}\mathbf{\Phi}\mathbf{(o)}^{B}=\tilde{\omega}\tilde{\mathbf{k}}$ so that $\bar{\omega}$ is no less “the”angular speed than is .]
|
||||
where $\theta$ is theradianmeasure of the anglebetween aline $L_{A}$ whose orientationis fixed in $A$ andaline $L_{B}$ similarlyfixed in $B$ (seeFig.2.2.1),both lines are perpendicularto $\mathbf{k}$ and $\theta$ isregarded aspositivewhen the anglecanbegeneratedbyarotation of $B$ relative to $A$ duringwhich aright-handed screwrigidly attachedto $B$ and parallel to $\mathbf{k}$ advances in the direction of $\mathbf{k}$ The scalarquantity $\omega$ is called an angularspeedof $B$ in $A$ .The indefinite article“an”is usedhere because,if $\tilde{\mathbf{k}}$ and $\bar{\omega}$ are defined as ${\dot{\mathbf{k}}}\ {\triangleq}\ -\mathbf{k}$ and $\tilde{\omega}\triangleq-\omega$ then Eq. (1) can be written ${}^{A}\mathbf{\Phi}\mathbf{(o)}^{B}=\tilde{\omega}\tilde{\mathbf{k}}$ so that $\bar{\omega}$ is no less “the”angular speed than is .
|
||||
|
||||
Derivation Let $\mathbf{a}_{1}$ $\mathbf{a}_{2}$ $\mathbf{a}_{3}$ bearight-handed set ofmutually erpendicular unit vectorsfixed in $A$ with $\mathbf{a}_{1}$ parallelto line $L_{A}$ and ${\bf a}_{3}={\bf k}$ and let ${\mathfrak{b}}_{1}$ ${\bf b}_{2}$ ${\mathfrak{b}}_{3}$ bea similar set ofunitvectorsfixed in $\boldsymbol{B}$ with $\mathfrak{b}_{_{1}}$ parallelto $L_{B}$ and $\mathbf{b}_{3}=\mathbf{k}.$ Then
|
||||
|
||||
@ -1190,7 +1188,7 @@ $$
|
||||
A_{\mathfrak{a}}P\triangleq{\frac{\mathfrak{A}d^{A}\mathbf{v}^{P}}{d t}}
|
||||
$$
|
||||
|
||||
Example In Fig. 2.6.1, $P_{1}$ and $\scriptstyle P_{2}$ designate two points connected by a line of length $\pmb{L}$ and free to move in a plane $\pmb{B}$ that is rotating at a constant rate $\omega$ about a line $\pmb{Y}$ fixed both in $\pmb{B}$ and in a reference frame A. The velocities $\pmb{A}_{\mathbf{y}}\pmb{P}_{\mathbf{l}}$ and $A_{\mathbf{v}}P_{2}$ of $\boldsymbol{P}_{1}$ and $\boldsymbol{P}_{2}$ in $A$ are to be expressed in terms of the quantities $q_{1},\,q_{2},\,q_{3}$ , their time-derivatives $\dot{q}_{1},\,\dot{q}_{2},\,\dot{q}_{3}$ , and the mutually perpendicular unit vectors $\pmb{\mathrm{e}}_{x},\pmb{\mathrm{e}}_{y},\pmb{\mathrm{e}}_{z}$ shown in Fig. 2.6.1.
|
||||
Example In Fig. 2.6.1, $P_{1}$ and $P_{2}$ designate two points connected by a line of length $\pmb{L}$ and free to move in a plane $\pmb{B}$ that is rotating at a constant rate $\omega$ about a line $\pmb{Y}$ fixed both in $\pmb{B}$ and in a reference frame A. The velocities $\pmb{A}_{\mathbf{y}}\pmb{P}_{\mathbf{l}}$ and $A_{\mathbf{v}}P_{2}$ of $\boldsymbol{P}_{1}$ and $\boldsymbol{P}_{2}$ in $A$ are to be expressed in terms of the quantities $q_{1},\,q_{2},\,q_{3}$ , their time-derivatives $\dot{q}_{1},\,\dot{q}_{2},\,\dot{q}_{3}$ , and the mutually perpendicular unit vectors $\pmb{\mathrm{e}}_{x},\pmb{\mathrm{e}}_{y},\pmb{\mathrm{e}}_{z}$ shown in Fig. 2.6.1.
