刚体的动量矩

假设刚体在某一时刻以角速度$\omega$作定点转动,取任一质点$P_{i}$,对$O$的位矢是$r_{i}$,设它的速度为$v_{i}$,质量为$m_{i}$,则$P_{i}$对定点$O$的动量矩为:

\[ \begin{equation} r_{i}\times m_{i}v_{i}\ \end{equation} \] 式中,$v_{i} = \omega \times r_{i}$
整个刚体对$O$点的动量矩为:
\[ \begin{equation} \begin{split} J &= \sum_{i=1}^{n}(r_{i}\times m_{i}v_{i})\\ &=\sum_{i=1}^{n}m_{i}[r_{i}\times (\omega \times r_{i})] \\ &=\sum_{i=1}^{n}m_{i}[\omega r_{i}^{2}-r_{i}(\omega \cdot r_{i})] \end{split} \end{equation} \]
用矢量分量的形式来表示$J$
\[ r_{i} = x_{i}i+y_{i}j+z_{i}k, \hspace{0.2cm}\omega = \omega_{x}i+\omega_{y}j+\omega_{z}k \]
所以有:
\[ \begin{equation} \begin{split} J_{x} &= \sum_{i=1}^{n}m_{i}[\omega_{i}(x_{i}^{2}+y_{i}^{2}+z_{i}^{2})-x_{i}(\omega_{x}x_{i}+\omega_{y}y_{i}+\omega_{z}z_{i})]\\ &=\omega_{x}\sum_{i=1}^{n}m_{i}(y_{i}^{2}+z_{i}^{2})-\omega_{y}\sum_{i=1}^{n}m_{x}x_{i}y_{i}-\omega_{z}\sum_{i=1}^{n}m_{i}x_{i}z_{i} \end{split} \end{equation} \]
同理:
\[ \begin{equation} J_{y} = \omega_{y}\sum_{i=1}^{n}m_{i}(x_{i}^{2}+z_{i}^{2})-\omega_{x}\sum_{i=1}^{n}m_{x}x_{i}y_{i}-\omega_{z}\sum_{i=1}^{n}m_{i}y_{i}z_{i} \end{equation} \]
\[ \begin{equation} J_{z} = \omega_{z}\sum_{i=1}^{n}m_{i}(x_{i}^{2}+y_{i}^{2})-\omega_{x}\sum_{i=1}^{n}m_{x}x_{i}z_{i}-\omega_{y}\sum_{i=1}^{n}m_{i}y_{i}z_{i} \end{equation} \]

惯量张量

引入符号:

\[ \begin{equation} \begin{split} I_{xx} &= \sum_{i=1}^{n}m_{i}(y_{i}^{2}+z_{i}^{2})\\ I_{yy} &= \sum_{i=1}^{n}m_{i}^(x_{i}^{2}+z_{i}^{2})\\ I_{zz} &= \sum_{i=1}^{n}m_{i}(x_{i}^{2}+y_{i}^{2}) \end{split} \end{equation} \]
\[ \begin{equation} \begin{split} I_{yz} =I_{zy} &= \sum_{i=1}^{n}m_{i}y_{i}z_{i}\\ I_{zx} =I_{xz} &= \sum_{i=1}^{n}m_{i}z_{i}x_{i}\\ I_{xy} =I_{yx} &= \sum_{i=1}^{n}m_{i}x_{i}y_{i} \end{split} \end{equation} \]
带入到各分量之中有:
\[ \begin{equation} \begin{split} J_{x} &= I_{xx}\omega_{x}-I_{xy}\omega_{y}-I_{xz}\omega_{z}\\ J_{y} &= I_{yy}\omega_{y}-I_{yx}\omega_{x}-I_{yz}\omega_{z}\\ J_{z} &= I_{zz}\omega_{z}-I_{zx}\omega_{x}-I_{zy}\omega_{y} \end{split} \end{equation} \]

把三个轴转动惯量和六个惯性积(只有三个是独立的)作为一个统一的物理量，来表示刚体转动时惯性的量度

可以写成矩阵的形式:

\[ \begin{equation} \begin{gathered} \begin{bmatrix} J_{x}\\ J_{y}\\ J_{z} \end{bmatrix} \end{gathered} = \begin{gathered} \begin{bmatrix} I_{xx} & -I_{xy} & -I_{xz}\\ -I_{yx} & I_{yy} & -I_{yz}\\ -I_{zx} & -I_{zy} & I_{zz}\\ \end{bmatrix} \end{gathered} \begin{gathered} \begin{bmatrix} \omega_{x}\\ \omega_{y}\\ \omega_{z} \end{bmatrix} \end{gathered} \end{equation} \]

把该矩阵叫做对$O$点的惯性张量,对角元素分别是刚体绕$x$轴、$y$轴、$z$轴的转动惯量,其他元素称为惯量积

取惯量主轴时的动量矩

对于式(7)，是可正可负也可为零的,取决于坐标系的选定

例如:对一圆柱体,如果将圆柱体的轴线作为$z$轴,那么式(7)均为0

能使惯量张量的三个惯量积均为0的坐标轴称为惯量主轴

从而可以使动量矩简化为:

\[ \begin{equation} \begin{gathered} \begin{bmatrix} J_{x}\\ J_{y}\\ J_{z} \end{bmatrix} \end{gathered} = \begin{gathered} \begin{bmatrix} I_{xx} & 0 & 0\\ 0 & I_{yy} & 0\\ 0 & 0 & I_{zz}\\ \end{bmatrix} \end{gathered} \begin{gathered} \begin{bmatrix} \omega_{x}\\ \omega_{y}\\ \omega_{z} \end{bmatrix} \end{gathered} \end{equation} \]

即:

\[ \begin{equation} J = I_{xx}\omega_{x}i+I_{yy}\omega_{y}j+I_{zz}\omega{z}k \end{equation} \]
从而刚体的动能可以表示为:
\[ \begin{equation} T = \frac{1}{2}J\cdot \omega = \frac{1}{2}(I_{xx}\omega_{x}^{2}+I_{yy}\omega_{y}^{2}+I_{zz}\omega_{z}^{2}) \end{equation} \]
由式(11)可推出Euler动力学方程