diff --git a/力学书籍/力学/Dynamics of Multibody Systems (Shabana A.A.) (Z-Library)/auto/Dynamics of Multibody Systems (Shabana A.A.) (Z-Library).md b/力学书籍/力学/Dynamics of Multibody Systems (Shabana A.A.) (Z-Library)/auto/Dynamics of Multibody Systems (Shabana A.A.) (Z-Library).md index fa75954..595231e 100644 --- a/力学书籍/力学/Dynamics of Multibody Systems (Shabana A.A.) (Z-Library)/auto/Dynamics of Multibody Systems (Shabana A.A.) (Z-Library).md +++ b/力学书籍/力学/Dynamics of Multibody Systems (Shabana A.A.) (Z-Library)/auto/Dynamics of Multibody Systems (Shabana A.A.) (Z-Library).md @@ -6516,14 +6516,15 @@ $$ Since the assumption of rigidity of the body $i$ implies that the distance between two arbitrary points on the body remains constant, one may conclude that the length of the vector $\bar{{\mathbf{u}}}^{i}$ remains constant and, as such, the components of this vector relative to the body coordinate system remain unchanged. Similar comments apply for the spatial analysis. 假设物体 $i$ 具有刚体性质，这意味着该物体上任意两点之间的距离保持不变。因此，可以得出结论，向量 $\bar{{\mathbf{u}}}^{i}$ 的长度保持不变，并且相对于该物体的坐标系，该向量的分量也保持不变。 空间分析也适用类似的论述。 -When deformable bodies are considered, the distance between two arbitrary points on the deformable body does not, in general, remain constant because of the relative motion between the particles forming the body. In this case, the vector $\bar{\mathbf{u}}^{i}$ can be written as 当考虑可变形体时，由于构成该体的粒子之间的相对运动，两个在可变形体上任意两点之间的距离通常不会保持恒定。 在这种情况下，向量 $\bar{\mathbf{u}}^{i}$ 可以表示为： +When deformable bodies are considered, the distance between two arbitrary points on the deformable body does not, in general, remain constant because of the relative motion between the particles forming the body. In this case, the vector $\bar{\mathbf{u}}^{i}$ can be written as +当考虑可变形体时，由于构成该体的粒子之间的相对运动，两个在可变形体上任意两点之间的距离通常不会保持恒定。 在这种情况下，向量 $\bar{\mathbf{u}}^{i}$ 可以表示为： $$ \bar{\mathbf{u}}^{i}=\bar{\mathbf{u}}_{o}^{i}+\bar{\mathbf{u}}_{f}^{i}=\bar{\mathbf{u}}_{o}^{i}+\mathbf{S}^{i}\mathbf{q}_{f}^{i} $$ where $\bar{\mathbf{u}}_{o}^{i}$ is the position of point $P$ in the undeformed state, $\mathbf{S}^{i}=\mathbf{S}^{i}(x_{1}^{i},x_{2}^{i},x_{3}^{i})$ is a space-dependent shape matrix, and ${\bf q}_{f}^{i}$ is the vector of time-dependent elastic generalized coordinates of the deformable body $i$ . One can then write the global position of an arbitrary point $P$ on body $i$ in the planar or the spatial case as -其中，$\bar{\mathbf{u}}_{o}^{i}$ 为点 $P$ 在未变形状态下的位置，$\mathbf{S}^{i}=\mathbf{S}^{i}(x_{1}^{i},x_{2}^{i},x_{3}^{i})$ 是与空间相关的形变矩阵，${\bf q}_{f}^{i}$ 是可变形体 $i$ 的随时间变化的弹性广义坐标向量。 那么，在平面或空间情况下，可以写出任意点 $P$ 在体 $i$ 上的全局位置为： +其中，**$\bar{\mathbf{u}}_{o}^{i}$ 为点 $P$ 在未变形状态下的位置**，**$\mathbf{S}^{i}=\mathbf{S}^{i}(x_{1}^{i},x_{2}^{i},x_{3}^{i})$ 是与空间相关的形变矩阵，${\bf q}_{f}^{i}$ 是可变形体 $i$ 的随时间变化的弹性广义坐标向量。** 那么，在平面或空间情况下，可以写出任意点 $P$ 在体 $i$ 上的全局位置为： $$ @@ -6544,7 +6545,7 @@ $$ $$ where $\mathbf{R}^{i}$ and ${\boldsymbol{\Theta}}^{i}$ are the reference coordinates and $\mathbf{q}_{f}^{i}$ is the vector of elastic coordinates. Note that the vector $\bar{\mathbf{u}}_{o}^{i}$ of Eq. 6 can be written as -其中，$\mathbf{R}^{i}$ 和 ${\boldsymbol{\Theta}}^{i}$ 分别为参考坐标，$\mathbf{q}_{f}^{i}$ 为弹性坐标矢量。