|
||||
|
||||
$^\dagger$ The discussion of velocities and accelerations in Secs. 2.7 and 2.8 involves the concept of angular velocity. Hence, to come into position to present this material without a break in continuity, one must deal with angular velocity before taking up velocity and acceleration. Conversely, as Secs. 2.1-2.4 show, angular velocity can be discussed without any reference to velocity or acceleration. Therefore, it is both natural and advantageous to treat these topics in the order used here, that is, angular velocity before velocity and acceleration, rather than in the reverse order.
|
||||
|
||||
@ -1274,7 +1272,7 @@ $$
|
||||
If $P$ and $Q$ are two points fixed on a rigid body $\pmb{B}$ having an angular velocity $\mathbf{\Omega_{op}}^{B}$ in $\pmb{A}$ , then the velocity $\pmb{A}_{\mathbf{v}}\pmb{P}$ of $\pmb{P}$ in $\pmb{A}$ and the velocity $\pmb{A}_{\mathbf{v}}\pmb{Q}$ of $Q$ in $\pmb{A}$ are related to each other as follows:
|
||||
|
||||
$$
|
||||
^{A}\mathbf{v}^{P}={^{A}\mathbf{v}^{Q}}+{^{A}\!\mathbf{\omega}^{B}}\times\mathbf{\omega}^{\!}
|
||||
^{A}\mathbf{v}^{P}={^{A}\mathbf{v}^{Q}}+{^{A}\!\mathbf{\omega}^{B}}\times\mathbf{r}
|
||||
$$
|
||||
|
||||
where r is the position vector from $Q$ to $\pmb{P}$ . The relationship between the acceleration $A_{\mathbf{a}}P$ of $\pmb{P}$ in $\pmb{A}$ and the acceleration $\pmb{a}_{\mathbf{a}}\pmb{Q}$ of $Q$ in $\dot{\pmb{A}}$ involves the angular acceleration $A_{\pmb{\alpha}}B$ of $\pmb{B}$ in $\pmb{A}$ and is given by
|
||||
@ -1454,21 +1452,21 @@ The configuration of a set S of $\pmb{\nu}$ particles $P_{1},...,P_{\nu}$ in a r
|
||||
|
||||
If the motion of $\boldsymbol{s}$ is affected by the presence of bodies that come into contact with one or more of $P_{1},...,P_{\nu}$ , restrictions are imposed on the positions that the affected particles may occupy, and $\pmb{s}$ is said to be subject to configuration constraints;
|
||||
|
||||
an equationexpressing sucharestrictionis calleda holonomicconstraintequation. $\mathbf{a}_{x}$ $\mathbf{a}_{y}$ ， ${\mathfrak{a}}_{z}$ are mutually perpendicular unit vectors fixed in $A$ and $x_{i},y_{i},z_{i},$ called Cartesiancoordinatesof $P_{i}$ in $A$ , are defined as
|
||||
an equation expressing such a restriction is called a holonomic constraint equation.If $\mathbf{a}_{x}$ $\mathbf{a}_{y}$ ， ${\mathfrak{a}}_{z}$ are mutually perpendicular unit vectors fixed in $A$ and $x_{i},y_{i},z_{i},$ called Cartesian coordinates of $P_{i}$ in $A$ , are defined as
|
||||
|
||||
$$
|
||||
x_{i}\triangleq\mathbf{p}_{i}\cdot\mathbf{a}_{x}\qquad y_{i}\triangleq\mathbf{p}_{i}\cdot\mathbf{a}_{y}\qquad z_{i}\triangleq\mathbf{p}_{i}\cdot\mathbf{a}_{z}\qquad(i=1,\ldots,\nu)
|
||||
$$
|
||||
|
||||
where $\mathbf{p}_{i}$ is the position vector from a point $\upsilon$ fixed in $A$ to thepoint $P_{i}$ thena holonomic constraint equation has the form
|
||||
where $\mathbf{p}_{i}$ is the position vector from a point $\upsilon$ fixed in $A$ to the point $P_{i}$ then a holonomic constraint equation has the form
|
||||
|
||||
$$
|
||||
f(x_{1},y_{1},z_{1},\dots,x_{v},y_{v},z_{v},t)=0
|
||||
$$
|
||||
|
||||
where $t$ isthe time.Holonomic constraintequations areclassifiedasrheonomicor scleronomic, according towhether the function $f$ does, or does not, contain t explicitly.
