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力学书籍/Kane-Dynamics-Theory-Applications/auto/Kane-Dynamics-Theory-Applications.md

@@ -3617,23 +3617,26 @@ and the ellipsoid appears as shown in Fig. 3.9.2. Furthermore, the moment of ine
 
 # GENERALIZED FORCES  
 
-The necessity to cross-multiply a vector $\mathbf{v}$ with the position vector ${\mathfrak{p}}^{A B}$ from a point $A$ to a point $B$ arises frequently [see, for example, Eqs. (2.7.1) and (3.5.27)]. Now, ${\bf p}^{A B}\mathrm{\bf~\times~}{\bf v}={\bf p}^{A C}\mathrm{\bf~\times~}{\bf v},$ where $\mathfrak{p}^{A C}$ is the position vector from $A$ to any point C of the line L that is parallel to v and passes through B; and, when C is chosen properly, it may be easier to evaluate $\mathsf{\pmb{p}}^{A C}$ x v than $\boldsymbol{\mathsf{p}}^{A B}\,\times\,\boldsymbol{\mathsf{v}}.$ This fact provides the motivation for introducing the concepts of “bound" vectors and “moments" of such vectors as in Sec. 4.1. The terms “couple” and “torque,” which have to do with special sets of bound vectors, are defined in Sec. 4.2, and the concepts of "equivalence" and “replacement,” each of which involves two sets of bound vectors, are discussed in Sec. 4.3. This material then is used throughout the rest of the chapter to facilitate the forming of expressions for quantities that play a preeminent role in connection with dynamical equations of motion, namely, two kinds of generalized forces. Sections 4.4-4.10 deal with generalized active forces, which come into play whenever the particles of a system are subject to the actions of contact and/or distance forces. Generalized inertia forces, which depend on both the motion and the mass distribution of a system, are discussed in Sec. 4.11. Mastery of the material brings one into position to formulate dynamical equations for any system possessing a finite number of degrees of fredom, as may be ascertained by reading Sec. 6.1.  
+The necessity to cross-multiply a vector $\mathbf{v}$ with the position vector ${\mathfrak{p}}^{A B}$ from a point $A$ to a point $B$ arises frequently [see, for example, Eqs. (2.7.1) and (3.5.27)]. Now, ${\bf p}^{A B}\mathrm{\bf~\times~}{\bf v}={\bf p}^{A C}\mathrm{\bf~\times~}{\bf v},$ where $\mathfrak{p}^{A C}$ is the position vector from $A$ to any point C of the line L that is parallel to v and passes through B; and, when C is chosen properly, it may be easier to evaluate $\mathsf{\pmb{p}}^{A C}$ x v than $\boldsymbol{\mathsf{p}}^{A B}\,\times\,\boldsymbol{\mathsf{v}}.$ This fact provides the motivation for introducing the concepts of “bound" vectors and “moments" of such vectors as in Sec. 4.1. The terms “couple” and “torque,” which have to do with special sets of bound vectors, are defined in Sec. 4.2, and the concepts of "equivalence" and “replacement,” each of which involves two sets of bound vectors, are discussed in Sec. 4.3. This material then is used throughout the rest of the chapter to facilitate the forming of expressions for quantities that play a preeminent role in connection with dynamical equations of motion, namely, two kinds of generalized forces. Sections 4.4-4.10 deal with generalized active forces, which come into play whenever the particles of a system are subject to the actions of contact and/or distance forces. Generalized inertia forces, which depend on both the motion and the mass distribution of a system, are discussed in Sec. 4.11. Mastery of the material brings one into position to formulate dynamical equations for any system possessing a finite number of degrees of freedom, as may be ascertained by reading Sec. 6.1.  
 
 # 4.1 MOMENT ABOUT A POINT, BOUND VECTORS, RESULTANT  
 
 Of the infinitely many lines that are parallel to every vector v, a particular one, say，, $L$ , called the line of action of v, must be selected before M, the moment of $\mathbf{v}$ aboutapoint $P$ , can be evaluated, for M is defined as  
+$$
+M\triangleq{p}\times{v}
+$$
 
-wherepis thepositionvector from $P$ to any point on $L$ Once $L$ hasbeenspecified, $\gamma$ is said to be a bound vector, and it is customary to show $\mathbf{v}$ on $L$ in pictorial representations of $\gamma$ A vector for which no line of action is specified is called a free vector.  
+where $p$ is the position vector from $P$ to any point on $L$ Once $L$ has been specified, $v$ is said to be a bound vector, and it is customary to show $\mathbf{v}$ on $L$ in pictorial representations of $v$. A vector for which no line of action is specified is called a free vector.  
 