需要注意的是，方程 6 中的矢量 $\bar{\mathbf{u}}_{o}^{i}$ 可以表示为 +其中，**$\mathbf{R}^{i}$ 和 ${\boldsymbol{\Theta}}^{i}$ 分别为参考坐标，$\mathbf{q}_{f}^{i}$ 为弹性坐标矢量。需要注意的是，方程 6 中的矢量 $\bar{\mathbf{u}}_{o}^{i}$ 可以表示为 $$ \bar{\mathbf{u}}_{o}^{i}=\left[x_{1}^{i}\quad x_{2}^{i}\quad x_{3}^{i}\right]^{\mathrm{T}} @@ -6581,13 +6582,15 @@ $$ \mathbf{S}=\left[\begin{array}{l l}{\xi}&{0}\\ {0}&{3(\xi)^{2}-2(\xi)^{3}}\end{array}\right] $$ -The location and orientation of the beam reference is defined by using the Cartesian coordinates $\mathbf{q}_{r}=[R_{1}~R_{2}~\theta]^{\mathrm{T}}$ . Therefore, the total vector of the beam coordinates $\mathbf{q}=[\mathbf{q}_{r}^{\mathrm{T}}\mathbf{\Lambda}\mathbf{q}_{f}^{\mathrm{T}}]^{\mathrm{T}}$ is defined as +The location and orientation of the beam reference is defined by using the Cartesian coordinates $\mathbf{q}_{r}=[R_{1}~R_{2}~\theta]^{\mathrm{T}}$ . Therefore, the total vector of the beam coordinates $\mathbf{q}=[\mathbf{q}_{r}^{\mathrm{T}}\mathbf{q}_{f}^{\mathrm{T}}]^{\mathrm{T}}$ is defined as +beam参考位置和方向由笛卡尔坐标 $\mathbf{q}_{r}=[R_{1}~R_{2}~\theta]^{\mathrm{T}}$ 定义。因此，beam坐标总矢量 $\mathbf{q}=[\mathbf{q}_{r}^{\mathrm{T}}\mathbf{q}_{f}^{\mathrm{T}}]^{\mathrm{T}}$ 被定义为 $$ \mathbf{q}=\left[\mathbf{q}_{r}^{\mathrm{T}}\quad\mathbf{q}_{f}^{\mathrm{T}}\right]^{\mathrm{T}}=\left[R_{1}\quad R_{2}\quad\theta\quad q_{f1}\quad q_{f2}\right]^{\mathrm{T}} $$ At a given instant of time $t$ , let the components of the vector $\mathbf{q}$ have the following numerical values: +在某一时刻 $t$ ，向量 $\mathbf{q}$ 的分量具有以下数值： $$ \mathbf{q}=[1.0\quad0.5\quad30^{\circ}\quad0.001\quad0.01]^{\mathrm{T}} @@ -6598,7 +6601,7 @@ Determine the global position of the tip point and the center of mass of the bea Solution At this instant of time, the transformation matrix A of Eq. 5 is given by $$ -\mathbf{A}={\left[\cos\theta\quad-\sin\theta\right]}={\left[\begin{array}{l l}{0.8660}&{-0.500{\big]}}\\ {0.500}&{0.8660}\end{array}\right]} +\mathbf{A}={\begin{bmatrix}{\cos\theta^{i}}&{-\sin\theta^{i}}\\ {\sin\theta^{i}}&{\cos\theta^{i}}\end{bmatrix}}={\left[\begin{array}{l l}{0.8660}&{-0.500{\big]}}\\ {0.500}&{0.8660}\end{array}\right]} $$ The global position of point $A$ can then be written as @@ -6607,7 +6610,7 @@ $$ \mathbf{r}_{A}=\mathbf{R}+\mathbf{A}\bar{\mathbf{u}}_{A} $$ -where the vector $\bar{\mathbf{u}}_{A}$ is the local position of the tip point and can be written by using Eq. 6 as $\bar{\mathbf{u}}_{A}=\bar{\mathbf{u}}_{o}+\bar{\mathbf{u}}_{f}$ , where $\bar{\ensuremath{\mathbf{u}}}_{o}$ is the undeformed position of point $A$ given by +where the vector $\bar{\mathbf{u}}_{A}$ is the local position of the tip point and can be written by using Eq. 6 as $\bar{\mathbf{u}}_{A}=\bar{\mathbf{u}}_{o}+\bar{\mathbf{u}}_{f}$ , where $\bar{{\mathbf{u}}}_{o}$ is the undeformed position of point $A$ given by $$ \bar{\mathbf{u}}_{o}=\left[\begin{array}{l}{l}\\ {0}\end{array}\right]=\left[\begin{array}{l}{0.5}\\ {0}\end{array}\right]