|
||||
where $t$ is the time. Holonomic constraint equations are classified as rheonomic or scleronomic, according to whether the function $f$ does, or does not, contain t explicitly.
|
||||
|
||||
ExampleFigure2.9.1showstwosmall blocks, $P_{1}$ and $P_{2}$ ,connected bya thin rod $R$ of length $L$ ,andconstrained toremain between two parallel panesof glass thatareattached toeachother,formingarigidbody $B_{\cdot}$ Thisbodyis madetorotateataconstantrate $\omega$ aboutaline $\boldsymbol{Y}$ fixedbothin $B$ andina reference frame $A$ Treating $P_{1}$ and $P_{2}$ as a set ${\cal{S}}$ of twoparticles,andletting $\mathbf{p}_{1}$ and ${\bf p}_{2}$ betheirpositionvectorsrelativetothepoint $o$ shown in Fig.2.9.1, one can express $\mathbf{p}_{1}$ and $\mathfrak{g}_{2}$ as
|
||||
Example Figure2.9.1showstwosmall blocks, $P_{1}$ and $P_{2}$ ,connected by a thin rod $R$ of length $L$ ,and constrained to remain between two parallel panes of glass that are attached to each other, forming a rigid body $B_{\cdot}$ This body is made to rotate at a constant rate $\omega$ about a line $\boldsymbol{Y}$ fixed both in $B$ and in a reference frame $A$ Treating $P_{1}$ and $P_{2}$ as a set ${\cal{S}}$ of two particles, and letting $\mathbf{p}_{1}$ and ${\bf p}_{2}$ be their position vectors relative to the point $o$ shown in Fig.2.9.1, one can express $\mathbf{p}_{1}$ and $\mathbf{p}_{2}$ as
|
||||
|
||||
$$
|
||||
{\bf p}_{i}=x_{i}{\bf a}_{x}+y_{i}{\bf a}_{y}+z_{i}{\bf a}_{z}\qquad(i=1,2)
|
||||
@ -1519,13 +1517,13 @@ Since in this equation,in contrast with Eq. (7), $t$ doesnotappear explicitly, E
|
||||
|
||||
# 2.10 GENERALIZED COORDINATES
|
||||
|
||||
When a set $s$ $\upnu$ particles $P_{1},\dots,P_{v}$ is subject to constraints (see Sec.2.9) represented by $M$ holonomic constraint equations, only
|
||||
When a set $s$ of $v$ particles $P_{1},\dots,P_{v}$ is subject to constraints (see Sec.2.9) represented by $M$ holonomic constraint equations, only
|
||||
|
||||
$$
|