 The resultant $\mathbf{R}$ of a set $\boldsymbol{S}$ of vectors $\mathbf{\nu}_{1},\dots,\mathbf{\nu}_{v}$ is defined as  
 
 $$
-\mathbb{R}\triangleq\sum_{i=1}^{\nu}\mathbb{v}_{i}
+{R}\triangleq\sum_{i=1}^{\nu}{v}_{i}
 $$  
 
 and,if $\mathbf{v}_{1},\ldots,\mathbf{v}_{\nu}$ are bound vectors, the sum of their moments about a point $P$ is called the moment of S about P.  
 
-At times, it is convenient to regard the resultant $\mathbf{k}$ of a set ${\cal{S}}$ of boundvectors $\mathbf{\nu}_{1},\dots,\mathbf{v}_{\nu}$ as a bound vector. Suppose, for example, that $\mathbf{M}^{\mathbf{s}/P}$ and ${\bf{M}}^{s/Q}$ denote the moments of ${\cal{S}}$ about points $P$ and $\mathcal{Q}$ respectivly, and $\mathbf{R}$ isregarded as abound vector whose line of action passesthrough $\boldsymbol{Q}$ . Then one can find ${\bf M}^{{\bf S}/{P}}$ simply by adding to $\mathbf{M}^{S/Q}$ themoment of Rabout $P$ for  
+At times, it is convenient to regard the resultant $R$ of a set ${\cal{S}}$ of boundvectors $\mathbf{\nu}_{1},\dots,\mathbf{v}_{\nu}$ as a bound vector. Suppose, for example, that $\mathbf{M}^{\mathbf{S}/P}$ and ${\bf{M}}^{S/Q}$ denote the moments of ${\cal{S}}$ about points $P$ and $\mathcal{Q}$ respectivly, and $\mathbf{R}$ isregarded as abound vector whose line of action passesthrough $\boldsymbol{Q}$ . Then one can find ${\bf M}^{{\bf S}/{P}}$ simply by adding to $\mathbf{M}^{S/Q}$ themoment of Rabout $P$ for  
 
 $$
 \mathbf{M}^{S/P}=\mathbf{M}^{S/Q}+\mathbf{r}^{P Q}\times\mathbf{R}
@@ -3856,7 +3859,7 @@ The method just used to arrive at Eq. (14) has one major faw, which is that it i
 
 # 4.4 GENERALIZED ACTIVE FORCES  
 
-Ⅱ $\mathbf{\dot{\boldsymbol{u}}}_{1},\ldots,\mathbf{\boldsymbol{u}}_{n}$ are generalized speeds for a simple nonholonomic system $s$ possessing $p$ degrees o freedom in a reference frame $A$ (see Sec.2.13), $p$ quantities $\boldsymbol{\tilde{F}}_{1},...,\boldsymbol{\tilde{F}}_{p}$ callednonholonomicgeneralizedactiveforcesfor $s$ .n $A$ and $n$ quantities $F_{1},\ldots,F_{n}$ called holonomic generalized active forces for $s$ in $A$ , are defined as  
+If $\mathbf{{\boldsymbol{u}}}_{1},\ldots,\mathbf{\boldsymbol{u}}_{n}$ are generalized speeds for a simple nonholonomic system $s$ possessing $p$ degrees of freedom in a reference frame $A$ (see Sec.2.13), $p$ quantities $\boldsymbol{\tilde{F}}_{1},...,\boldsymbol{\tilde{F}}_{p}$ callednonholonomicgeneralizedactiveforcesfor $s$ .n $A$ and $n$ quantities $F_{1},\ldots,F_{n}$ called holonomic generalized active forces for $s$ in $A$ , are defined as  
 
 $$
 \widetilde{F}_{r}\triangleq\sum_{i\,=\,1}^{\nu}\widetilde{\mathbf{v}}_{r}^{\,\,P_{i}}\cdot{\mathbf{R}}_{i}\qquad(r=1,\ldots,p)

多体+耦合求解器/思路.canvas

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+	"nodes":[
+		{"id":"4b7faf7953451d56","x":-140,"y":-160,"width":250,"height":60,"type":"text","text":"1、debug转速"},
+		{"id":"24b6d0ae8c62a4eb","x":-140,"y":-20,"width":250,"height":80,"type":"text","text":"2、Kane原理理解\n- 模态叠加法理解"},
+		{"id":"d0a44ee1cb295fda","x":-140,"y":120,"width":250,"height":100,"type":"text","text":"3、增加自由度，对应的广义主动力、惯性力如何推导"},
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-		{"id":"38d3d1a313c094ee","x":-280,"y":340,"width":250,"height":60,"type":"text","text":"广义坐标"},
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-		{"id":"8ec17237cebe7433","x":60,"y":340,"width":250,"height":60,"type":"text","text":"广义速率"}
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