||||
\boldsymbol{n}\triangleq\boldsymbol{3}\nu-\boldsymbol{M}
|
||||
$$
|
||||
|
||||
of the $3\upnu$ Cartesian coordinates $x_{i},y_{i},z_{i}\,(i=1,\ldots,\nu)$ of S in a reference frame $\pmb{A}$ are independent of each other. Under these circumstances one can express each of $x_{i},y_{i},z_{i}\,(i=1,\ldots,\nu)$ as a single-valued function of the time $t$ and $\pmb{n}$ functions of $t$ say, $q_{1}(t),\ldots,q_{n}(t),$ in such a way that the constraint equations are satisfied identically for all values of $t$ and $q_{1},\ldots,q_{n}$ in a given domain. The quantities $q_{1},\ldots,q_{n}$ are called generalized coordinates for $s$ in $\pmb{A}$
|
||||
of the $3v$ Cartesian coordinates $x_{i},y_{i},z_{i}\,(i=1,\ldots,\nu)$ of S in a reference frame $\pmb{A}$ are independent of each other. Under these circumstances one can express each of $x_{i},y_{i},z_{i}\,(i=1,\ldots,\nu)$ as a single-valued function of the time $t$ and $\pmb{n}$ functions of $t$ say, $q_{1}(t),\ldots,q_{n}(t),$ in such a way that the constraint equations are satisfied identically for all values of $t$ and $q_{1},\ldots,q_{n}$ in a given domain. The quantities $q_{1},\ldots,q_{n}$ are called generalized coordinates for $s$ in $\pmb{A}$
|
||||
|
||||
Example For the set $s$ in the example in Sec. 2.9, $\nu=2$ and $M=3.$ Hence $n=3$ . Three generalized coordinates for $s$ in $\pmb{A}$ may be introduced by expressing $x_{i},y_{i},z_{i}\,(i=1,2)$ as
|
||||
|
||||
@ -1849,7 +1847,7 @@ $$
|
||||
{\mathbf{v}}=\sum_{r=1}^{n}{\mathbf{v}}_{r}u_{r}+{\mathbf{v}}_{t}
|
||||
$$
|
||||
|
||||
where $\omega_{r},\,\mathbf{v}_{r}\,(r=1,\ldots,n),$ $\mathbf{\varrho}_{t}$ , and $\mathbf{y}_{t}$ are functions of $q_{1},\ldots,q_{n}$ , and the time $t$ The vector $\mathbf{\omega}_{r}$ is called the rth holonomic partial angular velocity of $\pmb{B}$ in $\pmb{A}$ , and $\mathbf{v}_{r}$ is referred to as the rth holonomic partial velocity of $P$ in $A$
|
||||
where $\omega_{r},\,\mathbf{v}_{r}\,(r=1,\ldots,n),$ $\mathbf{\varrho}_{t}$ , and $\mathbf{v}_{t}$ are functions of $q_{1},\ldots,q_{n}$ , and the time $t$ The vector $\mathbf{\omega}_{r}$ is called the rth holonomic partial angular velocity of $\pmb{B}$ in $\pmb{A}$ , and $\mathbf{v}_{r}$ is referred to as the rth holonomic partial velocity of $P$ in $A$
|
||||
|
||||
The vectors $\mathbf{\omega}$ and $\mathbf{v}$ can also be expressed uniquely as
|
||||
|
||||
|
||||
@ -1,3 +1,85 @@
|
||||
第一章 绪论
|
||||
|
||||
- 本章主要介绍了动力学的基本概念和符号1。
|
||||
- 强调了在分析复杂系统时，使用详细的符号的重要性，例如，在分析由多个刚体组成的系统时，需要区分不同刚体在不同参考系中的角速度和速度1。
|
||||
- 通过一个例子展示了如何使用偏导数来处理在不同参考系中定义的向量，并说明了如何计算向量在给定参考系中对时间的普通导数23。
|
||||
- 介绍了如何利用公式推导向量的导数关系，例如，通过将标量函数和向量进行关联，推导出了向量的导数公式4。
|
||||
|
||||
第二章 运动学
|
||||
|
||||
- 本章讨论了运动学，即研究物体运动而不考虑力的作用5。
|
||||
- 介绍了一个点在刚体上运动时的速度和加速度的计算方法，包括科里奥利加速度的概念6。
|
||||
- 通过例子展示了如何使用速度和加速度的定义来推导表达式5。
|
||||
- 讨论了广义坐标的概念，并使用例子说明了如何验证一组坐标是否为广义坐标78。
|
||||
- 介绍了完整系统和非完整系统的概念，以及简单非完整系统的定义9。
|
||||
- 阐述了部分角速度和部分速度的概念，区分了完整和非完整系统的部分速度和角速度，并介绍了它们之间的关系101112。
|
||||
- 讨论了加速度和部分速度之间的关系，以及如何使用广义速度来表示加速度1314。
|
||||
|
||||
第三章 质量分布
|
||||
- 本章主要关注质量分布的描述15。
|
||||
- 介绍了惯性标量的概念，包括转动惯量和惯性积16。
|
||||
- 讨论了如何使用惯性张量来表示刚体的惯性特性1718。
|
||||
- 介绍如何利用质心来简化计算19。
|
||||
- 讨论了主惯性轴的概念，以及如何使用主惯性轴来简化转动惯量表达式20。
|
||||
|
||||
第四章 广义力
|
||||
- 本章介绍了广义力的概念，以及如何计算作用在系统上的力所产生的广义力21。
|
||||
- 讨论了接触力和距离力对广义力的贡献22。
|
||||
- 探讨了相互作用力在广义力中的贡献2324。
|
||||
- 介绍了如何计算重力对广义力的贡献2526。
|
||||
- 通过例子展示了如何计算特定情况下的广义力2728。
|
||||
- 讨论了势能和广义力之间的关系，并解释了如何使用积分来找到势能函数293031。
|
||||
- 介绍了如何用势能来计算广义力3233。
|
||||
- 讨论了耗散力以及如何定义一个耗散函数来描述这些力3435。
|
||||
|
||||
第五章 能量函数
|
||||
- 本章主要介绍了能量函数，包括动能和势能35。
|
||||
- 讨论了如何计算一组粒子的动能35。
|
||||
- 介绍了如何使用质心速度和角速度来计算刚体的动能36。
|
||||
- 通过例子展示了如何验证动能的计算公式37。
|
||||
- 讨论了动能和广义速度之间的关系，以及如何使用动能来表示广义惯性力37。
|
||||
|
||||
第六章 动力学方程
|
||||
- 本章探讨了动力学方程的建立，即如何将力和运动联系起来38。
|
||||
- 使用牛顿定律推导了动力学方程38。
|
||||
- 通过一个例子展示了如何建立受约束的单摆的运动方程38394041...。
|
||||
- 讨论了如何使用线性化方法来简化运动方程444546。
|
||||
- 通过一个例子展示了如何处理有多个约束的系统，并使用偏速度来计算广义力47。
|
||||
|
||||
第七章 动量积分
|
||||
- 本章介绍了动量积分的概念，以及如何利用动量积分来简化运动方程48。
|
||||
- 讨论了线性动量和角动量的守恒原理48。
|
||||
- 介绍了如何选择广义速度和广义坐标来描述无约束运动的系统49。
|
||||
- 通过一个例子展示了如何利用动量积分来求解运动方程5051。
|
||||
- 讨论了如何分析具有阻尼器的系统的运动52。
|
||||
- 推导了刚体动力学的方程5354。
|
||||
- 介绍了广义冲量的概念，以及如何通过冲量和广义动量的变化来分析碰撞555657。
|
||||
- 讨论了碰撞期间的速度变化58。
|
||||
|
||||
- 推导了碰撞后的速度关系5960。
|
||||
|
||||
第八章 振动
|
||||
- 本章讨论了振动的概念，包括自由振动和受迫振动61。
|
||||
- 介绍了模态矩阵和正规坐标的概念61。
|
||||
- 讨论了如何利用模态叠加来分析系统的振动62。
|
||||
- 介绍如何对系统的运动进行模态截断近似63。
|
||||
- 展示了如何用计算机求解振动问题63。
|
||||
|
||||
问题集
|
||||
|
||||
本部分包含了一系列问题，涵盖了前面各章节所介绍的知识点64656667...。
|
||||
- 这些问题涉及到运动学，质量分布，广义力，能量函数，动力学方程，动量积分和振动等多个方面。
|
||||
- 问题涵盖了曲线运动，刚体运动，碰撞，阻尼和振动等多种情况。
|
||||
- 问题形式多样，包括证明题，计算题和概念题，可以帮助读者加深对理论知识的理解和应用。
|
||||
|
||||
附录
|
||||
- 提供了一些常用的几何形状的惯性特性数据，可以方便读者进行计算108。
|
||||
|
||||
索引
|
||||
- 提供了书中相关概念的索引，方便读者查找相关内容109
|
||||
|
||||
总的来说，这份资料涵盖了动力学的基本概念，分析方法和应用，从简单的质点运动到复杂的刚体运动，从运动学到动力学，都有详细的介绍。通过这些内容，读者可以系统地学习和理解动力学原理。
|
||||
|
||||
# 向量的微分
|
||||
|
||||
动力学这一学科涉及各种变化，例如粒子在参考系中位置的变化、机械系统配置的变化等。为了描述这些变化的方式，我们使用向量微积分，这可以视作对通常教授的标量函数微积分材料的扩展。这种扩展主要是为了适应参考系在动力学中许多感兴趣的向量问题中起着核心作用的事实。例如，设 $\pmb{A}$ 和 $\pmb{B}$ 是彼此相对移动但始终有一个共同点 $o$ 的参考系，并设 $\pmb{P}$ 是固定在 $\pmb{A}$ 中的一点，因而在 $\pmb{B}$ 中是运动的。那么，在 $\pmb{A}$ 中 $\pmb{P}$ 的速度为零，而在 $\pmb{B}$ 中 $\pmb{P}$ 的速度不为零。现在，这两个速度都是相同向量 ${\mathfrak{r}}^{o r}$ 关于时间的导数，即从 $^o$ 到 $\pmb{P}$ 的位置矢量。因此，不能简单地谈论 ${\mathsf{r}}^{o P}$ 关于时间的导数。显然，用来微分向量的计算必须允许我们区分在参考系 $\pmb{A}$ 中关于标量变量的微分和在相同变量中但是参考系 $\pmb{B}$ 的微分。
|
||||
|
||||
BIN
力学书籍/input/计算多体系统动力学 (洪嘉振著, 洪嘉振, 1944-) (Z-Library).pdf
Normal file
BIN
力学书籍/input/计算多体系统动力学 (洪嘉振著, 洪嘉振, 1944-) (Z-Library).pdf
Normal file
Binary file not shown.
@ -1 +1,51 @@
|
||||
把主动力和惯性力向偏速度方向投影得到动力学方程
|
||||
|
||||
|
||||
刚柔混合复杂多体系统的动力学计算方法通常有两种： 非线性有限单元法和刚柔耦合多体动力学方法［１ ２ ４]。**非线性有限元的变形描述是相对于惯性坐标系或单元共旋坐标系， 以单元节点坐标为广义坐标， 自由度多， 对于小变形计算效率低， 对大变形的计算效率尚可， 自然包含了动力钢化效应， 对于刚柔混合系统的处理受到限制， 且不利于机构控制**。刚柔耦合多体动力学方法在浮动坐标系中描述变形， 广义坐标可以是模态坐标或有限元节点坐标， 对于小变形可按通常的线性方法来处理， 如可进行模态展开和截断等， 故可减少自由度， 计算效率高， 对于大变形计算效率低， 适合于处理刚柔混合多体系统， 有利于机构控制。
|
||||
|
||||
凯恩方法利用**广义速率代替广义坐标**描述系统的运动， 直接利用达朗伯原理建立动力学方程， 并将矢量形式的力与达朗伯惯性力直接向特定的基矢量方向投影以消除理想约束力， 兼有矢量力学和分析力学的特点
|
||||
|
||||
|
||||
|
||||
# 广义坐标
|
||||
广义坐标：一个由 $n$ 个质点组成的质点系，若受到 $\boldsymbol{s}$ 个完整约束作用，则其在空间中的 $3n$ 个坐标不是彼此独立的。由这些约束方程可以将其中的 $s$ 个坐标表示成其余 $3n-s$ 个坐标的函数，这样该质点系在空间中的位置就可以用 $N\!=$$3n-s$ 个独立参数完全确定下来。描述质点系在空间中位置的独立参数，称为广义坐标。对于完整系统，广义坐标的数目等于系统的自由度数。
|
||||
|
||||
[[理论力学（II） (哈尔滨工业大学理论力学教研室 编) (Z-Library)#1-1 自由度和广义坐标]]
|
||||
|
||||
[[Kane-Dynamics-Theory-Applications#2.10 GENERALIZED COORDINATES]]
|
||||
|
||||
# 广义速度
|
||||
|
||||
|
||||
通过引入 $n$个量 $( u_{1},\ldots,u_{n}$)，称为参考系 $\pmb{A}$中系统 \( S \) 的广义速度，可以使刚体的角速度和系统 \( S \) 点在参考系 \( \pmb{A} \) 中配置由 \( n \) 个广义坐标 \( q_{1},\ldots,q_{n} \)（见第2.10节）所特征化时的速度表达式形式更为有利。这些量是通过以下形式的方程定义的：
|
||||
|
||||
广义速度被定义为描述系统运动的变量，它们可以被用来表示系统的速度，而无需直接使用广义坐标的时间导数
|
||||
广义速度，通常用 u 表示，与广义坐标 q 的时间导数（q̇）相关
|
||||
|
||||
广义速度的主要作用 是：
|
||||
|
||||
- 简化运动分析: 广义速度提供了一种比直接使用广义坐标的时间导数更灵活的方式来表达系统的运动。它们允许我们以更简洁的方式表达速度，尤其是在处理非完整约束系统时56。
|
||||
|
||||
- 处理约束: 广义速度可以帮助我们处理运动约束1。例如，在非完整系统中，广义速度可能不完全是广义坐标的时间导数，而是满足某些约束关系的变量56。
|
||||
|
||||
- 建立动力学方程: 广义速度在建立系统的动力学方程中起着重要作用，特别是在使用拉格朗日方程或凯恩方程时78。 **它们可以用来表达动能，从而推导出运动方程**29。
|
||||
|
||||
- 表达偏速度和偏角速度: 广义速度被用来定义偏速度和偏角速度56。
|
||||
|
||||
# 运动约束条件
|
||||
|
||||
完整系统
|
||||
非完整系统
|
||||
|
||||
有时，由于物理原因，系统 \( S \) 在参考系 \( A \) 中的广义速度 \( u_1, \ldots, u_n \)（见第2.4节）之间不是独立的。在这种情况下，\( s \) 被称为受到运动约束条件，并且将 \( u_1, \ldots, u_n \) 互相联系起来的方程被称为非完整系统（nonholonomic）约束方程。
|
||||
|
||||
当系统 \( S \) 不受运动约束时，\( s \) 被称为具有 \( n \) 自由度在参考系 \( \pmb{A} \) 中的完整系统（holonomic）系统。如果 \( s \) 受到运动约束，则被称为非完整系统（nonholonomic）系统。
|
||||
|
||||
当所有非完整系统约束方程都可以表示为以下 \( m \) 个关系时：
|
||||
|
||||
|
||||
|
||||
# 风电机组整机建模- 广义速率
|
||||
选取个自由度速度为广义速率 $u_r = \dot{q}_r, r=1,2,...,22$。
|
||||
|
||||
计算广义偏角速度，偏线速度
|
||||
|
||||
16
多体+耦合求解器/理论框架.canvas
Normal file
16
多体+耦合求解器/理论框架.canvas
Normal file
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||||
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